Heat Bath in a Quantum Circuit.

We discuss the concept and realization of a heat bath in solid state quantum systems. We demonstrate that, unlike a true resistor, a finite one-dimensional Josephson junction array or analogously a transmission line with non-vanishing frequency spacing, commonly considered as a reservoir of a quantum circuit, does not strictly qualify as a Caldeira-Leggett type dissipative environment. We then consider a set of quantum two-level systems as a bath, which can be realized as a collection of qubits. We show that only a dense and wide distribution of energies of the two-level systems can secure long Poincare recurrence times characteristic of a proper heat bath. An alternative for this bath is a collection of harmonic oscillators, for instance, in the form of superconducting resonators.

Ultimate Accuracy of Frequency to Power Conversion by Single-Electron Injection.

We analyze theoretically the properties of the recently introduced and experimentally demonstrated converter of frequency to power. The system is composed of a hybrid single-electron box with normal island and superconducting lead, and the detector of the energy flow using a thermometer on a normal metal bolometer. Here, we consider its potential for metrology. The errors in power arise mainly from inaccuracy of injecting electrons at the precise energy equal to the energy gap of the superconductor. We calculate the main systematic error in the form of the excess average energy of the injected electrons and its cumulants, and that due to subgap leakage. We demonstrate by analytic and numerical calculations that the systematic error in detection can, in principle, be made much smaller than the injection errors, which also, with proper choice of system parameters, can be very small, <1%, at low enough temperature. Finally, we propose a simplified configuration for metrological purposes.

Local and Nonlocal Two-Electron Tunneling Processes in a Cooper Pair Splitter.

We measure the rates and coupling coefficients for local Andreev, nonlocal Andreev, and elastic cotunneling processes. The nonlocal Andreev process, giving rise to Cooper pair splitting, exhibits the same coupling coefficient as the elastic cotunneling whereas the local Andreev process is more than 2 orders of magnitude stronger than the corresponding nonlocal one. Theory estimates describe the findings and explain the large difference in the nonlocal and local coupling arising from competition between electron diffusion in the superconductor and tunnel junction transparency.

Thermodynamics of Gambling Demons.

We introduce and realize demons that follow a customary gambling strategy to stop a nonequilibrium process at stochastic times. We derive second-law-like inequalities for the average work done in the presence of gambling, and universal stopping-time fluctuation relations for classical and quantum nonstationary stochastic processes. We test experimentally our results in a single-electron box, where an electrostatic potential drives the dynamics of individual electrons tunneling into a metallic island. We also discuss the role of coherence in gambling demons measuring quantum jump trajectories.

Active Quasiparticle Suppression in a Non-Equilibrium Superconductor.

Quasiparticle (qp) poisoning is a major issue that impairs the operation of various superconducting devices. Even though these devices are often operated at temperatures well below the critical point where the number density of excitations is expected to be exponentially suppressed, their bare operation and stray microwave radiation excite the non-equilibrium qp's. Here we use voltage-biased superconducting junctions to demonstrate and quantify qp extraction in the turnstile operation of a superconductor-insulator-normal metal-insulator-superconductor single-electron transistor. In this operation regime, excitations are injected into the superconducting leads at a rate proportional to the driving frequency. We reach a reduction of density by an order of magnitude even for the highest injection rate of 2.4 × 108 qp's per second when extraction is turned on.

Quantum Trajectory Analysis of Single Microwave Photon Detection by Nanocalorimetry.

We apply quantum trajectory techniques to analyze a realistic setup of a superconducting qubit coupled to a heat bath formed by a resistor, a system that yields explicit expressions of the relevant transition rates to be used in the analysis. We discuss the main characteristics of the jump trajectories and relate them to the expected outcomes ("clicks") of a fluorescence measurement using the resistor as a nanocalorimeter. As the main practical outcome, we present a model that predicts the time-domain response of a realistic calorimeter subject to single microwave photons, incorporating the intrinsic noise due to the fundamental thermal fluctuations of the absorber and finite bandwidth of a thermometer.

Universal First-Passage-Time Distribution of Non-Gaussian Currents.

We investigate the fluctuations of the time elapsed until the electric charge transferred through a conductor reaches a given threshold value. For this purpose, we measure the distribution of the first-passage times for the net number of electrons transferred between two metallic islands in the Coulomb blockade regime. Our experimental results are in excellent agreement with numerical calculations based on a recent theory describing the exact first-passage-time distributions for any nonequilibrium stationary Markov process. We also derive a simple analytical approximation for the first-passage-time distribution, which takes into account the non-Gaussian statistics of the electron transport, and show that it describes the experimental distributions with high accuracy. This universal approximation describes a wide class of stochastic processes, and can be used beyond the context of mesoscopic charge transport. In addition, we verify experimentally a fluctuation relation between the first-passage-time distributions for positive and negative thresholds.

