The control of electrons at the level of the elementary charge e was demonstrated experimentally already in the 1980s. Ever since, the production of an electrical current ef, or its integer multiple, at a drive frequency f has been a focus of research for metrological purposes. This review discusses the generic physical phenomena and technical constraints that influence single-electron charge transport and presents a broad variety of proposed realizations. Some of them have already proven experimentally to nearly fulfill the demanding needs, in terms of transfer errors and transfer rate, of quantum metrology of electrical quantities, whereas some others are currently ''just'' wild ideas, still often potentially competitive if technical constraints can be lifted. The important issues of readout of singleelectron events and potential error correction schemes based on them are also discussed. Finally, an account is given of the status of single-electron current sources in the bigger framework of electric quantum standards and of the future international SI system of units, and applications and uses of single-electron devices outside the metrological context are briefly discussed.

We show that the effect of a high-temperature environment in current transport through a normal metalinsulator-superconductor tunnel junction can be described by an effective density of states in the superconductor. In the limit of a resistive low-Ohmic environment, this density of states reduces into the well-known Dynes form. Our theoretical result is supported by experiments in engineered environments. We apply our findings to improve the performance of a single-electron turnstile, a potential candidate for a metrological current source. DOI: 10.1103/PhysRevLett.105.026803 PACS numbers: 73.40.Gk, 06.20.Jr, 72.70.+m, 73.20.At Introduction.-The density of states (DOS) of the carriers governs the transport rates in a mesoscopic conductor [1], e.g., in a tunnel junction. Understanding the current transport in a junction in detail is of fundamental interest, but it plays a central role also in practical applications, for instance, in the performance of superconducting qubits [2], of electronic coolers and thermometers [3], and of a singleelectron turnstile to be discussed in this Letter [4]. When one or both of the contacts of a junction are superconducting, the one-electron rates at small energy bias should vanish at low temperatures because of the gap in the Bardeen-Cooper-Schrieffer (BCS) DOS [5]. Yet, a small linear in voltage leakage current persists in the experiments [3,6-10] that can often be attributed to the Dynes DOS, a BCS-like expression with lifetime broadening [11,12]. A junction between two leads admits carriers to pass at a rate that depends on the DOS of the conductors, the occupation of the energy levels, and the number of conduction channels in the junction [13]. In general, basic one-electron tunneling coexists with many-electron tunneling, for instance, cotunneling in multijunction systems [14], or Andreev reflection in superconductors [15,16]. However, when the junction is made sufficiently opaque, a common situation in practice, only one-electron tunneling governed by the Fermi golden rule should persist. We demonstrate experimentally that the subgap current in a high-quality opaque tunnel junction between a normal metal and a superconductor can be ascribed to photon-assisted tunneling. We show theoretically that this leads exactly to the Dynes DOS with an inverse lifetime of e 2 k B T env R=@ 2 , where T env and R are the temperature and effective resistance of the environment.We employ a tunnel junction with a normal metalinsulator-superconductor (NIS) structure; see Fig. 1(a). The essentially constant DOS in the normal metal renders

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We investigate the dynamics of individual quasiparticle excitations on a small superconducting aluminum island connected to normal metallic leads by tunnel junctions. We find the island to be free of excitations within the measurement resolution. This allows us to show that the residual heating, which typically limits experiments on superconductors, has an ultralow value of less than 0.1 aW. By injecting electrons with a periodic gate voltage, we probe electron-phonon interaction and relaxation down to a single quasiparticle excitation pair, with a measured recombination rate of 16 kHz. Our experiment yields a strong test of BCS theory in aluminum as the results are consistent with it without free parameters.

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