Abstract

Quantum thermodynamics is emerging both as a topic of fundamental research and as a means to understand and potentially improve the performance of quantum devices1,2,3,4,5,6,7,8,9,10. A prominent platform for achieving the necessary manipulation of quantum states is superconducting circuit quantum electrodynamics (QED)11. In this platform, thermalization of a quantum system12,13,14,15 can be achieved by interfacing the circuit QED subsystem with a thermal reservoir of appropriate Hilbert dimensionality. Here we study heat transport through an assembly consisting of a superconducting qubit16 capacitively coupled between two nominally identical coplanar waveguide resonators, each equipped with a heat reservoir in the form of a normal-metal mesoscopic resistor termination. We report the observation of tunable photonic heat transport through the resonator–qubit–resonator assembly, showing that the reservoir-to-reservoir heat flux depends on the interplay between the qubit–resonator and the resonator–reservoir couplings, yielding qualitatively dissimilar results in different coupling regimes. Our quantum heat valve is relevant for the realization of quantum heat engines17 and refrigerators, which can be obtained, for example, by exploiting the time-domain dynamics and coherence of driven superconducting qubits18,19. This effort would ultimately bridge the gap between the fields of quantum information and thermodynamics of mesoscopic systems.

Main

Mesoscopic normal-metal (N) resistors are natural candidates for the role of heat reservoirs for superconducting circuit QED experiments. Their geometry and transport properties can be adapted to provide a controllable amount of dissipation by virtue of the electron–photon interaction20,21,22. Furthermore, either clean or tunnel-type interfaces with the surrounding circuit elements enable control of the impedance mismatch for a given microwave design. With their fast internal thermalization timescales23 and slow electron–phonon relaxation11,24 at subkelvin temperatures, reservoirs formed of normal metal electrodes have been demonstrated as effective broadband microwave detectors25 and sources26. Their thermal properties, as well as the experimental techniques required for temperature manipulation and readout, are well established and understood27.

In this work we consider heat transmitted between two such mesoscopic reservoirs, each of which is tied to the photon occupation number of a microwave resonator by the temperature-dependent Johnson–Nyquist current fluctuations of the resistor. Here, the two resonators are designed to have identical resonant frequencies fr and are coupled to each other via a tunable oscillator, a transmon-type qubit. This resonator–qubit–resonator assembly constitutes a quantum heat valve (QHV). The thermal conductance of the QHV, conceptually depicted in Fig. 1a, is expected to depend on the reservoir–resonator and resonator–qubit couplings (respectively γ and g, both normalized with respect to fr) and on the ratio r between the level spacing of the qubit and the eigenfrequency of the resonators (fq ≡ rfr). For the transmon qubit, fq depends on the magnetic flux Φ as

$$f_{\mathrm{q}}(\it{\Phi }) = \frac{{\sqrt {{\rm{8}}E_{\mathrm{J}}({{\Phi }})E_{\mathrm{C}}} - E_{\mathrm{C}}}}{h}$$
(1)

where EC and EJ(Φ) = \(E_{{\mathrm{J}}0}\left| {{\mathrm{cos}}\left( {\uppi {{\it{\Phi} /\it{\Phi} }}_0} \right)} \right|\) \(\sqrt {1 + d^2{\mathrm{tan}}^2\left( {\uppi {{\it{\Phi} /\it{\Phi} }}_0} \right)}\) are the charging and Josephson energies of the transmon, respectively; here, Φ0 = h/2e is the magnetic flux quantum, and critical current asymmetry in the superconducting quantum interference device (SQUID) junctions is accounted for by the parameter d. The static dependence of the electron temperature TD in the drain (D) reservoir is determined by the temperature of the source (S) reservoir TS and the qubit detuning with respect to the resonators.

