{'fraction_non_alphanumeric': 0.031465201465201466, 'fraction_numerical': 0.009706959706959707, 'mean_word_length': 4.481027906042963, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Single-Quantum-Dot Heat Valve

In the emerging field of quantum thermodynamics, heat transport and dissipation in a quantum electronic device is a fundamentally important topic. Gate-tunable single-quantum dot junctions are paradigmatic test benches for quantum transport. Whereas the study of charge transport therein has already allowed exploring a large palette of physical effects at play, heat transport and thermoelectric properties have only very recently started to be explored. Here, we demonstrate gate control of electronic heat flow in a single-quantum-dot junction. Electron temperature maps, as a function of the gate and bias voltages applied to the junction, reveal clearly defined Coulomb diamond structures with a maximum cooling right at the degeneracy point. This heat valve features a non-trivial bias and gate dependence revealing both the quantum nature of the dot at the heart of device and its strong coupling to leads.

As opposed to charge transport processes, the understanding of electronic heat transport and generation across a quantum device is, experimentally, still in its infancy [10][11][12]. The main reason for this is that local thermometry inside a quantum device remains a delicate issue. The temperature dependence of the critical current of a superconducting weak link was used in scanning probe experiments to reveal for instance the scattering sites in high-mobility graphene [13,14]. Yet, to date, these experiments are limited to relatively high temperatures, well above 1 K. At milliKelvin temperatures, local thermometry can be performed in quantum devices formed in a two-dimensional electron gas (2DEG) by a variety of methods [15] that have recently been pushed to quantitative accuracy [16]. This achievement has been followed by a series of spectacular experiments on the heat transport properties of edge channels in 2DEGs under high magnetic fields [17,18]. In metallic devices, electronic thermometry is usually based on the temperature dependence of charge transport in superconducting hybrids, either in the tunnelling regime for Normal metal-Insulator-Superconductor (NIS) junctions [19,20] or at higher transparencies allowing for superconducting correlations [21][22][23]. A photonic heat valve was realized with a superconducting qubit coupled to heat reservoirs (probed by NIS probes) through coplanar waveguide resonators [24].
The single electron transistor (SET) is an essential brick for the emerging field of quantum caloritronics [25]. Building on the NIS thermometry technique, the thermal conductance of a metallic SET was measured [26]. Despite the continuous density of states in the metallic island, electron interactions readily lead to striking deviations from the Wiedemann-Franz law [27]. Going beyond this simple case, two questions arise: (i) how does such a SET behave thermally beyond equilibrium, that is, at finite voltage bias and/or at large temperature difference where both Joule heat and heat transport are to be taken into account, and (ii), if the central island is replaced by a quantum dot (QD), how would the discrete nature of its energy spectrum manifest in the thermal properties of the device? In a QD transistor in the weak coupling regime, the discreteness of the electronic states spectrum makes electronic transport processes strongly selective in energy. Right at the degeneracy point, the heat flow is zero since electrons flow exactly at the common Fermi level of source and drain. Still at zero bias, cooling is predicted when the QD energy level (tuned by the gate) is positioned just above or below the Fermi level of both source and drain, so that high-energy electrons in the hot lead can escape through the dot, or low-energy electrons can be injected there [15,28]. The heat valve is maximally open when the QD level is shifted from the Fermi level by about the thermal energy k B T e . Figures 1(a,b) display a colored scanning electron micrograph of the device reported here, while Fig. 1(c) shows a thermal diagram of the same, with the corresponding color code for each device element. A bowtie shaped Pt electromigration junction forms the central part of the device on which 5-10 nm diameter Au nanoparticles were deposited, forming a dense layer of quantum dots, see Fig. 1(b). Here we have chosen Pt as to ensure the source local density of states at the QD contact to be free of superconducting correlations induced by the nearest Al contact. Indeed, and as opposed to Au, Pt suppresses the superconducting proximity effect extremely efficiently [29]. The electromigration junction is connected on one side to a bulky drain electrode made of Au, in fairly good contact to the thermal bath at a temperature T b , and on the other side to a narrow source electrode, again made of Au [7]. Four Al leads provide contacts to the source through a transparent interface. At temperatures well below the superconducting critical temperature of Al, these leads are thermally insulating. The source is therefore fairly thermally decoupled from its environment. In the standard hot electron assumption, electron-electron equilibration is much faster than any other dynamical process. The source electrons are thus in a quasi-equilibrium state described by a Fermi-Dirac distribution at a temperature T e that can significantly differ from T b . The closely spaced pair of Al leads to the source forms an SNS junction with a temperaturedependent critical current that will be used as an electronic thermometer. Conversely, the widely-spaced pair of Al leads forms instead a junction with a vanishing critical current, which allows it to be used as an ohmic heater. In contrast to prior work [26], we have chosen here transparent rather than tunnel contacts to the source, for two reasons. First, SNS junctions can provide less invasive thermometers than NIS junctions that are biased at a voltage V b ∼ ±∆/e, which in turn can lead to significant heating and cooling effects [20]. Second, electromigration requires low access resistances, which is inherently incompatible with tunnel contacts.
