Gate-tunable zero-frequency current cross-correlations of the quartet mode 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 $\epsilon_0$ is connected to three superconductors $S_a$, $S_b$ and $S_c$ with gap $\Delta$, biased at $V_a=V$, $V_b=-V$ and $V_c=0$, 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 $\epsilon_0/\Delta\gg 1$, which is demonstrated with a semi-analytical perturbative expansion of the cross-correlations. Noise reveals the absence of granularity of the superflow splitting from $S_c$ towards $(S_a,S_b)$ in the nonresonant dot regime, in spite of finite voltage. In the resonant dot regime $\epsilon_0/\Delta< 1$, cross-correlations measured in the $(V_a,V_b)$ 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 $F_\varphi$ are introduced, defined as the ratio between the amplitudes of phase modulations of the noise and the currents. At low bias, the Fano factors $F_\varphi$ are of order unity in the resonant dot regime $\epsilon_0/\Delta< 1$, and they are vanishingly small in the nonresonant dot regime $\epsilon_0/\Delta\gg 1$.