Figure 4.17: Current and differential conductance for equilibrated [(a) and (c)] and unequilibrated vibration [(b) and (d)] at polarization pr = −pl = 0.5, T = 10ω, ε0 = 0, Γ = 0.2ω,λ= 0.2ω, and γ0 = 10−5ω. In (b) the current ateV <0 decreases since the oscillator approaches the mechanical instability and the phonon occupation strongly increases. At eV > 0, the current is suppressed compared to the current in (a) since ¯n < nB(ω). In (c) and (d), we show the differential conductance dI/dV corresponding to (a) and (b), respectively.
4.8 Summary
For a suspended CNTQD in a spin-valve geometry, we studied the spin-dependent current through two spin levels and the steady-state phonon occupation for a vibra-tional flexural mode in presence of a spin-vibration interaction. Such a spin-vibration interaction can have the origin in the spin-orbit coupling or a magnetic gradient.
We have shown that even weakly spin-polarized currents allow the control of the phonon occupation ¯n in a way that a flexural mode can be cooled [¯n nB(ω)]
or heated [¯n nB(ω)] or even driven towards a mechanical instability regime in which the mechanical damping becomes negative. Such a control can be achieved by manipulating several parameters of the system. In particular, it can be obtained using electrical fields, viz. varying the bias-voltage polarity or the gate voltage, or using magnetic fields, viz. by changing the orientation of the magnetic polarization of the ferromagnetic contacts or tuning the energy separation of the dot’s spin levels.
The current shows characteristic features of the nonequilibrium phonon occupa-tion and directly can be exploited to demonstrate the presence of the spin-vibraoccupa-tion interaction and the non-thermal phonon occupation of the oscillator.
5 Ground-state cooling by
noise-interference in Andreev reflection
We study the ground-state cooling of a mechanical oscillator linearly coupled to the charge of a quantum dot inserted between a normal metal and a superconducting con-tact. Such a system can be realized e.g. by a suspended carbon nanotube quantum dot with capacitive coupling to a gate. Applying a bias-voltage, inelastic Andreev reflections and inelastic quasiparticle tunneling will drive the resonator to a nonequi-librium state.
We analyze the junction in the regime of a small coupling between the quantum dot and the resonator. As main result of this chapter we obtain that the vibration-assisted reflections can occur through two distinct interference paths. The interference deter-mines the ratio between the rates for absorption and emission processes of vibrational energy quanta. We show that ground-state cooling of the mechanical oscillator can be achieved in a wide parameter range and due to the interference even for many oscillator’s modes simultaneously.
The chapter is organized as follows. In Sec. 5.1, we give a brief introduction to the topic. The subgap transport regime in which inelastic vibration-assisted Andreev reflections drive the phonon occupation to a nonequilibrium state is discussed in Sec. 5.2. In Sec. 5.3, we briefly consider the effect of the quasiparticles above the gap on the damping and the phonon occupation. Finally, we draw our conclusions in Sec.5.4.
5.1 Introduction
Nanoelectromechanical and optomechanical systems promise to manipulate mechani-cal motion in the quantum regime using, respectively, electrons [134,184] or photons [30], for the realization of fundamental tests of quantum mechanics in a hitherto un-accessible parameter regime. This goal requires that the mechanical oscillator to be in or near its quantum ground state, viz. T ω, with the mechanical frequencyω. Ground-state cooling, i.e. the average vibrational quantan1, has been achieved in some nanomechanical devices, for instance, by direct cooling of an oscillator of GHz frequency using standard dilution refrigeration [28]. In another example, ground-state cooling was obtained using the so-called sideband method [185,186] in which the res-onator is coupled to a superconducting microwave electromagnetic cavity acting as an effective low-temperature bath ofT .1mK [27].
Reducing the size as well the frequency of nanomechanical oscillators have the dis-tinct advantages to increase the sensitivity as nanoscale detectors, the quality factor Qand the amplitude of the zero-point fluctuations. For this reason, suspended carbon
Figure 5.1: Sketch of a suspended carbon nanotube quantum dot between a normal and superconducting lead of gap ∆ with a capacitive coupling between the dot’s charge and the flexural mechanical modes. Applying a voltage V, inelastic vibration-assisted Andreev reflections (ARs) can cool one or several mechanical modes towards the quantum ground state.
nanotube quantum dots are considered good candidates: (i) their mechanical modes can reach elevated high quality factors Q ∼ 106 without detriment of the electron transport properties [31,125] and (ii) recent experiments also showed unprecedented control of the tunability of both electron transport and electromechanical coupling [135]. For such systems, approaching the quantum regime is still an open challenge due to the low-frequency spectrum of the flexural modes (f ≤100MHz) which corre-sponds to demanding temperatures for electronic circuits.
Several and interesting theoretical proposals have been analyzed for achieving cool-ing or ground-state coolcool-ing by uscool-ing electron transport [109,168,172,174,187–197], exploiting the interaction between the oscillator with a given quantum conductor.
Most of them are closely related to the mechanism of the side-band cooling employed in opto-mechanical devices and, in a scattering picture, are based on an enhanced phonon absorption between two levels of energy difference ∆E. This enhancement is caused by a resonance in the density of states in the electronic system. As conse-quence, cooling is obtained in a limited range of the system parameters which satisfy a particular resonant condition for the oscillator’s frequency ω = ∆E. We remark that cooling by pure electron transport has been experimentally achieved so far using a superconducting single electron transistor [198–200].