Optimal Probabilistic Work Extraction beyond the Free Energy Difference with a Single-Electron Device.

We experimentally realize protocols that allow us to extract work beyond the free energy difference from a single-electron transistor at the single thermodynamic trajectory level. With two carefully designed out-of-equilibrium driving cycles featuring kicks of the control parameter, we demonstrate work extraction up to large fractions of k_{B}T or with probabilities substantially greater than 1/2, despite the zero free energy difference over the cycle. Our results are explained in the framework of nonequilibrium fluctuation relations. We thus show that irreversibility can be used as a resource for optimal work extraction even in the absence of feedback from an external operator.

Experimental Determination of Dynamical Lee-Yang Zeros.

Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body 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.

Experimental realization of a Szilard engine with a single electron.

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 kBT 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.

Full counting statistics of Andreev tunneling.

We employ a single-charge counting technique to measure the full counting statistics of Andreev events in which Cooper pairs are either produced from electrons that are reflected as holes at a superconductor-normal-metal interface or annihilated in the reverse process. The full counting statistics consists of quiet periods with no Andreev processes, interrupted by the tunneling of a single electron that triggers an avalanche of Andreev events giving rise to strongly super-Poissonian distributions.

Bolometric detection of Josephson radiation.

One of the most promising approaches towards large-scale quantum computation uses devices based on many Josephson junctions. Yet, even today, open questions regarding the single junction remain unsolved, such as the detailed understanding of the quantum phase transitions, the coupling of the Josephson junction to the environment or how to improve the coherence of a superconducting qubit. Here we design and build an engineered on-chip reservoir connected to a Josephson junction that acts as an efficient bolometer for detecting the Josephson radiation under non-equilibrium, that is, biased conditions. The bolometer converts the a.c. Josephson current at microwave frequencies up to about 100 GHz into a temperature rise measured by d.c. thermometry. A circuit model based on realistic parameter values captures both the current-voltage characteristics and the measured power quantitatively. The present experiment demonstrates an efficient, wide-band, thermal detection scheme of microwave photons and provides a sensitive detector of Josephson dynamics beyond the standard conductance measurements.

Microwave quantum diode.

The fragile nature of quantum circuits is a major bottleneck to scalable quantum applications. Operating at cryogenic temperatures, quantum circuits are highly vulnerable to amplifier backaction and external noise. Non-reciprocal microwave devices such as circulators and isolators are used for this purpose. These devices have a considerable footprint in cryostats, limiting the scalability of quantum circuits. As a proof-of-concept, here we report a compact microwave diode architecture, which exploits the non-linearity of a superconducting flux qubit. At the qubit degeneracy point we experimentally demonstrate a significant difference between the power levels transmitted in opposite directions. The observations align with the proposed theoretical model. At - 99 dBm input power, and near the qubit-resonator avoided crossing region, we report the transmission rectification ratio exceeding 90% for a 50 MHz wide frequency range from 6.81 GHz to 6.86 GHz, and over 60% for the 250 MHz range from 6.67 GHz to 6.91 GHz. The presented architecture is compact, and easily scalable towards multiple readout channels, potentially opening up diverse opportunities in quantum information, microwave read-out and optomechanics.

Fluctuations of heat in driven two-state systems: Application to a single-electron box with superconducting gap.

By using trajectory averaging to analyze the statistics of energy dissipation in the nonequilibrium energy-state transitions of a driven two-state system, we show that the average energy dissipation induced by external driving is connected to its fluctuations about equilibrium through the simple relation 2k_{B}T〈Q〉=〈δQ^{2}〉, which is preserved by an adiabatic approximation scheme. We use this scheme to obtain the heat statistics of a single-electron box with a superconducting lead in the slow-driving regime, where the dissipated heat becomes normally distributed with a relatively high probability to be extracted from the environment rather than dissipated. We also discuss the validity of heat fluctuation relations beyond driven two-state transitions and the slow-driving regime.

Bolometric detection of Josephson inductance in a highly resistive environment.

