(Color online) Schematic diagram of the system considered. A suspended carbon nanotube of length is coupled to two superconducting leads biased at a voltage . Coupling between the Andreev states for the electronic degrees of freedom and the mechanical vibrations of the nanowire is achieved through the external transverse magnetic field which enables transition between the electronic branches through the emission/absorption of a vibrational quantum (see text).
Time evolution of the Andreev states (full lines) over one period . The state of the total system, , depends on the population of the two electronic branches, |+⟩, corresponding to the upper and lower branch, respectively, as well as on the quantum state of the oscillating nanowire, . Due to the large separation in energies between and , transitions between the electronic branches is only possible in the small resonance-window (modeled through the scattering matrix ) where the electronic state of the system can change through the emission/absorption of one vibrational quantum. In the above, the probability for the state initially in to scatter into the state after passing through the resonance depends on the state of the oscillator through the coefficient (see text). After one period the partially filled Andreev levels join the continuum, a process which is here represented by dashed arrows, and the electronic states are reset (filled and empty circles).
(Color online). Evolution of the distribution of the mechanical modes, , as a function of the quantum state for different number of periods . Initially is thermally distributed with . Here, , , , , and . Inset shows as a function of for the same parameters.
DC current as a function of inverse quality factor when the mechanical subsystem has been driven into the stationary regime. As can be seen, in the limit of very high quality factor, corresponding to complete ground state cooling of the oscillating nanowire, the dc current goes zero as no inter-Andreev level scattering is possible in this regime . In the opposite limit, , the dc current approaches a constant value which depends on the external temperature, . Here, the system parameters are the same as in Fig. 3.
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