Fig. 1: Quantum heat valve design.
Fig. 1

A transmon qubit, whose level spacing fq is tunable by magnetic flux Φ, is capacitively embedded between two superconducting transmission lines of identical length = 4.6 mm, each terminated by a mesoscopic normal-metal reservoir. We study the temperature of the drain reservoir TD as a function of the temperature of the source reservoir TS and of the ratio r ≡ fq/fr, where fr is the fundamental resonant frequency of the transmission lines. a, Conceptual depiction of the heat valve. b, Thermal model. c, Lumped-element idealization of the device; capacitors Cg couple the transmon to each LrCr resonator. d, Scanning electron micrograph of a waveguide termination, including three tunnel electrodes. A copper resistor (pink, online) is in clean contact with aluminium leads (light blue, online) connecting to the patterned niobium film (light grey) on a sapphire substrate (dark grey). The inset shows a magnified orthogonal view of the area spanned by the normal-metal element; the scale bar corresponds to 3 μm. e, Scanning electron micrograph of the SQUID element in the transmon structure; the scale bar corresponds to 10 μm.

Figure 1b summarizes the thermal model between the source and drain reservoirs. By voltage-biasing a pair of normal metal–insulator–superconductor junctions (SINIS) attached to the source reservoir, one can control its temperature. At sub-gap voltages, evacuation of hot quasiparticles from the source reservoir lowers TS below its unbiased value, while, above it, the biasing provides conventional Joule heating27. Under fixed experimental conditions, the electrons in each normal-metal reservoir are in local thermal equilibrium. In the detailed thermal balance we consider the interaction of the electron system (of temperature Tel) with the environment (resulting in an effective power Penv which includes the influence of SINIS biasing where appropriate) and with the phonon bath (whose temperature Tph is assumed to be uniform and equal to the temperature of the cryostat). The latter mechanism is modelled by the conventional normal-metal electron–phonon interaction \(P_{{\mathrm{el}} - {\mathrm{ph}}} = {{\Sigma }}{\cal V}\left( {T_{{\mathrm{el}}}^5 - T_{{\mathrm{ph}}}^5} \right)\) that for small temperature differences can be linearized with the thermal conductance \(G_{{\mathrm{el}} - {\mathrm{ph}}} = 5{{\Sigma }}{\cal V}T_{{\mathrm{el}}}^4\). Here, \({\cal V}\) is the volume of the normal-metal reservoir and Σ is the corresponding electron–phonon coupling constant. In the experiment, the source-to-drain heating power (PD = −PS by energy conservation) is determined by the response in TD under the assumption of the electron–phonon interaction dominating the thermal relaxation of the electrons in the drain reservoir. The lumped-element circuit representing the device is schematically illustrated in Fig. 1c. Each resonator is terminated at one end by a capacitor to the transmon (Cg ≈ 8.6 fF) and at the other end by the normal-metal resistor to the Nb ground plane (Fig. 1d). This configuration results in a quarter-wave resonator, with expected eigenfrequency fr = 6.4 GHz and quality factor Qr = Z0/RN ≈ 20, where Z0 = πZ/4 is the resonance impedance and Z = 50 Ω is the design impedance of the coplanar waveguide. Here RN ≈ 2 Ω is the nominal resistance of the N termination; depending on the transparency of the metallic interfaces, additional dissipation can significantly decrease the effective quality factor. In our design the relaxation to the reservoir is the dominant source of losses in the resonator, so that its quality factor Qr ≡ 1/γ.

In modelling the system, one can consider the photonic reservoir–reservoir coupling to be relatively weak, which allows us to apply standard perturbation theory to describe it. We expect the total thermal conductance between the reservoirs to be three orders of magnitude lower than the quantum of thermal conductance of a single channel \(G_{\mathrm{Q}} = \left( {\uppi k_{\mathrm{B}}^2{\mathrm{/}}6\hbar } \right)T\) at temperature T (refs 20,28). Here we explore two photonic weak-coupling models, each based on the formalism appropriate to the impact of reservoir-induced dissipation compared to the qubit coupling rate. We call these the quasi-Hamiltonian (QH) model for \(\gamma \simeq g\), and non-Hamiltonian (NH) model applicable when \(\gamma \gg g\), respectively. Conceptually, these two models showcase a different location for the Heisenberg cut (that is, the separation between the quantum subsystem and its classical environment): either at the qubit–resonator to reservoir boundaries or at the qubit–resonator interfaces, respectively. In both models, the power to each reservoir is given by

$$P_{{\mathrm{S/D}}} = \mathop {\sum }\limits_{k,l} {\kern 1pt} \rho _{kk}{\kern 1pt} E_{kl}{\kern 1pt} {{\Gamma }}_{k \to l,{\mathrm{S}}/{\mathrm{D}}}$$
(2)

where ρ is the density matrix and Ek,l, Γkl,S/D are the transition energy and rate for each respective reservoir, and the sum runs over all the eigenstate indices k, l.

