Abstract: Calculations have been performed on superconducting constrictions with hyperbolic geometry. Stationary Ginzburg-Landau equations are used, neglecting magneticfields. Emphasis is placed on the difference between two-and three -dimensional constrictions, which is related to the difference between uniform-thickness (UT) and variable-thickness (VT) superconducting microbridges. The width of the constriction w, normalized to the coherence length ξ is indicated by the parameter A (â‰<192> w/2ξ). It is found that small (A < 0.1), three-dimensional constrictions and VT bridges have a sinusoidal current-phase relation, linear temperature dependence of the critical current I c, and an I cR product (Ris the normal state resistance) equal to the Ambegaokar-Baratoff expression for Josephson junctions near T c. Two-dimensional constrictions behave as if they consist of an inner core with junction properties, in series with the films on both sides. The core consists of the region within a coherence length from the center of the structure. This size is temperature dependent. The core shows a sinusoidal current-phase relation and IcR according to Ambegaokar and Baratoff. For the whole constriction neither the phase difference nor R is finite. Two-dimensional constrictions have linear temperature dependence only when they are extremely narrow (A < 0.001). In two-dimensionalbridges the order parameter is depressed cover a distance of approximately the coherence length; in small three-dimensional constrictions this distance is approximately equal to the width. In narrow constrictions (and short microbridges) the current is not homogeneously distributedover the cross section. The effect has been investigated that occurs when in three-dimensional constrictions the width w is not much larger than l 0, the electron mean free path in the basic material. To this purpose a Ginzburg-Landau equation is derived from the Zaitsev boundary conditions which is valid for continuously changing material parameters. The critical current is decreased, but the IcR product remains constant.The results of the calculations are compared with experimental results for superconducting microbridges.

Abstract: Optical-to-electrical conversion, which is the basis of the operation of optical detectors, can be linear or nonlinear. When high sensitivities are needed, single-photon detectors are used, which operate in a strongly nonlinear mode, their response being independent of the number of detected photons. However, photon-number-resolving detectors are needed, particularly in quantum optics, where n-photon states are routinely produced. In quantum communication and quantum information processing, the photon-number-resolving functionality is key to many protocols, such as the implementation of quantum repeaters1 and linear-optics quantum computing2. A linear detector with single-photon sensitivity can also be used for measuring a temporal waveform at extremely low light levels, such as in long-distance optical communications, fluorescence spectroscopy and optical time-domain reflectometry. We demonstrate here a photon-number-resolving detector based on parallel superconducting nanowires and capable of counting up to four photons at telecommunication wavelengths, with an ultralow dark count rate and high counting frequency.