Array programming with NumPy.

Array programming provides a powerful, compact and expressive syntax for accessing, manipulating and operating on data in vectors, matrices and higher-dimensional arrays. NumPy is the primary array programming library for the Python language. It has an essential role in research analysis pipelines in fields as diverse as physics, chemistry, astronomy, geoscience, biology, psychology, materials science, engineering, finance and economics. For example, in astronomy, NumPy was an important part of the software stack used in the discovery of gravitational waves1 and in the first imaging of a black hole2. Here we review how a few fundamental array concepts lead to a simple and powerful programming paradigm for organizing, exploring and analysing scientific data. NumPy is the foundation upon which the scientific Python ecosystem is constructed. It is so pervasive that several projects, targeting audiences with specialized needs, have developed their own NumPy-like interfaces and array objects. Owing to its central position in the ecosystem, NumPy increasingly acts as an interoperability layer between such array computation libraries and, together with its application programming interface (API), provides a flexible framework to support the next decade of scientific and industrial analysis.

Nonreciprocal Josephson Linear Response.

We consider the finite-frequency response of multiterminal Josephson junctions and show how nonreciprocity in them can show up at linear response, in contrast to the static Josephson diodes featuring nonlinear nonreciprocity. At finite frequencies, the response contains dynamic contributions to the Josephson admittance, featuring the effects of Andreev bound state transitions along with Berry phase effects, and reflecting the breaking of the same symmetries as in Josephson diodes. We show that outside exact Andreev resonances, the junctions feature nonreciprocal reactive response. As a result, the microwave transmission through those systems is nondissipative, and the electromagnetic scattering can approach complete nonreciprocity. Besides providing information about the nature of the weak link energy levels, the nonreciprocity can be utilized to create nondissipative and small-scale on-chip circulators whose operation requires only rather small magnetic fields.

Author Correction: SciPy 1.0: fundamental algorithms for scientific computing in Python.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

SciPy 1.0: fundamental algorithms for scientific computing in Python.

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments.

Electron-phonon coupling in suspended graphene: supercollisions by ripples.

Using electrical transport experiments and shot noise thermometry, we find strong evidence that "supercollision" scattering processes by flexural modes are the dominant electron-phonon energy transfer mechanism in high-quality, suspended graphene around room temperature. The power law dependence of the electron-phonon coupling changes from cubic to quintic with temperature. The change of the temperature exponent by two is reflected in the quadratic dependence on chemical potential, which is an inherent feature of two-phonon quantum processes.

Superconducting spintronic heat engine.

Heat engines are key devices that convert thermal energy into usable energy. Strong thermoelectricity, at the basis of electrical heat engines, is present in superconducting spin tunnel barriers at cryogenic temperatures where conventional semiconducting or metallic technologies cease to work. Here we realize a superconducting spintronic heat engine consisting of a ferromagnetic insulator/superconductor/insulator/ferromagnet tunnel junction (EuS/Al/AlOx/Co). The efficiency of the engine is quantified for bath temperatures ranging from 25 mK up to 800 mK, and at different load resistances. Moreover, we show that the sign of the generated thermoelectric voltage can be inverted according to the parallel or anti-parallel orientation of the two ferromagnetic layers, EuS and Co. This realizes a thermoelectric spin valve controlling the sign and strength of the Seebeck coefficient, thereby implementing a thermoelectric memory cell. We propose a theoretical model that allows describing the experimental data and predicts the engine efficiency for different device parameters.

Induced superconductivity in the three-dimensional topological insulator HgTe.

A strained and undoped HgTe layer is a three-dimensional topological insulator, in which electronic transport occurs dominantly through its surface states. In this Letter, we present transport measurements on HgTe-based Josephson junctions with Nb as a superconductor. Although the Nb-HgTe interfaces have a low transparency, we observe a strong zero-bias anomaly in the differential resistance measurements. This anomaly originates from proximity-induced superconductivity in the HgTe surface states. In the most transparent junction, we observe periodic oscillations of the differential resistance as a function of an applied magnetic field, which correspond to a Fraunhofer-like pattern. This unambiguously shows that a precursor of the Josephson effect occurs in the topological surface states of HgTe.

Theory of microwave-assisted supercurrent in diffusive SNS junctions.

The observation of very large microwave-enhanced critical currents in superconductor-normal-metal-superconductor (SNS) junctions at temperatures well below the critical temperature of the electrodes has remained without a satisfactory theoretical explanation for more than three decades. Here we present a theory of the supercurrent in diffusive SNS junctions under microwave irradiation based on the quasiclassical Green's function formalism. We show that the enhancement of the critical current is due to the energy redistribution of the quasiparticles in the normal wire induced by the electromagnetic field. The theory provides predictions across a wide range of temperatures, frequencies, and radiation powers, both for the critical current and the current-phase relationship.

Thermodynamic cycles in Josephson junctions.