In the absence of dissipation, a fully Hamiltonian (FH) description considers that the qubit and the two resonators form a system of three coupled harmonic oscillators with level spacing hfq, hfr. This neglects both nonlinear SQUID dynamics and occupation of higher resonator harmonics, under the justification of quasi-static qubit drive and low temperatures in the two reservoirs \(\left[ {\beta _{{\mathrm{S}}/{\mathrm{D}}}hf_{\mathrm{r}} \equiv hf_{\mathrm{r}}{\mathrm{/}}\left( {k_{\mathrm{B}}T_{{\mathrm{S}}/{\mathrm{D}}}} \right) \gg 1} \right]\), respectively. The second-quantized Hamiltonian of the hybrid system reads as

$$\begin{array}{*{20}{l}} {\hat H} \hfill & = \hfill & {hf_{\mathrm{r}}\left[ {\left( {\hat a_{\mathrm{D}}^\dagger \hat a_{\mathrm{D}} + \hat a_{\mathrm{S}}^\dagger \hat a_{\mathrm{S}}} \right) + r\hat b^\dagger \hat b} \right.} \hfill \cr {} \hfill & {} \hfill & { + g\left( {\hat b\hat a_{\mathrm{D}}^\dagger + \hat b^\dagger \hat a_{\mathrm{D}} + \hat b\hat a_{\mathrm{S}}^\dagger + \hat b^\dagger \hat a_{\mathrm{S}}} \right)} \hfill \cr {} \hfill & {} \hfill & {\left. { + \tilde g\left( {\hat a_{\mathrm{D}}\hat a_{\mathrm{S}}^\dagger + \hat a_{\mathrm{D}}^\dagger \hat a_{\mathrm{S}}} \right)} \right]} \hfill \end{array}$$
(3)

where \(\hat{a}_{S/D}^{\dagger}\) and \(\hat{a}_{S/D}\) are the creation and annihilation operators for the S/D-resonators, while \(\hat{b}^{\dagger}\) and \(\hat{b}\) are the creation and annihilation operators for the qubit; additionally, \(\tilde g\) quantifies direct resonator-to-resonator coupling. Following the low-temperature argument above, we choose the minimal four-level basis of \(\left\{ {\left| {000} \right\rangle ,\left| {100} \right\rangle ,\left| {010} \right\rangle ,\left| {001} \right\rangle } \right\}\), where the entries in each state refer to the S-resonator, the qubit, and the D-resonator, respectively. With the addition of the parameter \(a = {\mathrm{\Delta }}f{\mathrm{/}}f_{\mathrm{r}} \ll 1\) (quantifying possible minor asymmetry Δf between the eigenfrequencies of the two resonators), this choice of basis results in the matrix representation

$$\hat H = hf_{\mathrm{r}}\left( {\begin{array}{*{20}{c}} 0 & 0 & 0 & 0 \cr 0 & {1 + a{\mathrm{/}}2} & g & {\tilde g} \cr 0 & g & r & g \cr 0 & {\tilde g} & g & {1 - a{\mathrm{/}}2} \end{array}} \right)$$
(4)

