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.

The most succinct manifestation of the second law of thermodynamics is the limitation imposed by the Landauer principle on the amount of heat a Maxwell demon (MD) can convert into free energy per single bit of information obtained in a measurement. We propose and realize an electronic MD based on a single-electron box operated as a Szilard engine, where k B T ln 2 of heat is extracted from the reservoir at temperature T per one bit of created information. The information is encoded in the position of an extra electron in the box.T he work of Maxwell introduced what is now known as the "Maxwell demon" (MD) (1). This idea was quantified later on by Szilard (2), initiating interest in the relationship between information and thermodynamics; see, e.g., refs. 3-6. MD extracts heat from a thermal reservoir at temperature T by observing a thermodynamic system to make a spontaneous, thermally induced, transition into a state with larger-than-average free energy (either because of a larger internal energy or a smaller entropy) and using the feedback to collect this extra free energy as work. Szilard demonstrated that by obtaining a single bit of information as a measurement result of the state of the system, one could collect up to k B T ln 2 useful work, where k B is the Boltzmann constant. Such a direct conversion of heat into work would by itself violate the second law of thermodynamics, because both the measurement and the feedback part of MD operation can in principle be done without generating extra entropy. In particular, a classical measurement can be viewed as a process of copying the state of the system into the memory of the detector. The only fundamentally unavoidable thermodynamic costs of conversion of heat into work by an MD is the creation of information about the state of the measured system. According to the Landauer principle (7-9), erasure of this information generates at least the extracted amount of heat, k B T ln 2 per bit, restoring the agreement with the second law.Whereas these general principles of MD operation are well understood in theory [see, e.g., recent discussions (10-12)], only few experimental realizations of an MD exist (13), and thus far none demonstrates a quantitative connection between the MD output and the obtained information. The goal of this work is to realize a system that demonstrates explicitly the extraction of k B T ln 2 of heat from a thermal reservoir by an MD per one bit of created information. The operating cycle we use is close to the thought experiment suggested by Szilard. It realizes the MD operation using feedback-controlled molecule in a box as the working system. Fig. 1A, from left to right, shows the steps of the operation of such a Szilard engine (SE). The molecule is in equilibrium at temperature T, and the box is divided initially into two equal sections. After the measurement establishes which section the molecule is in, the container is allowed to expand into the full volume, lifting a weight tied to the dividing wall. In doing so, it extracts work from t...

We validate experimentally a fluctuation relation known as generalized Jarzynski equality governing the work distribution in a feedback-controlled system. The feedback control is performed on a single electron box analogously to the original Szilard engine. In the generalized Jarzynski equality, mutual information is treated on an equal footing with the thermodynamic work. Our measurements provide the first evidence of the role of mutual information in the fluctuation theorem and thermodynamics of irreversible processes.

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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Statistical physics provides the concepts and methods to explain the phase behavior of interacting manybody systems. Investigations of Lee-Yang zeros-complex singularities of the free energy in systems of finite size-have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize nonequilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such nonequilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.

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.

We explore the microwave radiation emitted from a biased double quantum dot due to the inelastic tunneling of single charges. Radiation is detected over a broad range of detuning configurations between the dot energy levels, with pronounced maxima occurring in resonance with a capacitively coupled transmission line resonator. The power emitted for forward and reverse resonant detuning is found to be in good agreement with a rate equation model, which considers the hybridization of the individual dot charge states.