Absence of a Dissipative Quantum Phase Transition in Josephson Junctions

Half a century after its discovery, the Josephson junction has become the most important nonlinear quantum electronic component at our disposal. It has helped reshape the International System of Units around quantum effects and is used in scores of quantum devices. By itself, the use of Josephson junctions in volt metrology seems to imply an exquisite understanding of the component in every aspect. Yet, surprisingly, there have been long-standing subtle issues regarding the modeling of the interaction of a junction with its electromagnetic environment. Here, we find that a Josephson junction connected to a resistor does not become insulating beyond a given value of the resistance due to a dissipative quantum phase transition, as is commonly believed. Our work clarifies how this key quantum component behaves in the presence of a dissipative environment and provides a comprehensive and consistent picture, notably regarding the treatment of its phase.

Popular Summary

Scores of quantum devices make use of Josephson junctions. In these junctions, where two superconducting materials are coupled via some weak connection, electric current can flow with no applied voltage. Despite being discovered more than half a century ago, there are still subtle unresolved issues regarding how the junction interacts with its electromagnetic environment. Here, we find that a Josephson junction connected to a resistor does not become insulating above some value of the resistance due to a dissipative quantum phase transition, as is commonly believed.

In 1983, researchers predicted that when a Josephson junction is connected in series with a resistance greater than about $6.5\mathrm{k}\mathrm{\Omega }$—calculated from a ratio of fundamental constants—the junction should become insulating as its temperature is reduced toward absolute zero. To test this prediction, we connect junctions to resistances of 8 and $\text{12\hspace{0.17em}\hspace{0.17em}}\mathrm{k}\mathrm{\Omega }$—and observe that they do not become insulating at lower temperatures. This contradiction of a long-standing prediction leads to a substantial modification of our understanding of this key quantum electronic component. Our analysis indicates that the junction’s supercurrent is actually resilient to any amount of dissipation, and that the originally predicted phase transition arises only in nonsuperconducting systems.

Our work invites researchers to revisit theoretical predictions of phase transitions in Josephson-junction systems or other superconducting systems that were derived from, or are similar to, the originally predicted superconducting-to-insulating transition.

Figure 1

(a) A Josephson junction connected to a resistor $R$ (abbreviated as $\mathrm{JJ}+R$). The junction’s capacitance $C$ determines the charging energy $E_C=(2e{)}^2/2C$, while the transparency of the tunnel barrier and the superconducting gap set its Josephson coupling energy $E_J$. (b) Sketch of the Schmid-Bulgadaev phase diagram for the circuit in (a). In the phase $I$ ($S$), the junction is predicted to be insulating (superconducting) at zero temperature. The insulating phase is paradoxical because the left axis (green line, where $R=\infty$) is the location of the Cooper pair box family of superconducting qubits for which it is well known that the junction is superconducting. Similarly, our samples $S_1$ and $S_2$ are found to remain superconducting when lowering the temperature, even though they are supposed to be well inside the insulating phase.

Figure 2

Top: Simplified schematics of the experimental setup. Bottom: One of the samples measured. Two SEM micrographs are stitched to show the entire central part and colorized to evidence the different metals used (see Appendix pp2 for fabrication details).

Figure 3

Measured modulus of the transmission $S_{21}$ (as a power ratio) for sample 1 (top panels) and 2 (bottom panels). Left panels: $|S_{21}|$ as a function of the flux through the SQUIDs at the base temperature. The modulation is periodic with the flux (data not shown), as usual for a SQUID; only half a period is represented. Note that the position of the zero flux is different in the top and bottom panels. Right panels: $|S_{21}|$ for several flux values (using the same colors as on the left panels) as a function of the temperature. For sample 2, the error bars are smaller than the symbols used (note the larger vertical scale).

Figure 4

(a) Reinterpreted Schmid-Bulgadaev phase diagram, in which the junction is superconducting everywhere, except for $E_J=0$. In the $\mathcal{S}$ parts, the junction is superconducting with a fully delocalized charge and, correspondingly, a fluctuationless (classical) compact phase. Partial charge (phase) localization occurs in the $\mathcal{CL}$ ($\mathcal{PL}$) part. The classical phases $\mathcal{S}$ are artifacts which disappear when improving the model (see text and Appendix pp7). (b) Final description of the junction’s behavior in the parameter space. Drawings are sketches of the diagonal elements of the junction’s reduced density matrix in an extended description (red, charge representation; green, phase representation). Close to the left (right) half of the upper (lower) axis, they nearly take the form of Dirac combs where the phase is almost a classical variable. In the lower left (upper right) sector, partial charge (phase) localization occurs, as in (a). The density matrix evolves continuously, interpolating between these limits, without any phase transition.

Figure 5

Variations of the transmission through the samples (top, sample 1; bottom, sample 2), as a function of the power at the sample input, for different values of the flux through the SQUIDs, at the lowest temperature (variations are taken with respect to the value at $P_{\mathrm{in}}=-80\mathrm{dBm}$). The size of the error bars does not vary monotonically because the averaging time was increased when reducing the power. The dashed lines indicate the power levels that were chosen to take the data shown in Fig. 3.

Figure 6

Quadrupole model of the on-chip components for the calculation of the transmission $S_{21}$. In the ideal case where the external circuit can be fully calibrated by measuring reference samples, $S_{21}$ would depend only on $C_c$, $R$, and $Y$, the admittance of the junction. In our experiment, this full calibration is not performed, and a weak stray admittance $|Y_{\text{stray}}|\ll |i{C}_c\omega |$ very likely dominates our measurements at lower values of $S_{21}$, when the junction admittance is large $(|Y|R\gg 1)$.