In the a → 0, r → 0 limit, the photon cavity modes contribute a pair of eigenstates corresponding to the symmetric and antisymmetric combinations of the eigenmodes localized in each resonator. They are in general non-degenerate due to \(\tilde g \ne 0\), and only the symmetric combination interacts with the qubit via g. These features are evident in the dispersion of the eigenenergies shown in Fig. 2b, where the dominant transitions between the levels are also indicated. To directly probe the flux-dependent spectrum of eigenstates of the QHV in the FH limit \(\left( {\gamma \ll g} \right)\), we use a design where the CPWs in the source and drain resonators are connected directly to the ground plane without resistors. In this design, a diagnostic resonator (fd ≈ 7.4 GHz) is capacitively coupled (Cd ≈ 3.4 fF) to the top arm of the transmon island and inductively coupled to a microwave feedline. Typical two-tone spectroscopic data, obtained by standard29 transmission readout of the diagnostic resonator, are shown in Fig. 2a. Inspection of the transition branches indicates that the coupling capacitance Cg induces a 216-MHz-wide avoided crossing with the symmetric resonator eigenmode, consistent with g = 0.02. Additionally, a small (<1%) asymmetry in resonator eigenfrequencies allows the interaction between the qubit and the antisymmetric S/D resonator eigenmode, visible as a minor avoided crossing at f2nd ≈ 5.47 GHz. These figures set the typical power scale of the qubit-mediated heat transfer to \(hf_{\mathrm{r}}^2g \approx 0.4{\kern 1pt} {\mathrm{fW}}\).

Fig. 2: Fundamental excitations of the resonator-qubit-resonator assembly.
Fig. 2

a, Two-tone transmission spectroscopy data centred in the fr ≈ fq region for a sample in the fully Hamiltonian limit. Thin lines, representing eigenvalues derived from equation (4), are superimposed to the experimental data set; optimal matching is obtained with parameters listed under column FH in Supplementary Table 1 with the addition of resonator asymmetry a = 0.008 to reproduce the secondary avoided crossing visible at f2nd ≈ 5.47 GHz. b, Overview of the dominant steady-state transitions between eigenstates contributing to heat transport in the quasi-Hamiltonian limit according to equation (5). For visual clarity, here \(g = - \tilde g = 0.1\), \(a = 0.05\).

We now consider the effect of introducing moderate dissipation to the system via the S/D reservoirs—that is, the quasi-Hamiltonian (QH) regime. Equation (2) allows us to determine the power from the S-reservoir to the D-reservoir as

$$P_{\mathrm{D}} = \frac{{2\pi hf_{\mathrm{r}}^2}}{{Q_{\mathrm{r}}}}\mathop {\sum }\limits_{k,l} {\kern 1pt} \rho _{kk}\frac{{\left| {\left\langle k \right|\hat a_{\mathrm{D}} - \hat a_{\mathrm{D}}^\dagger \left| l \right\rangle } \right|^2}}{{1 + Q_{\mathrm{r}}^2\left( {f_{kl}{\mathrm{/}}f_{\mathrm{r}} - f_{\mathrm{r}}{\mathrm{/}}f_{kl}} \right)^2}}\frac{{\left( {E_{kl}{\mathrm{/}}hf_{\mathrm{r}}} \right)^2}}{{1 - \mathrm{e}^{ - \beta _{\mathrm{D}}E_{kl}}}}$$
(5)

Here, the steady-state balance of the transition rates Γkl determines the level populations ρkk. In this model, PD is: limited by the reciprocal quality factor \(Q_{\mathrm{r}}^{ - 1} \equiv \gamma\); non-vanishing at all values of flux even far away from the resonance; affected, around fq = fr, by fast variation of populations, energy splitting, and matrix elements. Experimental data for a QH-type sample recorded at Tph = 45 mK are presented in Fig. 3a. Here, different traces, representing the estimate for the power absorbed by the drain reservoir, correspond to different thermal biases applied between the source reservoir (TS, controlled in the 100–330 mK range) and the drain reservoir (unbiased temperature TD ≈ 100 mK). The traces show a sizeable amount of flux-independent power transmitted to the drain reservoir. This is particularly impressive for complete resonator–qubit detuning for an applied flux corresponding to half-integer values of Φ0. The origin of this power flow between the reservoirs lies in the role of the two mixed S/D resonator eigenmodes spanning the whole resonator–qubit–resonator assembly. Remarkably, approaching the fq > fr condition near integer flux bias values results in an initial increase of the absorbed power, followed by a step-like decrease and a partial revival when reaching integer Φ/Φ0 values, where fq(Φ) is maximal. The comparison with the theoretical prediction provided by equation (5) with the nominal Qr = 20 value indicates that the model captures all these features quantitatively. In this case, optimal reproduction of experimental data is found, according to the estimates presented in Supplementary Table 1, with g ≈ 0.019 and \(\tilde g \approx - 0.020\). These values compare well to the ones directly measured from the two-tone spectroscopy of the FH-type samples; notably, g/γ = gQr ≈ 0.4.