The nanometer-sized gap was created within the Pt constriction by means of electromigration at a temperature of 4 K [7,30]. The device was further cooled in situ down to the cryostat base temperature of about 70 mK. Figure 1(d) shows a differential conductance map of the QD junction as a function of bias and gate voltages, V b and V g respectively, with no additional heating. From the observed Coulomb diamonds, one finds a charging energy E c = 4 meV. We estimate the tunnel coupling Γ ≈ 0.9 - 1.4 meV from the FWHM of the conductance map lines. In a weakly coupled QD device, high-conductance lines running parallel to the Coulomb diamond edges are expected, in correspondence to single electron levels. Here these features are absent, presumably washed out due to the large tunnel couplinghΓ k B T . Nevertheless, no Kondo effect is observed.
We now move to the description of the electron thermometers. The critical current I C (i.e. the maximum supercurrent) of an SNS junction is highly sensitive to the electronic temperature T e in N. The relevant energy scale is the Thouless energy th =hD/L 2 , where D is the diffusion constant in N and L is the junction length [31]. For T e > th /k B , I C decreases rapidly with increasing temperature, allowing it to be used as a secondary electron thermometer [21,22]. In a single IV characteristic, the switching current is defined as the value of the current at which the voltage is larger than a threshold voltage chosen above the noise level of the measurement. Figure 2 values. The histogram width increases with the temperature, consistently with a Josephson energy fluctuating by 2k B T . In Fig. 2(b), the variation of the critical current dependence with the bath temperature fits nicely the theoretical expectation [31], the latter being used as the calibration of the thermometer. The low Thouless energy th ∼ 5 µeV was chosen in order to avoid a saturating calibration down to the base temperature. The thermometer thus remains sensitive in the low temperature range of interest here, where the thermal effects across the QD junction dominate other heat relaxation processes.
In the experiment, we heat up the source by applying a constant heating powerQ H = 6 fW to the heater junction. The drain is biased at a potential V b , the source side being grounded via one of the SNS thermometer contacts. Figure 3 ing charge current due to the related Joule power. Right at the degeneracy point, the source temperature is lower than in the rest of the map. The higher resolution temperature map of Fig. 4(a) shows a clear cooling region of ellipsoidal shape, with slightly canted axes. Individual gate traces of T e at two different bias voltages are shown in Fig. 3(b). At zero bias and at gate-closed position, there is neither Joule power nor heat flow through the QD. The source is overheated up to T e = 163.5 mK due to the balance between the applied powerQ H and the main thermal leakage channel, namely the electronphonon coupling. Still at zero bias, but near a charge degeneracy point, heat flows through the QD. This shows up (blue curve in Fig. 3(b)) as a temperature T e drop by several mK at the degeneracy point. At higher bias, this cooling contribution is overcome by the Joule heat. A temperature maximum is thus observed at values of the gate potential close to the degeneracy point (red curve in Fig. 3(b)). The gate-controlled QD junction acts as a heat valve in both cases. The mere observation of cooling at the degeneracy point is in clear contradiction with the theoretical prediction in the sequential tunneling regime. Indeed the present experiment deals with a strong tunnel coupling between the QD and the leads, with a ratio Γ/k B T e ≈ 70. Therefore cotunneling transport processes largely predominate over sequential ones, rendering the above-described picture inapplicable. For an illustration, Fig. 3(c) shows a schematic of the system at the bias levels of Fig. 3(b), respecting the ratios between the bias V b , the level broadening Γ and the thermal smearing k B T .
To go beyond the sequential tunneling approximation we resort to a different approach [4,33]. Thanks to the extremely high charging energy, in the vicinity of a degeneracy point, the device can be described as a non-interacting single energy level. We use a non-interacting Anderson impurity model within a nonequilibrium Green's functions (NEGFs) approach. The Hamiltonian of the system isĤ =Ĥ 0 + α=s,dV kαĉ kα , the first term is the free Hamiltonian of the dot and the second is the Hamiltonians of the source (s) and drain (d) leads. The term V (α) is the coupling Hamiltonian between the lead α and the QD. We are interested in exploring the properties of the leads at stationarity and in particular their electronic temperature; in the NEGFs framework this is possible via the so-called inbedding technique [32,34]. It is worth to mention that it is not based on a full heat balance model accounting for the heat flow via phonons and the superconducting leads. We instead assume that the electron-phonon coupling strength itself does not change appreciably within the temperature range of the map, which is equivalent to assume that the main particle and energy redistribution processes in the lead are dominated by electron-electron interactions. By including in the theory the measured temperature (163.5 mK) of the source when decoupled from the QD, we effectively take into account its thermal coupling to the bath.
