To illustrate the emergence of Coulomb blockade from coherent quantum phase-slip processes in thin superconducting wires, we propose and theoretically investigate two elementary setups, or devices. The setups are derived from the Cooper-pair box and Cooper-pair transistor, so we refer to them as the QPS box and QPS transistor, respectively. We demonstrate that the devices exhibit sensitivity to a charge induced by a gate electrode, this being the main signature of Coulomb blockade. Experimental realization of these devices will unambiguously prove the Coulomb blockade as an effect of coherence of phase-slip processes. We analyze the emergence of discrete charging in the limit of strong phase slips. We have found and investigated six distinct regimes that are realized depending on the relation between three characteristic energy scales: inductive energy, charging energy, and phase-slip amplitude. For completeness, we include a brief discussion of dual Josephson-junction devices.

We show theoretically the possibility of quantum synchronization of Josephson and Bloch oscillations in a superconducting device. One needs an LC oscillator to achieve exponentially small rate of synchronization errors. The synchronization leads to quantization of transresistance similar to that in (Fractional) Quantum Hall Effect.One of the most interesting discoveries of XX century was the perfect (fractional) quantization of Hall transresistance in rather imperfect 2DEG semiconducting samples [1]. The resistance as a function of electron density and magnetic field tends to be close to plateaus with valuesn, m being integer numbers. The accuracy is so good as to enable numerous metrological applications [2,3]. The physical explanation of the effect is the commensurability of electron density and density of the magnetic flux penetrating the sample, this taking place any time the ratio of numbers of elementary charges and flux quanta in the structure is a rational fraction n/m. Quantum Hall samples are macroscopic involving infinitely many degrees of freedom. Shortly after the discovery, Likharev and Zorin [4] hypothesized that similar resistance quantization may occur in a Josephsonjunction superconducting device encompassing only few quantum degrees of freedom. They foresaw it as a result of quantum synchronization of Bloch [5] and Josephson [6] oscillations in two junctions. The Josephson frequency ω J = 2eV O / is proportional to the average voltage dropping at one of the junctions while the Bloch frequency ω B = πI O /e is proportional to the average current in another junction. A synchronization condition of the two oscillations, nω J = mω B results inThe resistance quantum is modified in comparison with Eq. 1 manifesting the double charge 2e of Cooper pairs in superconductors. Unfortunately, the original device suggestion [4] does not work. The reason of the failure seems fundamental. The quantities to be synchronized, the charge and flux in the device are canonically conjugated variables. Quantum mechanics forbids them to be simultaneously certain, and the synchronization is expected to be destroyed by quantum fluctuations. A recent outburst of theoretical and experimental activities concerns quantum-coherent phase slips in thin nanowires [7]. On theoretical side, a concept of phase-slip (PS) junction has emerged [8,9]. Such junction is exactly dual to a common Josephson junction with respect to charge-flux conjugation. This inspired the proposals of novel superconducting devices [10][11][12]. Very recently, a PS qubit on InO nanowires has been realized [13]. Relevant experimental developments include observation of the predicted phenomena: phase-slips in Josephson junction chains [14,15], Bloch oscillations [16], and charge sensitivity [17].In this Letter, we demonstrate that combining PS and Josephson junctions in a single device solves the problem of quantum synchronization. A necessary element of the device appears to be an LC oscillator with high quality factor Q. With this, one can make the rate Γ ...

Non-linear effects on driven oscillations are important in many fields of physics, ranging from applied mechanics to optics. They are instrumental for quantum applications [1,2]. A limitation is that the non-linearities known up to now are featureless functions of the number of photons N in the oscillator. Here we show that the nonlinearities found in an oscillator where superconducting inductance is subject to coherent phaseslips, are more interesting. They oscillate as a function of number of photons N with a period of the order of √ N , which is the spread of the coherent state. We prove that such non-linearities result in multiple metastable states encompassing few photons and study oscillatory dependence of various responses of the resonator. A phase-slip process in a superconducting wire is a topological fluctuation of the superconducting order parameter whereby it reaches zero at certain time moment and in certain point of the wire [3].Such a process results in a ±2π change of the superconducting phase difference between the ends of the wire; this produces a voltage pulse. Incoherent thermallyactivated phase slips were shown to be responsible for residual resistance of the wire slightly below critical temperature [4,5]. At lower temperatures and in thinner wires phase slips are quantum fluctuations. Although resistance measurements indicate the quantum nature of the phase-slips [6], they cannot prove a possible quantum coherence of phase-slip events. A set of other nanodevices [7,8] have been proposed to verify the coherence experimentally. To facilitate this verification was the initial motivation of our research.The inductance L of the wire brings about the inductive energy scale E L = Φ 0 /L 2 , where Φ 0 = πh/e is the flux quantum withh the Planck constant and e the electron charge. It is usually assumed that experimental observation of coherent quantum phase slips requires the phase slip amplitude E S to be comparable with E L [8]. The phase-slip amplitude E S depends exponentially on the wire parameters, so its value can hardly be predicted and it may be small. This is why it is important to be able to detect arbitrary small values of E S . Our idea is to use a driven oscillator. We prove that in this case the detectable values of E S are only limited by damping of the oscillator E S ≈hΓ hω 0 . There is an outburst of activity in applying super conducting oscillators for quantum manipulation purposes [9]. The inductance of such an oscillator may be either a thin superconducting wire [10,11] or a chain of Josephson junctions [12,13]. The multi-junction chains also exhibit phase slips and for our purposes are very similar to a wire. Typical experimental values for the main frequency and dissipation rate are ω 0 10 10 Hz and Γ 10 5 Hz.This brings us to the system under consideration: the phase slip oscillator. The setup is shown in Fig. 1a and the equivalent circuit in Fig. 1b. For simplicity, we neglect the effects of the capacitance distribution along the wire attributing all the capacitance C to the...

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This paper characterizes, with static and roving GNSS receivers in the context of precision agriculture research, the hybrid ionospheric-geodetic GNSS model Wide-Area Real-Time Kinematics (WARTK), which computes and broadcasts real-time corrections for high-precision GNSS positioning and navigation within sparse GNSS receiver networks. This research is motivated by the potential benefits of the low-cost precise WARTK technique on mass-market applications such as precision agriculture. The results from two experiments summarized in this work, the second one involving a working spraying tractor, show, firstly, that the corrections from the model are in good agreement with the corrections provided by IGS (International GNSS Services) analysis centers computed in post-processing from global GNSS data. Moreover, secondly and most importantly, we have shown that WARTK provides navigation solutions at decimeter-level accuracy, and the ionospheric corrections significantly reduce the computational time for ambiguity estimation: up to convergence times for the 50%, 75% and 95% of cases equal or below 30 s (single-epoch), 150 s and 600 s approximately, vs. 1000 s, 2750 s and 4850 s without ionospheric corrections, everything for a roving receiver at more than 100 km far away from the nearest permanent receiver. The real-time horizontal position errors reach up to 3 cm, 5 cm and 12 cm for 50%, 75% and 95% of cases, respectively, by constraining and continuously updating the ambiguities without updating the permanent receiver coordinates, vs. the 6 cm, 12 cm and 32 cm, respectively, in the same conditions but without WARTK ionospheric corrections.