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Physical Review B

Phase states in dc SQUIDs with inhomogeneous junction parameters

Roberto De Luca, Francesco Romeo

Published Jun 19, 2006• 2 citations
Publisher

Abstract

By means of perturbation analysis, assuming small inductance values, the dynamical equations for the gauge-invariant superconducting phase differences in a dc superconducting quantum interference device (SQUID), containing junctions with different resistive and coupling parameters, are reduced to a single nonlinear differential equation. The resulting effective reduced potential shows that degenerate phase states exist for half-integer values of the applied flux number ${\ensuremath{\Psi}}_{\mathit{ex}}$ and for zero bias current. It can be also shown that, by small variations of the externally applied flux, this degeneracy can be removed and, for sufficiently low values of the SQUID parameter $\ensuremath{\beta}$, different magnetic states, characterized by opposite magnetic moments, can be reversibly realized.

DOI
10.1103/physrevb.73.214518

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By means of perturbation analysis, assuming small inductance values, the dynamical equations for the gauge-invariant superconducting phase differences in a dc superconducting quantum interference device (SQUID), containing junctions with different resistive and coupling parameters, are reduced to a single nonlinear differential equation. The resulting effective reduced potential shows that degenerate phase states exist for half-integer values of the applied flux number \ensuremathΨex{\ensuremath{\Psi}}_{\mathit{ex}}\ensuremathΨex​ and for zero bias current. It can be also shown that, by small variations of the externally applied flux, this degeneracy can be removed and, for sufficiently low values of the SQUID parameter \ensuremathβ\ensuremath{\beta}\ensuremathβ, different magnetic states, characterized by opposite magnetic moments, can be reversibly realized.
We study a 0-$\ensuremath{\pi}$ dc superconducting quantum interference device (SQUID) with asymmetric inductances and critical currents of the two Josephson junctions (JJs). By considering such a dc SQUID as a black box with two terminals, we calculate its effective current-phase relation ${I}_{s}(\ensuremath{\psi})$ and the Josephson energy $U(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is the Josephson phase across the terminals. We show that there is a domain of parameters where the black box has the properties of a $\ensuremath{\varphi}$ JJ with degenerate ground state phases $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The $\ensuremath{\varphi}$ domain is rather large, so one can easily construct a $\ensuremath{\varphi}$ JJ experimentally. We derive the current phase relation and show that it can be tuned in situ by applying an external magnetic flux resulting in a continuous transition between the systems with static solutions $\ensuremath{\psi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi},\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}={\ensuremath{\varphi}}_{0} ({\ensuremath{\varphi}}_{0}\ensuremath{\ne}0,\ensuremath{\pi})$ and even $\ensuremath{\psi}={\ensuremath{\varphi}}_{0}\ifmmode\pm\else\textpm\fi{}\ensuremath{\varphi}$. The dependence of ${\ensuremath{\varphi}}_{0}$ on applied magnetic flux is not $2\ensuremath{\pi}$ (one flux quantum) periodic.
We have measured the switching current versus applied magnetic flux of a highly asymmetric, hysteretic dc superconducting quantum interference device (SQUID) from 30 mK to 400 mK. The SQUID loop had an inductance of 1 nH, and the capacitances of the two Al-Al2O3-Al Josephson junctions comprising the SQUID were 100 fF and 5 fF. As expected, the switching current of the SQUID displayed a small, periodic modulation with applied flux, from which the switching currents of the large and small junctions at 30 mK were determined to be 846.3 ± 0.1 nA and 0.7 ± 0.08 nA, respectively. Measurements of an identically prepared small junction exhibited a highly suppressed switching current (IS < 10 pA), consistent with phase diffusion in this device. The observed modulation in the SQUID thus demonstrates that the quantum mechanical uncertainty in the phase difference across the small junction was significantly reduced by coupling to the large junction through the loop inductance. We discuss the sensitivity of this technique for determining the gauge-invariant phase difference of small Josephson junctions, and the implications for imaging the phase of superconductors at the atomic scale using scanning tunneling microscopy.
Integer and half-integer Shapiro step amplitudes in the current-voltage characteristics of symmetric superconducting quantum interference devices (SQUIDS), containing more than one Josephson junction on each arm, are analytically calculated for small values of the parameter β and in the absence of noise. This analysis is made possible by the reduction of the dynamical equations of the multiple-junction SQUID to the dynamical equation of a single Josephson junction possessing a nonconventional current-phase relation.
This study shows that we can obtain an enhanced mutual coupling by using a superconducting quantum interference device (SQUID) containing a π-phase shift Josephson junction (π-junction). The π-junction can increase the current flowing in the loop inductance of the SQUID, thereby enhancing the magnetic flux generated in the inductance. Therefore, the effective self-inductance of the SQUID can increase, resulting in an increase in the mutual coupling. We use a 0-π SQUID formed by a conventional Josephson junction (0-junction) and a π-junction. Our numerical analysis for a simplified structure revealed that the ratio of effective to original self-inductance increased to 50. To confirm this, we measured the modulation patterns of critical current in dc-SQUIDs in which part of the loop inductance was replaced with a series of 0-π SQUIDs. The observed modulation pattern revealed that the effective self-inductance became more than 10-fold larger than the original self-inductance under the conditions that the applied current is a few μA while the critical current of the 0-junction in 0-π SQUIDs is 35 μA.
The ability to switch the coupling between quantum bits (qubits) on and off is essential for implementing many quantum-computing algorithms. We demonstrated such control with two flux qubits coupled together through their mutual inductances and through the dc superconducting quantum interference device (SQUID) that reads out their magnetic flux states. A bias current applied to the SQUID in the zero-voltage state induced a change in the dynamic inductance, reducing the coupling energy controllably to zero and reversing its sign.
At an extremely low temperature of 20 mK, we measured the loop current in a tunable rf superconducting quantum interference device (SQUID) with a dc-SQUID. By adjusting the magnetic flux applied to the rf-SQUID loop (Φ f ) and the small dc-SQUID (Φ cjj f ), respectively, the potential shape of the system can be fully controlled in situ. Variation in the transition step and overlap size in the switching current with a barrier flux bias are analyzed, from which we can obtain some relevant device parameters and build a model to explain the experimental phenomenon.