Laser-like instabilities in quantum nano-electromechanical systems

We discuss negative damping regimes in quantum nano-electromechanical systems formed by coupling a mechanical oscillator to a single-electron transistor normal or superconducting. Using an analogy to a laser with a tunable atom-field coupling, we demonstrate how these effects scale with system parameters. We also discuss the fluctuation physics of both the oscillator and the single-electron transistor in this regime, and the degree to which the oscillator's motion is coherent. negative damping of the oscillator. 9-11 The resulting instabil- ity is ultimately cut off by a nonlinearity in the dynamics, and is characterized by an effective strong coupling between the mechanical and electronic dynamics. In this Rapid Communication, we investigate the proper- ties of both normal-state and superconducting SET NEMS systems in the negative damping regime. Using the analogy between these systems and a laser with a tunable atom-field coupling, we derive simple scaling relations for the station- ary state and fluctuations in the negative damping regime. We also discuss how noise measurements could be used to sensitively probe the physics in this regime. In particular, the critical slowing down associated with the transition to the "lasing" state can be seen clearly in the low-frequency cur- rent noise of the transistor. Note that the analogy between laser physics and a superconducting SET NEMS was also studied recently in Ref. 12 using a numerical approach. An alternate proposed mechanical analog of a laser was dis- cussed in Ref. 13. Model. We consider a standard SET-based NEMS where the oscillator acts as a voltage gate with an x-dependent cou- pling capacitance to the SET island; details may be found in Ref. 5. In the absence of any intrinsic damping of the oscil- lator, the total Hamiltonian is given by H= Hosc+ H0+ HC. Here, Hosc describes an oscillator with angular frequency and mass m. H0 describes the noninteracting part of the SET Hamiltonian, and includes the kinetic energies of electrons in the leads and island, as well as tunneling terms taking elec- trons to and from the leads; in the case of a superconducting SET SSET, the island and leads are described by BCS Hamiltonians. Finally, HC describes the Coulomb charging energy of the island in the presence of the oscillator: HC = ECn ˆ 2 2 nN0 + A 2EC x ˆ, 1