The theoretical temperature map around a degeneracy point is shown in Fig. 4(b) and reveals a nice agreement with the experimental data in Fig. 4(a). Here, the temperature of the drain T d is set to 100 mK and the coupling of the QD to the drain and source are |g s | 2 = |g d | 2 = Γ/2 = 0.45 meV. These best fit values allow us to reproduce faithfully the location and temperature profiles of the crossing region, see Fig. 4(c,d). The width in gate potential of the cooling region is independent of the temperature and increases with the coupling Γ [34]. Conversely its extension in bias depends weakly on the coupling strength and increases with the temperature difference across the junction.
The present case actually has some similarities with the regime of a metallic Single Electron Transistor where cooling at the degeneracy point was also found [7,27]. Nevertheless, an asymmetry in gate voltage is clearly observed in the experimental and theoretical temperature map. For a bias voltage V b around 22 µV , the source temperature can be tuned either below or above the temperature (163.5 mK) in the gate-closed state by acting on the gate voltage, see Fig. 4(c). This behavior is not to be expected in the case of a metallic island where electronhole symmetry in the density of states makes transport properties symmetric across the degeneracy point. Fig-ure 4(c) data is thus an unambiguous signature of the QD discrete energy spectrum. At a given bias, the value of the gate potential determines the mean energy of the tunneling electrons. This in turn affects the heat balance in the source and modifies the boundary of the cooling region in the temperature map. The extension in bias of this crossover zone, where one can switch from cooling to heating by finely adjusting with the gate, depends on both on the coupling Γ and the temperature difference between drain and source [34].
This work shows that electronic heat transport through a QD junction can be modulated by a gate potential, making it act as a gate-tunable heat valve. The Coulomb diamond structures in the temperature maps reveal the intimate relation between charge conductance on one hand and heat transport and dissipation on the other. Further experiments may allow a quantitative comparison of thermal effects to the charge transport properties, in a wide range of tunnel couplings. The ability of precision electron thermometry at the heart of a QDbased device demonstrated here opens wide perspectives in the field of heat transport and dissipation in quantum electronic devices, paving the way for quantitative tests of the Landauer principle in the quantum information regime [35].
Methods The device fabrication procedure is similar to our previous work [7]. The fabrication of the main part of the device is based on e-beam lithography, three-angle metallic thin film evaporation and lift-off. After the lift-off, we deposit a 1-2 nm thin Au layer on top of the whole device. Due to surface tension forces, this small amount of deposit leads to the formation of 5-10 nm diameter Au nanoparticles on the sample. As the switching to the resistive state is a stochastic process, a statistical measurement is useful to determine I C with the highest accuracy. We use a saw-tooth shaped a.c. current bias of the junction to measure 3000 switching events in about 10 sec. The critical current of the junction is determined as the gaussian maximum. The thermometer sensitivity is found to be 1.5 µA/K at 80 mK, with a noise level of about 200 µK/ √ Hz.
Collegium for Science and Medicine. Numerical simulations were performed at the Finnish CSC facilities under the Project no. 2000962 "Thermoelectric effects in nanoscale devices".

Supplemental Material: A Single-Quantum-Dot Heat Valve
In this supplemental material part, we discuss about the performance of our SN S thermometer junction as a bolometric detector, and the details of the theoretical approach used to explain our data.

PERFORMANCE OF A SNS JUNCTION AS A BOLOMETER
In order to measure the heat flow through a QD junction, which is described in the main text, one needs to be able to access a very small change in electronic temperature. Moreover, one needs an operating temperature of down to 100 mK or below, where the QD heat flow dominates over the other paths of heat relaxation such as electron-phonon coupling. These two requirements lead us to consider SNS proximity junction as a thermometer that can fulfill both of these requirements. We have optimized the sensitivity of the SNS thermometer with several repetitions of the junctions parameters such as the length and thickness of the normal metal. In this way, we reduced the Thouless energy ( th ) of the SNS junction, which basically determines the lowest saturation temperature of the thermometer [1].
