We present a Schwinger-Keldysh scheme theory for full counting statistics applicable to the solid-state entangler, which consists of two coupled superconducting/normal-conducting/normal-conducting capacitively coupled single-electron transistors (double S/N/N C-SET). We focus on the case when the superconducting gap energy is larger than the charging energy since flexible control of entanglement information can be expected by the charging effect for various bias conditions, and we study the double S/N/N C-SET with and without dissipative environment. We derive the cumulant generation function for the double S/N/N C-SET under current continuity conditions, from which arbitrary order of current noise cumulants are obtained. We explicitly show that the synchronized Coulomb oscillation, a celebrated experimental finding by Hofstetter et al. [Nature (London) 461, 960 (2009)], originate from the crossed Andreev current conveying quantum entanglement. We also investigate current in the double S/N/N C-SET, its current components, and the resulting cross-correlation of current noise $S_{\mathrm{LR}}(\omega =0)$ in the superconducting subgap region. It is shown that the bunching-antibunching nature strongly depends on the relative location of the Coulomb gap regions for the relevant current components since the bunching-antibunching nature is determined by the competition between the contributions of relevant current components. Depending on the bias conditions for each of the two C-SETs, $S_{\mathrm{LR}}(\omega =0)$ exhibits a sign crossover from positive (bunching) to negative (antibunching), which is followed by restoration to bunching correlations. The effect of the dissipative environment tends to reduce $S_{\mathrm{LR}}(\omega =0)$ because of a reduction in relevant currents. Although crossover and restoration become less conspicuous, the way the bunching-antibunching nature appears is essentially the same as in the case of the inductive environment.

• Received 25 September 2019
• Revised 1 April 2020
• Accepted 29 May 2020

DOI:https://doi.org/10.1103/PhysRevB.101.245312