Testing Kubo formula on a nonlinear quantum conductor driven far from equilibrium via power exchanges

TL;DR

This work tests the Kubo formula for a strongly nonlinear quantum conductor driven far from equilibrium by a DC bias, using a SIS tunnel junction coupled to a narrowband linear detector to separately extract emission and absorption noise via power exchange. By calibrating the detector occupation with a nearby NIN shot-noise source and comparing the noise-derived admittance to that measured by RF reflectometry, the authors validate that Re$Y(f)$ equals the current-fluctuation asymmetry $(S_{II}(-f)-S_{II}(f))/(2hf)$ even far from equilibrium. They further generalize the Lesovik–Loosen power-exchange framework to include strong back-action, showing that the spectral density of power exchanged with the detector reflects Joule dissipation through Re$Y(f)$ as long as the conductor acts as a current source and the environment remains non-singular. The results establish a robust route to accessing non-symmetrized noise in nonlinear quantum conductors and clarify the role of the detection circuit in quantum fluctuation–dissipation relations, with implications for open quantum systems and metrology of quantum transport.

Abstract

We present an experimental test of Kubo formula performed on a nonlinear quantum conductor, a Superconductor-Insulator-Superconductor tunnel junction, driven far from equilibrium by a DC voltage bias. We implement the proposal of Lesovik and Loosen [1] and demonstrate experimentally that it is possible to extract both the emission and absorption noise of the conductor by measuring the power it exchanges with a linear detection circuit whose occupation is tuned close to vacuum levels. We then compare their difference to the real part of the admittance which is independently measured by coherent reflectometry, finding that Kubo formula holds within experimental accuracy. Last, we show theoretically that the spectral density of power exchanged between a quantum conductor and its linear detection circuit follows a Lesovik and Loosen like formula, even in the presence of strong detection back-action. This result applies as long as the conductor acts as a current source for the detection circuit and the detection circuit is not singular.

Figures (13)

• Figure 1: (a) A quantum conductor with current operator $\hat{I}$ is galvanically coupled to a linear circuit with impedance $Z_{det}$. The minimal QED coupling reads $\hat{H}_{QED}=\hat{I}\hat{\Phi}$, where $\hat{\Phi}$ is the electromagnetic flux $\hat{\Phi}(t)=\int_{-\infty}^t dt'\hat{V}(t')$ defined at the coupling node and $\hat{V}$ the corresponding voltage. (b) Scheme of the circuit used to test Kubo formula. The quantum conductor to be measured is a SIS junction (in green) connected to the circuit via a bias tee. Its inductive port is used to DC-voltage bias and measure it. The capacitive port is used to couple it to a RF detection circuit via a $660\, \mathrm{MHz}$ bandwidth cavity filter centered at $f_0=\unit{6.8}{\giga \hertz}$ (in blue) whose impedance is vanishing out of the coupled band. The photon occupation of this narrow band detection impedance can be externally tuned by the RF shot--noise emitted by a $50\, \mathrm{\Omega}$ matched DC--biased NIN junction (in orange) coupled non-reciprocally to the cavity via a $18\,\mathrm{dB}$ isolation circulator. The noise power contained in the cavity filter is routed with a pair of circulators to a cryogenic amplifier and detected at room temperature with a square law detector (red diode symbol). A $-20\,\mathrm{dB}$ directional coupler inserted after the cavity filter is used to shine a heavily attenuated RF tone generated by a VNA at the input of the SIS junction. The reflected signal is routed to the VNA detection port after amplification in order to measure the SIS admittance. (c)$dI/dV(V_{SIS})$ curve of the SIS junction obtained from the dV/dI measured in a three point configuration at its $50\,\mathrm{\Omega}$ shunted input. The tunneling resistance obtained is $R_{T}=\unit{6.7}{\kilo\ohm}$ and the superconducting gap is $2\Delta=\unit{400}{\micro\electronvolt}$.
• Figure 2: Power exchanged between the SIS junction and its linear detection circuit within the coupled bandwidth $\Delta f=\unit{660}{\mega\hertz}$ around frequency $f_0=\unit{6.8}{\giga \hertz}$. It is defined by Eq. \ref{['eq:powerExch']} and is normalized by the amplification chain noise $k_{\mathrm{B}}T_{amp}\Delta f=P(V_{SIS}= V_{NIN}=0)$ (roughly $\unit{2.9}{\kelvin}$ noise temperature). At zero NIN bias, the exchanged power is always positive: the SIS junction can only emit into the detection circuit. The onset for this spontaneous emission is found at $eV_{SIS}=2\Delta +hf_0$, where $hf_0=\unit{28}{\micro \electronvolt}$ is the energy quantum of the cavity filter. At finite NIN voltage bias $V_{NIN}$, the linear circuit is driven out of equilibrium by the incoherent radiation emitted by the NIN junction providing it with finite photon occupation. In this case the exchanged power can also be negative, meaning the SIS junction is absorbing some power from the detection circuit. The onset for this power absorption is $eV_{SIS}=2\Delta -hf_0$, while the exchanged power becomes positive again after the emission threshold.
• Figure 3: Emission and absorption noise power at frequency $f_0=\unit{6.8}{\giga \hertz}$ of the SIS junction extracted from the power exchanges shown in Figure \ref{['fig:PowerExch']} exploiting Eq. (\ref{['eq:Lesovik']}) and the protocol described in the article: The emission power is directly obtained from the zero occupation ($V_{NIN}=0$) measurement according to Eqs. (\ref{['eq:emP']}), while the absorption noise is obtained using Eq. (\ref{['eq:absP']}) for all the power exchanges measured at finite occupation $n(V_{NIN})$ ($V_{NIN}\neq0$). All absorption noise curves measured at different NIN biases collapse on the same shape. The black dashed curve is obtained by applying $\pm2hf_0=\unit{56}{\micro\electronvolt}$ horizontal offsets to the emission power curve at negative/positive biases, which according to Rogovin and Scalapino ROGOVIN1974 should reproduce the absorption noise in the $k_\mathrm{B}T\ll eV_{SIS}$ limit.
• Figure 4: Admittance of the SIS junction as a function of DC bias measured in the coupled band of the cavity filter. Blue line: real part of the admittance of the SIS tunnel junction $\operatorname{Re}Y_{Kubo}$ extracted from the emission and absorption noise spectral densities shown in Figure \ref{['fig:EmAbs']} using Kubo formula Eq. (\ref{['eq:reY']}). It is compared to the real (orange line) and imaginary (green line) parts of the admittance independently measured by coherent reflectometry with a Vector Network Analyzer $Y_{VNA}$ using Eq. (\ref{['eq:Gamma']}). Dashed lines are the corresponding predictions from BCS theory in the zero temperature and zero depairing limit detailed in the Appendix C.2.
• Figure B.1: Noise temperature measured in the coupled bandwidth with the square--law detector as a function of the temperature set to the mixing chamber plate. The noise temperature units are extracted from fitting its linear evolution to Johnson-Nyquist formula and thus assume a perfect matching between the line and the amplifier. The data is then compared to the Callen-Welton noise prediction.