Bolometric detection of Josephson inductance in a highly resistive environment

Abstract

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 R_Q = 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.

Fig. 1. Experimental setup and principle of the photonic heat transport in high ohmic environment.

a Colored scanning electron micrograph (scale bar: 5 μm) highlighting the (Cr) normal metal (blue and red) and the aluminum superconducting leads (light blue). The Josephson energy of the SQUID is tuned with an external magnetic field. Aluminum leads (vertical, light blue) are connected through an oxide tunnel barrier to the Cr-strip to cool down its electrons locally or as an electronic temperature sensor using a floating DC current source. b Schematic illustration of the thermal model of the system. The drain-source heat flow ${\stackrel{°}{Q}}_{\nu }$ is adjusted by the SQUID. The source and drain electron baths are thermally coupled to the phonon bath (which here is hotter), receiving a power ${\stackrel{°}{Q}}_{\mathrm{ep},\mathrm{S}}$ and ${\stackrel{°}{Q}}_{\mathrm{ep},\mathrm{D}}$, respectively. Wiggly lines highlight the strong interaction between the SQUID and the ohmic environment, mediated by photons.

Fig. 2. DC charge transport measurements.

a Scanning electron micrograph of one of the Replica samples (scale bar: 5 μm) with the schematics of the IV measurement. b, c A close-up of the IVC at the low-bias voltage for the two Replica samples measured at a cryostat temperature of 87 mK exhibiting the Coulomb blockade feature. The measurements were recorded at two magnetic flux values Φ = 0 (solid circles) and Φ = Φ₀/2 (open circles). The dashed lines in (b) and (c) are the theoretical results obtained by the standard P(E)-theory for two different magnetic flux values. For Replica I in (b), the fit parameters are: critical current I_C ~ 7 nA, Josephson energy E_J ~ 0.17 K, charging energy E_C ~ 0.6 K and, for Replica II in (c): I_C ~ 3 nA, E_J ~ 0.08 K, E_C = 1.4 K. Cr-strips' resistance is R_e = 11 kΩ for both samples. The inset in (b) and (c) show the effective temperature of the resistor at low voltage bias. d Illustration of the Schmid phase diagram for a Josephson junction attached to a resistive environment R at zero temperature. Here, R = R_S + R_D = 2R_e is the total resistance of the environment. Our samples are well placed in the insulating part, represented by the two points.

Fig. 3. Electronic refrigeration.

a, b Electronic temperature of the source T_S (blue points) and drain T_D resistor (red points) at Φ = 0 for samples I and II, respectively, as a function of the heating voltage V_H applied on the source resistor. The measurements were recorded at phonon temperature T₀ = 151 mK (Sample I) and 180 mK (Sample II). c, d Temperature drops of the source ΔT_S (circles) and the drain ΔT_D (stars) recorded at two magnetic flux values Φ = 0 (solid symbols) and Φ = Φ₀/2 (open symbols) against T₀. The error bars are not shown since their values are smaller than the symbols.

Fig. 4. Heat transport mediated by photons and the theoretical model proposed.

a, b Photonic heat current from the drain to the source by using the continuity Eq. (1). The error bars are given in their lower and upper parts, the combination of the thermometer calibration and electron-phonon coupling constant uncertainties, while the upper part also includes the parasitic heat leak on the resistors due to the NIS junctions, with a 0.4 fW upper bound estimate. The solid line represents the power transmitted through a single channel at the quantum limit ${\stackrel{°}{Q}}_{\mathrm{Q}}$ (see text). The dashed lines are obtained by solving Eq. (2) with a photon transmission probability calculated with the linear circuit model, with the circuit parameters obtained from fitting the IVC of the Replica samples in Fig. 2b, c. d, e Heat current modulation as a function of the reduced magnetic flux Φ/Φ₀ through the SQUID at a given temperature T₀. The dashed lines are the application of the linear model (see text), keeping the same circuit parameters as in (a) and (b). c Electric representation of the device in the linear model, where the SQUID is approximated by a re-normalized variable inductor L_eff(Φ) in parallel with the junction geometric capacitor C_J, and f P(E) model with an effective impedance $\tilde{Z}\left(\omega ,\mathrm{\Phi }\right)$ replacing the Josephson element.