Here we describe a test experiment, where we determine the sensitivity of our optimized SNS thermometer and test its operation as a bolometric detector. The SEM image of the device under test is shown in Fig. S1(a), where the normal metal Au is shown in red color and the superconducting Al leads in light-blue. The basic structure of the device is similar to that of the QD device described in the main text, with the only difference that there is no QD is placed in between source and drain after the electromigration. Therefore, the device can be essentially considered as a ∼ 5 µm long and ∼ 100 nm wide rectangular normal metallic island. Like the samples discussed in the main text it has a very long SNS junction to inject Joule heat into it and a short SNS junction to measure the electronic temperature.
We heat-up the island by applying a constant but variable d.c. power through the heater junction, using an 1.3 V isolated d.c. battery. The SNS thermometer is calibrated against the well known bath temperature, by measuring the histograms of its stochastic switching current, as described in the main text. Here we present an experiment, where the bath temperature is at T b = 90 mK and the heater junction is current-biased through a 200 MΩ biasing resistor, which leads to a heating powerQ H = 100 aW. We continuously monitor the electronic temperature of the island by measuring a histogram of 500 switching currents in about 1 sec. The real time temperature trace of the island is shown in Fig. S2. One can easily identify the change of the electronic temperature by a few mK w.r.t. the background temperature of about 92 mK, whenever the heater is turned on (off). Therefore the thermometer clearly detects an input heat as low as 100 aW, thus performing as a bolometric detector of very small heating power. The noise equivalent power is about 100 aW/ √ Hz. This observation of the island's electronic temperature can be determined by a heat-balance equation, as shown by a heat-balance model in the Fig. S1 where Σ is the material dependent constant, V is the volume of the island, T e and T b are the electron and the bath (phonon) temperatures respectively. Any parasitic heat source (sink) such as heat losses through the superconducting leads due to imperfect thermal insulation [2] or parasitic heating by the electromagnetic environment are taken into account within the injected heating powerQ H . The electronic temperature of the island can be extracted by solving the above heat-balance equation (S1). If we use an injected heating powerQ H = 100 aW, the material constant for Au Σ = 2.4 × 10 9 Wm −3 K −5 [3], the volume of the island V = 2 × 10 −20 m 3 and the bath temperature T b = 90 mK, we get an increase of the electronic temperature ∆T e ∼ 3 mK, which is consistent with the measured value. This justifies the analysis of the heat relaxation mechanism in the island as discussed above.

INBEDDING TECHNIQUE
Here, we give a brief overview of the inbedding technique used to compute the non-equilibrium steady-state of the leads and their electronic temperatures under the hot-electron assumption. The Hamiltonian of the total system (quantum dot plus source and drain reservoirs) is given by: QD the coupling Hamiltonian between the quantum dot, QD, and the α = s, d leads. Using the non-equilibrium Green's function approach [4], and assuming that the whole system reaches a (possibly non-equilibrium) stationary state, the state of the QD is completely characterized by the retarded and lesser single-particle Green's functions: here G A QD (ω) = [G R QD (ω)] † is the advanced component of the Green's functions. g R is the Fourier transform of the retarded Green's function of the isolated (non-coupled to the leads) quantum dot g QD (t, t ) = iθ(t − t ) [d(t),d † (t )] , withd(t) = e iĤQDtd e −iĤQDt . The embedding self-energy is defined as Σ K emb (ω) = α |g α | 2 g K α (ω), with K = R, < and g K α (ω) the Fourier transform of isolated leads' Green's functions. We work in the wide band limit approximation and therefore we have: Σ R emb (ω) = −i α |g α | 2 /2 = −iΓ/2. The first equation in Eqs. S5 is the Dyson equation for the retarded component of the single-particle Green's function, from which the spectral function of the QD can be computed as A QD (ω) = −π −1 ImG R (ω). The embedding self-energy approach accounts for the effect that the leads have on the physical properties of the system, but once the solution to Eqs. S5 are obtained, it is possible to find the Green's functions of the leads and explore the influence of the QD on the physical features of the reservoirs. By introducing the inbedding self-energy Σ K inb,α (ω) = |g α | 2 G K QD (ω) one has the following relations: The lesser Green's functions of the leads G < α (ω) are different from the isolated ones g < α (ω) = if α (ω), with f α (ω) the Fermi-Dirac distributions characterized by the initial chemical-potential and temperature µ α and T α . From the knowledge of G < α (ω) we can compute both the average particle number and energy of the leads: N α = i dω 2π G < α (ω) and E α = −i dω 2π ω G < α (ω). We now resort to the hot electron assumption that accounts for a neat separation of the energy relaxation time scales between system and leads. Basically, it assumes that the electron-electron interactions into the leads makes their equilibration time much faster than any other dynamical processes in the whole system. Therefore, we can assume that the leads at the steady state are described by a new Fermi-Dirac distribution f α (ω) with a different set of chemical potential and temperature µ α and T α . The two parameters are the solution of the set of non-linear equations: (S11)

PARTICLE AND ENERGY CURRENTS
Within the non-equilibrium Green's function formalism, particle, energy and heat currents of the source lead are given by the following expressions: For the set of parameters considered in the main text, the heat and particle currents for the source are shown in Fig. S3. It is interesting to notice the resemblance between the heat current map and the temperature map shown in the main text. It confirms that in the regime we explored the temperature changes in the source lead correspond indeed to a heat current to/from the source (panel (a) in Fig. S3). At large bias the source heats up; the system behaves as a heater, namely the energy of the bias is transformed into internal energy. At low bias we observe instead heat flow from the hot to the cold lead; the system behaves as a valve, meaning that it enables the natural flow of heat from the hot to the cold lead. It is also interesting to observe that there is a whole region in which heat and particle currents have opposite signs.