Fig. 3: Modulation of photonic heat transport.
Fig. 3

Total heating power absorbed by the drain reservoir as a function of the applied magnetic flux Φ. Different traces correspond to the source temperature values TS shown in the adjacent legend bar. The unbiased temperature of the drain reservoir is here marked by a triangle. In each plot, experimental data are juxtaposed to the optimal fit of the appropriate theoretical model. a,b, Plots correspond to the quasi-Hamiltonian (equation (5)) and the non-Hamiltonian (equation (6)) regimes, respectively. Relevant modelling parameters are listed in Supplementary Table 1. Residual reservoir–reservoir coupling, mediated by weak on-chip thermal conductance, is represented by an additional power-law contribution P0 = ξ[(TS/TD)n − 1], where ξ = 5.14 aW and n = 4.63 are empirical parameters.

The NH model is described in ref. 19. The power from S-reservoir to the D-reservoir reads

$$\begin{array}{*{20}{l}} {P_{\mathrm{D}}} \hfill & = \hfill & {\textstyle{{{\pi hgf_{\mathrm{r}}^2n( {\beta _{\mathrm{S}}hf_{\mathrm{q}}}) - n( {\beta _{\mathrm{D}}hf_{\mathrm{q}}})} \over {[ {1 + Q_{\mathrm{r}}^2( {r - 1{\mathrm{/}}r} )^2} ][ {{\mathrm{coth}}( {\beta _{\mathrm{S}}hf_{\rm{q}}{/}2} ) + {\mathrm{coth}}( {\beta _{\mathrm{D}}hf_{\mathrm{q}}{\mathrm{/}}2} )} ]}}}} \hfill \cr {} \hfill & + \hfill & {\pi h\kappa f_{\mathrm{r}}^2\mathop {\smallint }\limits_0^\infty {\kern 1pt} \frac{{n( {x{\kern 1pt} \beta _{\mathrm{S}}hf_{\mathrm{r}}} ) - n( {x{\kern 1pt} \beta _{\mathrm{D}}hf_{\mathrm{r}}} )}}{{[ {1 + Q_{\mathrm{r}}^2(x - 1{\mathrm{/}}x)^2} ]^2}}x^3{\mathrm{d}}x} \hfill \end{array}$$
(6)

where n(βS/Dhf) = 1/(exp(βS/Dhf) − 1) is the equilibrium mode population in each resonator; the second term describes direct resonator-to-resonator photon transfer, quantified by κ. Overall, PD is: limited by the couplings g, κ (as opposed to γ in the QH model); peaking when the qubit transition frequency matches the resonator eigenfrequency, fq = fr; inhibited when the qubit–resonator detuning exceeds the resonator linewidth \(\left| {f_{\mathrm{q}} - f_{\mathrm{r}}} \right|{\mathrm{/}}f_{\mathrm{r}} \gg Q_{\mathrm{r}}^{ - 1}\). The flux dependence of the power to the D-reservoir recorded at Tph = 55 mK for an NH-design device is shown in Fig. 3b. This dependence is consistent with the expectations based on equation (6). Here, different traces, representing the estimate of the power absorbed by the drain reservoir, correspond to different thermal biases applied between the source reservoir (TS, controlled in the 80–360 mK range) and the drain reservoir (unbiased temperature TD ≈ 120 mK). The modulation of all traces shows the clear presence of two broad peaks per flux period, corresponding to the condition fq = fr. The shape of the flux modulation appears independent of the sign of the thermal bias. This sign reversal can be observed in the traces corresponding to the three lowest values of TS, obtained by electron-cooling (instead of heating) of the source reservoir with an appropriate sub-gap voltage bias of a SINIS junction pair. In this figure, experimental data are juxtaposed to the best fit of equation (6), yielding the parameter estimates listed in Supplementary Table 1, in particular Qr = 3.15 ± 0.14. Such a low quality factor fully justifies the adoption of the NH model, even in the presence of a non-negligible coupling: g/γ = gQr ≈ 0.05.

