Gate-tunable zero-frequency current cross correlations of the quartet state in a voltage-biased three-terminal Josephson junction

A three-terminal Josephson junction biased at opposite voltages can sustain a phase-sensitive dc current carrying three-body static phase coherence, known as the ``quartet current.'' We calculate the zero-frequency current noise cross correlations and answer the question of whether this current is noisy (like a normal current in response to a voltage drop) or noiseless (like an equilibrium supercurrent in response to a phase drop). A quantum dot with a level at energy ${\ensuremath{\epsilon}}<sub>0</sub>$ is connected to three superconductors ${S}<sub>a</sub>,\phantom{\rule{0.16em}{0ex}}{S}<sub>b</sub>$, and ${S}<sub>c</sub>$ with gap $\mathrm{\ensuremath{\Delta}}$, biased at ${V}<sub>a</sub>=V,\phantom{\rule{0.16em}{0ex}}{V}<sub>b</sub>=\ensuremath{-}V$, and ${V}<sub>c</sub>=0$, and with intermediate contact transparencies. At zero temperature, nonlocal quartets (in the sense of four-fermion correlations) are noiseless at subgap voltage in the nonresonant dot regime ${\ensuremath{\epsilon}}<sub>0</sub>/\mathrm{\ensuremath{\Delta}}\ensuremath{\gg}1$, which is demonstrated with a semianalytical perturbative expansion of the cross correlations. Noise reveals the absence of granularity of the superflow splitting from ${S}<sub>c</sub>$ towards $({S}<sub>a</sub>,{S}<sub>b</sub>)$ in the nonresonant dot regime, in spite of finite voltage. In the resonant dot regime ${\ensuremath{\epsilon}}<sub>0</sub>/\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}1$, cross correlations measured in the $({V}<sub>a</sub>,{V}<sub>b</sub>)$ plane should reveal an ``anomaly'' in the vicinity of the quartet line ${V}_a+{V}_b=0$, related to an additional contribution to the noise, manifesting the phase sensitivity of cross correlations under the appearance of a three-body phase variable. Phase-dependent effective Fano factors ${F}_{\ensuremath{\varphi}}$ are introduced, defined as the ratio between the amplitudes of phase modulations of the noise and the currents. At low bias, the Fano factors ${F}_{\ensuremath{\varphi}}$ are of order unity in the resonant dot regime ${\ensuremath{\epsilon}}₀/\mathrm{\ensuremath{\Delta}}\ensuremath{\lesssim}1$, and they are vanishingly small in the nonresonant dot regime ${\ensuremath{\epsilon}}₀/\mathrm{\ensuremath{\Delta}}\ensuremath{\gg}1$.