Quantum nanoelectromechanics with electrons, quasi-particles and Cooper pairs: effective bath descriptions and strong feedback effects

Using a quantum-noise approach, we discuss the physics of both normal metal and superconducting single-electron transistors (SSETs) coupled to mechanical resonators. Particular attention is paid to the regime where transport occurs via incoherent Cooper-pair tunnelling (either via the Josephson quasi-particle (JQP) or double JQP (DJQP) process). We show that, surprisingly, the back-action of tunnelling Cooper pairs (or superconducting quasi-particles) can be used to significantly cool the oscillator. We also discuss the physical origin of negative-damping effects in this system and how they can lead to a regime of strong electromechanical feedback, where despite a weak SET-oscillator coupling, the motion of the oscillator strongly effects the tunnelling of the Cooper pairs. We show that in this regime, the oscillator is characterized by an energy-dependent effective temperature. Finally, we discuss the strong analogy between back-action effects of incoherent Cooper-pair tunnelling and ponderomotive effects in an optical cavity with a moveable mirror; in our case, tunnelling Cooper pairs play the role of the cavity photons.