It is worth to mention that this effect is not due to the onset of a thermovoltage which would make particles flow against the applied bias voltage without necessarily causing an inversion of heat current. The thermovoltage, although present, is very small compared to the extension in bias voltage where the mismatch in sign is observed. This is shown in Fig. S3 panel (c) where we plot the thermovoltage V th (V g ), together with the corresponding thermopower, defined as the bias voltage at which the particle current vanishes.
The observed significant thermal conductance is a signature of the strong coupling of the QD to the leads. We computed the Lorentz ratio L = κ/(T σ) where κ = ∂Q s /∂∆T | Is=0 and σ = ∂I s /∂V b | ∆T =0 are the thermal and electrical conductivities respectively. In panel (d) of Fig. S3 we plot the Lorentz number L/L 0 with L 0 = (π 2 /3)(k B /e) 2 at the degeneracy point V g = 0. The dashed line corresponds to the ratio Γ/k B T ≈ 70 considered in the system presented in the main text. It is clear that the deviation from the WF law is small because of the strong coupling to the leads.
The reason for the heat valve behavior has to be sought in the microscopic selection of particles from above and below the chemical potential of the lead. Particle current follows the chemical potential whereas the total energy carried depends upon the balance between energy subtracted from below and from above the chemical potential. Extracting particles from above (below) the chemical potential results in cooling (heating). This feature is highlighted in Fig. S4 where we consider a cut of the source temperature at a given V b = 22 µV. The sign of the heat current is shown with a red arrow for three specific gate voltages; the particle current is always finite and negative. Figure S5 shows the map of the calculated electronic temperature for couplings (panels (a) and (c)) Γ = 0.5 meV and (panels (b) and (d)) Γ = 1.5 meV and for the same drain temperature at closed gate voltage T d = 100 mK similarly to Fig. 3 of the main text. A change in the coupling changes the extension of the cooling region in the gate voltage but it does not affect dramatically the extension of the cooling region in the bias voltage nor the position of the transition from cooling to heating.

THE COOLING AND TRANSITION REGIONS
To give a more quantitative analysis we computed the extension of the cooling region in the bias voltage A b at V b = 0 and its width A g in gate voltage for different couplings and temperatures of the drain at closed gate. The results are plotted in Fig. S6 in panels (a) and (b) where it can be appreciated that the coupling constant does not change significantly the extension in bias which instead strongly depends upon the difference in the equilibrium temperatures between the drain and source. Indeed as the temperature of the drain increases towards the temperature of the source the extension in V b shrinks. Nevertheless the extension in the gate voltage is only determined by the coupling and does not present any significant dependence upon the closed gate temperature of the drain.
The transition region, namely the region where at fixed bias it is possible to obtain both heating and cooling by changing the gate potential, has a strong dependence on both the coupling and the temperature difference at closed gate. This is shown in Fig. S6 panels (c) and (d) where we plot the width ∆ g and extension ∆ b of the transition region; they have been determined by finding the curve V b (V g ) such that T e =163.5 mK and then taking ∆ b = Max(V b (V g )) − min(V b (V g )) whereas ∆ g is the difference between the voltage gates at which V b is larger than its values at closed gate plus 0.1∆ b on both sides. It can be observed that this region becomes smaller as the coupling is increased whereas its width increases with the coupling. The width also decreases steadily as the temperature of the drain increases whereas the behavior of its width with temperature is less trivial. It decreases at large couplings whereas it increases as the temperature increases at small couplings.