Microwave spectroscopy of the Schmid transition

Schmid transition was introduced first as a superconductor-insulator transition in the zero-frequency response of a shunted Josephson junction in equilibrium at zero temperature. As it is typical for a quantum impurity problem, at finite frequencies the transition is broadened to a crossover. Modern attempts to find Schmid transition rely on finite-frequency measurements of a quantum circuit. We predict the frequency dependence of the admittance and reflection phase shift for a high-impedance transmission line terminated by a Josephson junction for a wide variety of devices, from a charge qubit to a transmon. Our results identify the circuit parameters needed for the experimental observation of universal scaling of the responses with frequency. On the insulating side of the transition, the full crossover from weak to strong coupling can only be observed in a transmon. On the contrary, observation of such crossover on the superconducting side of the transition is possible only with the charge qubit. The frequency dependence gets weaker and vanishes upon approaching the Schmid transition from either side.

Figure 1

Two equivalent circuits. (a) A transmission line terminated by a Josephson junction and (b) a voltage driven Josephson junction in series with a resistor.

Figure 2

High- and low-frequency asymptotes of ${\delta }^{\prime }\left(\omega \right)$ and ${\sigma }_{\mathrm{in}}\left(\omega \right)=1-e^{-4{\delta }^{\prime \prime }\left(\omega \right)}$ in the scaling regime. For a transmon on the insulating side of the Schmid transition, $K<\frac{1}{2}$, the results are obtained from Eqs. (12), (16, 17, 18), and footnote [30]. The asymptotes for a charge qubit on the superconducting side of the transition, $K>\frac{1}{2}$, are obtained from these equations by the duality relation, see Eqs. (22), (6), and (7).

Figure 3

Frequency dependence of ${\delta }^{\prime }\left(\omega \right)$ and ${\sigma }_{\mathrm{in}}\left(\omega \right)=1-e^{-4{\delta }^{\prime \prime }\left(\omega \right)}$ for a transmon with $\lambda /{\omega }_0=0.2$ and various values of the environment impedance, using the classical result at $K=0.1$, Eq. (20) at the Toulouse point, $K=0.25$, and Eq. (21) at $K=0.4$ and 0.45. (Upper panel) ${\delta }^{\prime }\left(\omega \right)$ varies between 0 and $\pi /2$, and is plotted in log-log scale; (lower panel) ${\sigma }_{\mathrm{in}}\left(\omega \right)$ varies between 0 and 1, and is plotted in linear-log scale.

Figure 4

Variation of ${\delta }^{\prime }$ (upper panel) and ${\sigma }_{\mathrm{in}}$ (lower panel) as a function of $K$ varying around the Schmid transition point, $K=\frac{1}{2}$, using Eq. (C25), for a set of fixed frequencies $\omega <{\omega }_0$, in a transmon with $\lambda /{\omega }_0=0.2$.