A superconductor/normal metal/superconductor Josephson junction is a coherent electron system where the thermodynamic entropy depends on temperature and difference of phase across the weak-link. Here, exploiting the phase-temperature thermodynamic diagram of a thermally isolated system, we argue that a cooling effect can be achieved when the phase drop across the junction is brought from 0 to π in a iso-entropic process. We show that iso-entropic cooling can be enhanced with proper choice of geometrical and electrical parameters of the junction, i.e. by increasing the ratio between supercurrent and total junction volume. We present extensive numerical calculations using quasi-classical Green function methods for a short junction and we compare them with analytical results. Interestingly, we demonstrate that phase-coherent thermodynamic cycles can be implemented by combining iso-entropic and iso-phasic processes acting on the weak-link, thereby engineering the coherent version of thermal machines such as engines and cooling systems. We therefore evaluate their performances and the minimum temperature achievable in a cooling cycle.

High operating temperature in V-based superconducting quantum interference proximity transistors.

Here we report the fabrication and characterization of fully superconducting quantum interference proximity transistors (SQUIPTs) based on the implementation of vanadium (V) in the superconducting loop. At low temperature, the devices show high flux-to-voltage (up to 0.52 mV/Φ0) and flux-to-current (above 12 nA/Φ0) transfer functions, with the best estimated flux sensitivity ~ 2.6 µΦ0/(Hz)1/2 reached under fixed voltage bias, where Φ0 is the flux quantum. The interferometers operate up to T bath [Formula: see text] 2 K, with an improvement of 70% of the maximal operating temperature with respect to early SQUIPTs design. The main features of the V-based SQUIPT are described within a simplified theoretical model. Our results open the way to the realization of SQUIPTs that take advantage of the use of higher-gap superconductors for ultra-sensitive nanoscale applications that operate at temperatures well above 1 K.

0-π phase-controllable thermal Josephson junction.

Two superconductors coupled by a weak link support an equilibrium Josephson electrical current that depends on the phase difference ϕ between the superconducting condensates. Yet, when a temperature gradient is imposed across the junction, the Josephson effect manifests itself through a coherent component of the heat current that flows opposite to the thermal gradient for |ϕ| < π/2 (refs 2-4). The direction of both the Josephson charge and heat currents can be inverted by adding a π shift to ϕ. In the static electrical case, this effect has been obtained in a few systems, for example via a ferromagnetic coupling or a non-equilibrium distribution in the weak link. These structures opened new possibilities for superconducting quantum logic and ultralow-power superconducting computers. Here, we report the first experimental realization of a thermal Josephson junction whose phase bias can be controlled from 0 to π. This is obtained thanks to a superconducting quantum interferometer that allows full control of the direction of the coherent energy transfer through the junction. This possibility, in conjunction with the completely superconducting nature of our system, provides temperature modulations with an unprecedented amplitude of ∼100 mK and transfer coefficients exceeding 1 K per flux quantum at 25 mK. Then, this quantum structure represents a fundamental step towards the realization of caloritronic logic components such as thermal transistors, switches and memory devices. These elements, combined with heat interferometers and diodes, would complete the thermal conversion of the most important phase-coherent electronic devices and benefit cryogenic microcircuits requiring energy management, such as quantum computing architectures and radiation sensors.

Lindblad-equation approach for the full counting statistics of work and heat in driven quantum systems.

We formulate the general approach based on the Lindblad equation to calculate the full counting statistics of work and heat produced by driven quantum systems weakly coupled with a Markovian thermal bath. The approach can be applied to a wide class of dissipative quantum systems driven by an arbitrary force protocol. We show the validity of general fluctuation relations and consider several generic examples. The possibilities of using calorimetric measurements to test the presence of coherence and entanglement in the open quantum systems are discussed.

Supercurrent-induced temperature gradient across a nonequilibrium SNS Josephson junction.

Using tunneling spectroscopy, we have measured the local electron energy distribution function in the normal part of a superconductor-normal metal-superconductor (SNS) Josephson junction containing an extra lead to a normal reservoir. In the presence of simultaneous supercurrent and injected quasiparticle current, the distribution function exhibits a sharp feature at very low energy. The feature is odd in energy and odd under reversal of either the supercurrent or the quasiparticle current direction. The feature represents an effective temperature gradient across the SNS Josephson junction that is controllable by the supercurrent.

Thermopower induced by a supercurrent in superconductor-normal-metal structures.

We examine the thermopower Q of a mesoscopic normal-metal (N) wire in contact with superconducting (S) segments and show that even with electron-hole symmetry Q may become finite due to the presence of supercurrents. Moreover, we show how the dominant part of Q can be directly related to the equilibrium supercurrents in the structure. In general, a finite thermopower appears both between the N reservoirs and the superconductors and between the N reservoirs themselves. The latter, however, strongly depends on the geometrical symmetry of the structure.

Measuring non-Gaussian fluctuations through incoherent cooper-pair current.

We study a Josephson junction (JJ) in the regime of incoherent Cooper-pair tunneling, capacitively coupled to a nonequilibrium noise source. The current-voltage (I-V) characteristics of the JJ are sensitive to the excess voltage fluctuations in the source, and can thus be used for wideband noise detection. Under weak driving, the odd part of the I-V can be related to the second cumulant of noise, whereas the even part is due to the third cumulant. After calibration, one can measure the Fano factors for the noise source, and get information about the frequency dependence of the noise.