Landau-Zener population control and dipole measurement of a two-level-system bath

Tunneling two-level systems (TLSs), present in dielectrics at low temperatures, have been recently studied for fundamental understanding and superconducting device development. According to a recent theory by Burin et al. [Phys. Rev. Lett. 110, 157002 (2013)], the TLS bath of any amorphous dielectric experiences a distribution of Landau-Zener transitions if exposed to simultaneous fields. In this experiment we measure amorphous insulating films at millikelvin temperatures with a microwave field and a swept electric field bias using a superconducting resonator. We find that the maximum dielectric loss per microwave photon with the simultaneous fields is approximately the same as that in the equilibrium state, in agreement with the generic material theory. In addition, we find that the loss depends on the fields in a way which allows for the separate extraction of the TLS bath dipole moment and density of states. This method allows for the study of the TLS dipole moment in a diverse set of disordered films, and provides a technique for continuously inverting their population.

Figure 1

(a) Image of biased bridge resonator used. (b) Schematic of the resonator. (c) Steady-state equilibrium loss tangent, measured as a function of the microwave field at temperatures of 33 and 200 mK. (d) Time-dependent nonequilibrium loss tangent measured (main panel) with the electric field sweep rate magnitude (inset). (e) Comparison of linear-response equilibrium loss tangent (black triangles) to the maximum nonequilibrium loss tangent (blue circles) as a function of temperature. The solid curve shows a fit of the conventional theory of thermally saturated TLSs, $tan{\delta }_{0}tanh(\hslash \omega /2k_{B}T)$, to the equilibrium data.

Figure 2

(a) Loss tangent for three different input microwave fields (+, $\times$, $▵$) as a function of the bias rate ${\dot{E}}_{\text{bias}}$, taken with different techniques. Dashed curves are Monte Carlo fits with dipole moment $p=7.9$ D. (b) Loss tangent measurements for each ac drive, normalized to their maximum and minimum values, shown as a function of the dimensionless sweep rate $\xi$. The data collapse onto one curve.