The Josephson junction is a building block of quantum circuits. Its behavior, well understood when treated as an isolated entity, is strongly affected by coupling to an electromagnetic environment. In 1983, Schmid predicted that a Josephson junction shunted by a resistance exceeding the resistance quantum RQ = h/4e2 ≈ 6.45 kΩ for Cooper pairs would become insulating since the phase fluctuations would destroy the coherent Josephson coupling. However, recent microwave measurements have questioned this interpretation. Here, we insert a small Josephson junction in a Johnson-Nyquist-type setup where it is driven by weak current noise arising from thermal fluctuations. Our heat probe minimally perturbs the junction's equilibrium, shedding light on features not visible in charge transport. We find that the Josephson critical current completely vanishes in DC charge transport measurement, and the junction demonstrates Coulomb blockade in agreement with the theory. Surprisingly, thermal transport measurements show that the Josephson junction acts as an inductor at high frequencies, unambiguously demonstrating that a supercurrent survives despite the Coulomb blockade observed in DC measurements.

Micrometre-scale refrigerators.

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.

Single-mode heat conduction by photons.

The thermal conductance of a single channel is limited by its unique quantum value G(Q), as was shown theoretically in 1983. This result closely resembles the well-known quantization of electrical conductance in ballistic one-dimensional conductors. Interestingly, all particles-irrespective of whether they are bosons or fermions-have the same quantized thermal conductance when they are confined within dimensions that are small compared to their characteristic wavelength. The single-mode heat conductance is particularly relevant in nanostructures. Quantized heat transport through submicrometre dielectric wires by phonons has been observed, and it has been predicted to influence cooling of electrons in metals at very low temperatures due to electromagnetic radiation. Here we report experimental results showing that at low temperatures heat is transferred by photon radiation, when electron-phonon as well as normal electronic heat conduction is frozen out. We study heat exchange between two small pieces of normal metal, connected to each other only via superconducting leads, which are ideal insulators against conventional thermal conduction. Each superconducting lead is interrupted by a switch of electromagnetic (photon) radiation in the form of a DC-SQUID (a superconducting loop with two Josephson tunnel junctions). We find that the thermal conductance between the two metal islands mediated by photons indeed approaches the expected quantum limit of G(Q) at low temperatures. Our observation has practical implications-for example, for the performance and design of ultra-sensitive bolometers (detectors of far-infrared light) and electronic micro-refrigerators, whose operation is largely dependent on weak thermal coupling between the device and its environment.

An electron turnstile for frequency-to-power conversion.

Single-electron transport relates an operation frequency f to the emitted current I through the electron charge e as I = ef (refs. 1-5). Similarly, direct frequency-to-power conversion (FPC) links both quantities through a known energy. FPC is a natural candidate for a power standard resorting to the most basic definition of the watt: energy emitted per unit of time. The energy is traceable to Planck's constant and the time is in turn traceable to the unperturbed ground state hyperfine transition frequency of the caesium 133 atom. Hence, FPC comprises a simple and elegant way to realize the watt6. In this spirit, single-photon emission7,8 and detection9 at known rates have been proposed as radiometric standards and experimentally realized10-14. However, power standards are so far only traceable to electrical units, that is, to the volt and the ohm6,15-17. In this Letter, we demonstrate an alternative proposal based on solid-state direct FPC using a hybrid single-electron transistor (SET). The SET injects n (integer) quasi-particles (QPs) per cycle into the two superconducting leads with discrete energies close to their superconducting gap Δ, even at zero source-drain voltage. Furthermore, the application of a bias voltage can vary the distribution of the power among the two leads, allowing for an almost equal power injection nΔf into the two. While in single-electron transport current is related to a fixed universal constant (e), in our approach Δ is a material-dependent quantity. We estimate that under optimized conditions errors can be well below 1%.

Photonic heat transport in three terminal superconducting circuit.

We report an experimental realization of a three-terminal photonic heat transport device based on a superconducting quantum circuit. The central element of the device is a flux qubit made of a superconducting loop containing three Josephson junctions, which can be tuned by magnetic flux. It is connected to three resonators terminated by resistors. By heating one of the resistors and monitoring the temperatures of the other two, we determine photonic heat currents in the system and demonstrate their tunability by magnetic field at the level of 1 aW. We determine system parameters by performing microwave transmission measurements on a separate nominally identical sample and, in this way, demonstrate clear correlation between the level splitting of the qubit and the heat currents flowing through it. Our experiment is an important step towards realization of heat transistors, heat amplifiers, masers pumped by heat and other quantum heat transport devices.

Violation of the fluctuation-dissipation theorem in time-dependent mesoscopic heat transport.

We have analyzed the spectral density of fluctuations of the energy flux through a mesoscopic constriction between two equilibrium reservoirs. It is shown that at finite frequencies, the fluctuating energy flux is not related to the thermal conductance of the constriction by the standard fluctuation-dissipation theorem, but contains additional noise. The main physical consequence of this extra noise is that the fluctuations do not vanish at zero temperature together with the vanishing thermal conductance.