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Using a quantum noise approach, we discuss the physics of both normal metaland superconducting single electron transistors (SET) coupled to mechanicalresonators. Particular attention is paid to the regime where transport occursvia incoherent Cooper-pair tunneling (either via the Josephson quasiparticle(JQP) or double Josephson quasiparticle (DJQP) process). We show that,surprisingly, the back-action of tunneling Cooper pairs (or superconductingquasiparticles) can be used to significantly cool the oscillator. We alsodiscuss the physical origin of negative damping effects in this system, and howthey can lead to a regime of strong electro-mechanical feedback, where despitea weak SET - oscillator coupling, the motion of the oscillator strongly effectsthe tunneling of the Cooper pairs. We show that in this regime, the oscillatoris characterized by an energy-dependent effective temperature. Finally, wediscuss the strong analogy between back-action effects of incoherentCooper-pair tunneling and ponderomotive effects in an optical cavity with amoveable mirror; in our case, tunneling Cooper pairs play the role of thecavity photons.Comment: 27 pages, 7 figures; submitted to the New Journal of Physics focus  issue on Nano-electromechanical Systems; error in references correcte

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