In the NH case, the number of photonic excitations in each resonator is dominated by dissipative processes in the reservoirs. Under this hypothesis, the overdamping prevents the formation of the mixed S/D eigenmodes characteristic of the Hamiltonian limit. Notably, the presence of these excitations (spectroscopically probed in the FH sample, where \(g{\mathrm{/}}\gamma \gg 1\)) is required to quantitatively reproduce via equation (5) the heat modulation observed in the QH sample, in spite of its arguably low g/γ ≈ 0.4. In the NH limit, instead, the excitation of the qubit acts as an independent flux-tunable spectral filter between the photonic populations tied to the source and drain reservoirs (equation (6)). Figure 4 presents a comparison of QH and NH samples in terms of performance as a heat valve. We see that the highest modulation ratio is obtained in the NH sample for low temperatures, where the flux-independent ‘background’ contributions are small in comparison to the actual photonic power.

Fig. 4: Quantum heat valve performance.
Fig. 4

Valve modulation ratio (maxΦPD − minΦPD)/maxΦ |PD| as a function of the source temperature TS for non-Hamiltonian and quasi-Hamiltonian regimes. Dotted lines are intended as a visual aid.

The quantum heat valve presented here is a key platform dedicated to the investigation of quantum thermodynamic phenomena in hybrid mesoscopic/circuit QED systems. Planning devices including active thermal degrees of freedom requires matching resonator eigenenergies to the expected reservoir temperature. The principal heat transport bottleneck can be the resonator–qubit coupling, typically g 0.05 for coplanar elements. On the other hand, a comparably strong resonator–reservoir relaxation mechanism is required for the thermalization of the relevant photonic mode population. We find that the competition between qubit–resonator and reservoir–resonator couplings affects strongly not only the power scale of the heat transport, but also the locality of its physical origin.

Methods

Fabrication protocols

The devices were fabricated on 330-μm-thick sapphire substrates coated with 200-nm-thick sputtered niobium film. Broader features, such as coplanar waveguides, transmon island and electrode fanout were patterned by reactive ion etching on an electron-beam lithography-defined mask. The CPW design features a 20-μm-wide centreline spaced by 10 μm with respect to the ground plane, resulting in capacitance and inductance per unit length of 153 fF mm−1 and 403 pH mm−1, respectively. All chip layouts are available in the Supplementary Information. Nanostructures, including the tunnel junction elements, were realized in two steps with shadow-mask electron-beam lithography on a 1-μm-thick poly(methyl-metacrylate)/copolymer resist bilayer, followed by tilted thin film deposition in an electron-beam evaporator. In the first step, two offset depositions of 28-nm-thick Al layers (with intermediate oxidation) are performed to realize the transmon SQUID (Fig. 1e) with typical per-junction tunnel resistance RT ≈ 7 kΩ at cryogenic temperature. In the second step, the terminations of the resonators are realized by first depositing and oxidizing a 15-nm-thick Al layer, followed by a 50-nm-thick Cu layer, and finally by a 85-nm-thick Al layer in clean contact with the Cu layer. The typical tunnel resistance is RNIS ≈ 25 kΩ; during the experiment, these electrodes are connected to fanout lines for the setting and readout of the electron temperature in the reservoirs over a typical timescale of tens of milliseconds. In both steps, the contact between the Nb substrate and the deposited metal is facilitated by in situ Ar ion plasma milling, while tunnel junctions are realized by controlled oxidation (oxygen partial pressure ≈ 10 mbar for 8 min). After liftoff in acetone and cleaning in isopropyl alcohol, the substrates are diced to size (4 × 8 mm for QH/NH-type and 7 × 7 mm for FH-type chips) with a diamond-coated resin blade and wire-bonded to a custom-made brass chip carrier for the cryogenic characterization.

