Electrical conductance is quantized in units of σ Q = 2e 2 /h in ballistic one-dimensional conductors [1,2]. Similarly, thermal conductance at temperature T is expected to be limited by the quantum of thermal conductance of one mode, G Q = πk 2 B 6 T , when physical dimensions are small in comparison to characteristic wavelength of the carriers [3]. The relation between σ Q and G Q obeys the Wiedemann-Franz law [4] for ballistic electrons (apart from factor 2 in σ Q due to spin degeneracy) [5], but somewhat amazingly the same expression of G Q is expected to hold also for phonons and photons, or any other particles with arbitrary exclusion statistics [6,7]. The single-mode heat conductance is particularly relevant in nanostructures, e.g., when studying heat conduction by phonons in dielectric materials [8], or cooling of electrons in metals at very low temperatures [9]. Here we show, based on our experimental results, that at low temperatures heat is transferred by photon radiation, in our case along a superconducting line, when electron-phonon [10] as well as normal electronic heat conduction are frozen out. Thermal conductance is limited by G Q , approaching this value towards low temperatures. Our observation has implications on, e.g., performance and design of ultra-sensitive bolometers and electronic micro-refrigerators [11], whose operation is largely dependent on weak thermal coupling between the device and its environment.To get a picture of the radiative thermal coupling, we start by considering two resistors at temperatures T e1 and T e2 , whose resistances are R 1 and R 2 , respectively, connected via a frequency (ω/2π) dependent impedance Z(ω), see Fig. 1. For simplicity we assume Z(ω) to be fully reactive, so that only the two resistors emit and absorb noise heating. The net power flow P ν between the two resistors from 1 to 2 due to the electron-photon coupling is then given by [3, 9](1)Here, Z t (ω) ≡ R 1 + R 2 + Z(ω) is the total series impedance of the circuit, and−1 are the boson occupation factors at the temperatures of the resistors i = 1, 2. Specifically, for a lossless direct connection of the two resistors, Z(ω) ≡ 0, we can integrate (1) easily with the resultHere r 0 ≡ 4R 1 R 2 /(R 1 + R 2 ) 2 is the matching factor, which obtains its maximum value of unity, when R 1 = R 2 . Thermal conductance by the photonic coupling, G ν , defined as the linear response of P ν for small temperature difference ∆T ≡ T e1 − T e2 around T ≡ (T e1 + T e2 )/2 can then be obtained from (2) for the lossless connection asThus it attains the maximum value for a single transmission channel, the quantum of thermal conductance, in a matched circuit. This result is predicted to hold not only for such photon-mediated coupling, but much more generally for carriers of arbitrary exclusion statistics [12,13] from bosons to fermions [6,14,15].

We investigate hysteresis in the transport properties of superconductor -normal-metal -superconductor (S-N-S) junctions at low temperatures by measuring directly the electron temperature in the normal metal. Our results demonstrate unambiguously that the hysteresis results from an increase of the normal-metal electron temperature once the junction switches to the resistive state. In our geometry, the electron temperature increase is governed by the thermal resistance of the superconducting electrodes of the junction.

The basis of synchronous manipulation of individual electrons in solid-state devices was laid by the rise of single electronics about two decades ago 1-3 . Ultrasmall structures in a low-temperature environment form an ideal domain for addressing electrons one by one. In the so-called metrological triangle, voltage from the Josephson effect and resistance from the quantum Hall effect would be tested against current via Ohm's law for a consistency check of the fundamental constants of nature,h and e (ref. 4). Several attempts to create a metrological current source that would comply with the demanding criteria of extreme accuracy, high yield and implementation with not too many control parameters have been reported [5][6][7][8][9][10][11] . Here, we propose and prove the unexpected concept of a hybrid normal-metalsuperconductor turnstile in the form of a one-island singleelectron transistor with one gate, which demonstrates robust current plateaux at multiple levels of e f at frequency f .Synchronized sources, where current I is related to frequency by I = N ef and N is the integer number of electrons injected in one period, are the prime candidates for the devices to define one ampere in quantum metrology. The accuracy of these devices is based on the discreteness of the electron charge and the high accuracy of frequency determined from atomic clocks. Modern methods are replacing classical definitions of electrical quantities; voltage can be derived on the basis of the a.c. Josephson effect of superconductivity 12 and resistance by the quantum Hall effect 13,14 , but one ampere still needs to be determined via the mutual force exerted by the leads carrying the current. Early proposals of current pumps for quantum metrology were based on arrays of mesoscopic metallic tunnel junctions 5,6 , in which small currents could eventually be pumped at very low error rates 7 . However, these multijunction devices are hard to control and relatively slow 15 . Thus, the quest for feasible implementation with a possibility of parallel architecture for higher yield have led to alternative solutions such as surface-acoustic-wave-driven one-dimensional channels 8 , superconducting devices 11,16-21 and semiconducting quantum dots 22 . These do produce large currents in the nano-ampere range but their accuracy is still limited.Surprisingly, a simple hybrid single-electron transistor, with a small normal-metal (N) island and superconducting (S) leads, has been overlooked in this context. As demonstrated here, an SNS transistor, or alternatively an NSN transistor, see Fig. 1, presents a robust turnstile for electrons showing current plateaux at multiples of ef . We emphasize here that a one-island turnstile does not work even in principle without the hybrid design. An important feature in the present system is that hybrid tunnel junctions suppress tunnelling in an energy range determined by the gap ∆ in the density of states of the superconductor, see Fig. 1d bottom inset; current through a junction vanishes as long as |V J | ∼ < ∆/e...

A superconductor with a gap in the density of states or a quantum dot with discrete energy levels is a central building block in realizing an electronic on-chip cooler. They can work as energy filters, allowing only hot quasiparticles to tunnel out from the electrode to be cooled. This principle has been employed experimentally since the early 1990s in investigations and demonstrations of micrometre-scale coolers at sub-kelvin temperatures. In this paper, we review the basic experimental conditions in realizing the coolers and the main practical issues that are known to limit their performance. We give an update of experiments performed on cryogenic micrometre-scale coolers in the past five years.

We demonstrate radio-frequency thermometry on a micrometer-sized metallic island below 100 mK. Our device is based on a normal-metal-insulator-superconductor tunnel junction coupled to a resonator with transmission readout. In the first generation of the device, we achieve 90 μK= ffiffiffiffiffiffi Hz p noise-equivalent temperature with 10 MHz bandwidth. We measure the thermal relaxation time of the electron gas in the island, which we find to be of the order of 100 μs. Such a calorimetric detector, upon optimization, can be seamlessly integrated into superconducting circuits, with immediate applications in quantumthermodynamics experiments down to single quanta of energy.