Weak localization corrections to the thermal conductivity in ss-wave superconductors

We study the thermal conductivity in disordered $s$-wave superconductors. Expanding on previous works for normal metals, we develop a formalism that tackles particle diffusion as well as the weak localization (WL) and weak anti-localization (WAL) effects. Using a Green's functions diagrammatic technique, which takes into account the superconducting nature of the system by working in Nambu space, we identify the system's low-energy modes, the diffuson and the Cooperon. The time scales that characterize the diffusive regime are energy dependent; this is in contrast with the the normal state, where the relevant time scale is the mean free time $\tau_e$, independent of energy. The energy dependence introduces a novel energy scale $\varepsilon_*$, which in disordered superconductors ($\tau_e \Delta\ll 1$, with $\Delta$ the gap) is given by $\varepsilon_* = \sqrt{\Delta/\tau_e}$. From the diffusive behavior of the low-energy modes, we obtain the WL correction to the thermal conductivity. We give explicitly expressions in two dimensions. We determine the regimes in which the correction depends explicitly on $\varepsilon_*$ and propose an optimal regime to verify our results in an experiment.Comment: 15 pages, 6 figure

Zeeman-limited Superconductivity in Crystalline Al Films

We report the evolution of the Zeeman-mediated superconducting phase diagram (PD) in ultra-thin crystalline Al films. Parallel critical field measurements, down to 50 mK, were made across the superconducting tricritical point of films ranging in thickness from 7 ML to 30 ML. The resulting phase boundaries were compared with the quasi-classical theory of a Zeeman-mediated transition between a homogeneous BCS condensate and a spin polarized Fermi liquid. Films thicker than $\sim$20 ML showed good agreement with theory, but thinner films exhibited an anomalous PD that cannot be reconciled within a homogeneous BCS framework.Comment: 8 pages, 9 figure

Quasiparticles in superconducting qubits with asymmetric junctions

Designing the spatial profile of the superconducting gap -- gap engineering -- has long been recognized as an effective way of controlling quasiparticles in superconducting devices. In aluminum films, their thickness modulates the gap; therefore, standard fabrication of Al/AlOx/Al Josephson junctions, which relies on overlapping a thicker film on top of a thinner one, always results in gap-engineered devices. Here we reconsider quasiparticle effects in superconducting qubits to explicitly account for the unavoidable asymmetry in the gap on the two sides of a Josephson junction. We find that different regimes can be encountered in which the quasiparticles have either similar densities in the two junction leads, or are largely confined to the lower-gap lead. Qualitatively, for similar densities the qubit's excited state population is lower but its relaxation rate higher than when the quasiparticles are confined; therefore, there is a potential trade-off between two desirable properties in a qubit.Comment: Revised version. To be published in PRX Quantu

Long-range exchange interaction between spin qubits mediated by a superconducting link at finite magnetic field

Solid state spin qubits are promising candidates for the realization of a quantum computer due to their long coherence times and easy electrical manipulation. However, spin-spin interactions, which are needed for entangling gates, have only limited range as they generally rely on tunneling between neighboring quantum dots. This severely constrains scalability. Proposals to extend the interaction range generally focus on coherent electron transport between dots or on extending the coupling range. Here, we study a setup where such an extension is obtained by using a superconductor as a quantum mediator. Because of its gap, the superconductor effectively acts as a long tunnel barrier. We analyze the impact of spin-orbit (SO) coupling, external magnetic fields, and the geometry of the superconductor. We show that while spin non-conserving tunneling between the dots and the superconductor due to SO coupling does not affect the exchange interaction, strong SO scattering in the superconducting bulk is detrimental. Moreover, we find that the addition of an external magnetic field decreases the strength of the exchange interaction. Fortunately, the geometry of the superconducting link offers a lot of room to optimize the interaction range, with gains of over an order of magnitude from a 2D film to a quasi-1D strip. We estimate that for superconductors with weak SO coupling (\textit{e.g.}, aluminum) exchange rates of up to 100\,MHz over a micron-scale range can be achieved with this setup in the presence of magnetic fields of the order of 100\,mT

Shielding superconductors with thin films

Determining the optimal arrangement of superconducting layers to withstand large amplitude AC magnetic fields is important for certain applications such as superconducting radiofrequency cavities. In this paper, we evaluate the shielding potential of the superconducting film/insulating film/superconductor (SIS') structure, a configuration that could provide benefits in screening large AC magnetic fields. After establishing that for high frequency magnetic fields, flux penetration must be avoided, the superheating field of the structure is calculated in the London limit both numerically and, for thin films, analytically. For intermediate film thicknesses and realistic material parameters we also solve numerically the Ginzburg-Landau equations. It is shown that a small enhancement of the superheating field is possible, on the order of a few percent, for the SIS' structure relative to a bulk superconductor of the film material, if the materials and thicknesses are chosen appropriately.Comment: 7 pages, 5 figure

Dissipation by normal-metal traps in transmon qubits

Quasiparticles are an intrinsic source of relaxation and decoherence for superconducting qubits. Recent works have shown that normal-metal traps may be used to evacuate quasiparticles, and potentially improve the qubit life time. Here, we investigate how far the normal metals themselves may introduce qubit relaxation. We identify the ohmic losses inside the normal metal and the tunnelling current through the normal metal-superconductor interface as the relevant relaxation mechanisms. We show that the ohmic loss contribution depends strongly on the device and trap geometry, as a result of the inhomogeneous electric fields in the qubit. The correction of the quality factor due to the tunnelling current on the other hand is highly sensitive to the nonequilibrium distribution function of the quasiparticles. Overall, we show that even when choosing less than optimal parameters, the presence of normal-metal traps does not affect the quality factor of state-of-the-art qubits.Comment: 13 pages, 4 figure

Bogoliubov Quasiparticles in Superconducting Qubits

Extending the qubit coherence times is a crucial task in building quantum information processing devices. In the three-dimensional cavity implementations of circuit QED, the coherence of superconducting qubits was improved dramatically due to cutting the losses associated with the photon emission. Next frontier in improving the coherence includes the mitigation of the adverse effects of superconducting quasiparticles. In these lectures, we review the basics of the quasiparticles dynamics, their interaction with the qubit degree of freedom, their contribution to the qubit relaxation rates, and approaches to control their effect.Comment: 42 pages, 14 figures. Submitted to SciPost Lecture Notes. To appear in 'Quantum Information Machines; Lecture Notes of the Les Houches Summer School 2019', eds. M. Devoret, B. Huard, and I. Po