Distribution of two-level system couplings to strain and electric fields in glasses at low temperatures

The thermal, acoustic, and dielectric properties of glasses below 1 K are dictated by the interaction of two-level systems (TLS) with strain and electric fields. In a previous paper, we proposed a modified TLS model to quantitatively account for the universally small phonon scattering in glasses at low temperatures. A key ingredient of this model was a wide distribution of couplings between TLS and phonons, contrary to the standard model which assumes a single averaged value is sufficient. In this paper, we expand on this view and include couplings to strain as well as electric fields. We then compare our theoretical results to measurements obtained using superconducting qubits. We find that the predictions of the modified TLS model are more consistent with experiments than those of the standard model. For the distribution of couplings between TLS and the strain field, there is a better agreement with experiments if we include a random distribution of local strains. Such a distribution of local strains is consistent with those found from molecular dynamics simulations.

Figure 1

Energy relaxation time $T_1$ (gray scale) of a superconducting transmon qubit as a function of the qubit resonance frequency and the applied mechanical strain. Dark hyperbolic traces indicate the resonances of TLS defects which absorb energy from the qubit.

Figure 2

Raw distribution of $d_z$ obtained from Fig. 3 of Ref. [15] (Exp.). The squares are the prediction of the standard TLS model with $d_o=4$ (Std. TLS). The stars markers are the prediction of the modified TLS model, also with $d_o=4$ (Mod. TLS).

Figure 3

Prediction for ${\gamma }_z\alpha$ as measured in Ref. [12]. (a) TLS model prediction. (b) Modified TLS model. (c) Experimental data.

Figure 4

Prediction for the distribution of ${\gamma }_z{\alpha }_m$ using the TLS model and the modified TLS model, including intrinsic Gaussian strain fluctuations. (a) TLS model plus Gaussian strain fluctuations. (b) Modified TLS model plus Gaussian strain fluctuations. (c) Experimental data.

Figure 5

Computed distribution of internal strains for an external strain ${\epsilon }_{xx}^{\mathrm{ext}}=2\times {10}^{-3}$. Domain size is 4 Å.

Figure 6

Evolution of the internal strain variance as a function of domain size (in Å). The external strain is ${\epsilon }_{xx}^{\mathrm{ext}}=2\times {10}^{-3}$.

Figure 7

Prediction for the distribution of ${\gamma }_z{\alpha }_m$ using the TLS model and the modified TLS model, including an intrinsic strain gradient. (a) TLS model. (b) Modified TLS model. (c) Experimental data.