Measurements

The experiment has been performed in a custom-made dilution refrigerator able to reach base temperature values of approximately 50 mK. The bonded chip, shielded by two brass Faraday enclosures, is connected to the room-temperature breakout box with conventional cryogenic signal lines. Each line is filtered by a 1-m-long Thermocoax wire segment, resulting in an effective signal bandwidth of 0–10 kHz, for low-impedance loads. A magnetic field is applied perpendicular to the sample substrate by a superconducting magnet wound on the exterior of the insert vacuum can. The latter is inserted in a high-permeability magnetic shield.

Current and voltage electrical bias are applied by programmable voltage sources and function generators with appropriate room-temperature resistor networks. Current and voltage amplification is performed by room-temperature low-noise amplifiers (FEMTO Messtechnik GmbH, models DLPCA-200 and DLPVA-100). In order to minimize the impact of signal pickup and low-frequency drifts of the differential voltage amplifier output in the SINIS thermometer readout, temperature signals are derived from the first harmonic recorded by a lock-in amplifier synchronized to the square-wave modulation (42 Hz) of the voltage bias of the source reservoir SINIS circuit. The noise-equivalent spectral density obtained in this differential readout scheme is 0.1 mK/\(\sqrt {{\mathrm{Hz}}}\), corresponding to typical uncertainty δT ≈ 40 μK (r.m.s.) in the temperature estimates (effective integration bandwidth = 0.14 Hz for the lock-in measurement). Quantitative estimates of the bias-dependent power absorbed by the drain reservoir are obtained assuming that the electron–phonon interaction dominates the thermal relaxation, yielding

$$P_{\mathrm{D}} \approx {\mathrm{\Sigma }}{\cal V}T_{\mathrm{D}}^4{\mathrm{\Delta }}T_{\mathrm{D}}$$
(7)

where ΔTD is the peak-peak amplitude of the signal recorded by the drain thermometer.

Data availability

The data that support the plots within this article are available from the corresponding author upon reasonable request.

Additional information

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Acknowledgements

This work was funded through Academy of Finland grants 297240, 312057 and 303677 and from the European Union’s Horizon 2020 research and innovation programme under the European Research Council (ERC) programme and Marie Sklodowska-Curie actions (grant agreements 742559 and 766025). This work was supported by Centre for Quantum Engineering (CQE) at Aalto University. We acknowledge the facilities and technical support of Otaniemi research infrastructure for Micro and Nanotechnologies (OtaNano), and VTT Technical Research Center for sputtered Nb films. We acknowledge M. Meschke for technical help and O.-P. Saira for useful discussions in the initial stages of this work. We thank D. Golubev and Y. Galperin for helpful discussions.

Author information

Affiliations

  1. QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, Aalto, Finland

    • Alberto Ronzani
    • , Bayan Karimi
    • , Jorden Senior
    • , Yu-Cheng Chang
    • , Joonas T. Peltonen
    • , ChiiDong Chen
    •  & Jukka P. Pekola
  2. Department of Physics, National Taiwan University, Taipei, Taiwan, Republic of China

    • Yu-Cheng Chang
  3. Institute of Physics, Academia Sinica, Taipei, Taiwan, Republic of China

    • Yu-Cheng Chang
    •  & ChiiDong Chen

Authors

  1. Search for Alberto Ronzani in:

  2. Search for Bayan Karimi in:

  3. Search for Jorden Senior in:

  4. Search for Yu-Cheng Chang in:

  5. Search for Joonas T. Peltonen in:

  6. Search for ChiiDong Chen in:

  7. Search for Jukka P. Pekola in:

Contributions

The experiment was conceived by J.P. and B.K., with contributions from C.D.C. A.R. performed the experiment. A.R., J.S. and Y.-C.C. designed and fabricated the samples. Data analysis was performed by A.R. based on theoretical models conceived and solved by J.P. and B.K. Y.-C.C. performed the spectroscopy measurements. J.T.P. provided technical support in fabrication, low-temperature set-ups and measurements. All authors have been involved in the discussion of scientific results and implications of this work. The manuscript was written by A.R. with contributions from J.P., B.K. and J.S.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Alberto Ronzani.

Supplementary information

  1. Supplementary Information

    Supplementary notes, figures and references

About this article

Publication history

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DOI

https://doi.org/10.1038/s41567-018-0199-4