Figure 5.7: Total currentI (solid blue line) and elastic current (dashed orange line) as function of ε0 for Γn = 0.01ω, Γs = 0.05ω, T = 10ω, eV = 5ω, λ = 0.02ω for equilibrated (a) and unequilibrated vibrationγ0/ω= 10−4(b). The ratio between the areas underlying the two peaks is related to the phonon occupation.
then the experimental ratio between the areas underlying the two peaks, associated to the absorption and emission in the inelastic Andreev reflections corresponds to
∆Il/∆Ir = (2¯nl+ 1)/(2¯nr + 1). Eventually, other comparison with the theoretical predictions can be done whether one parameter of the systems as for instance Γs, can be varied.
5.3 Effects of quasiparticles
In the previous section, we focused on the subgap regime and discussed the damping and phonon occupation. In this section, we briefly consider the effects of a finite gap on the phonon occupation. The goal of this section is to show that ground-state cooling is achievable at finite ∆. To describe the effect of the quasiparticle we introduce the Hamiltonian of the normal-superconducting contact interacting with the resonator in Sec.5.3.1. Compared to the previous section, we start with the BCS Hamiltonian of the right lead and will not use the hybridized Hamiltonian for the quantum dot and the superconductor of Eq. (5.4).
5.3.1 Model for quantum dot and leads
The quantum dot between a normal and a superconducting lead interacting with the resonator is described by
Hˆ = ˆHn+ ˆHt+ ˆHs+ ˆHd+ ˆHm. (5.23) The Hamiltonians ˆHn and ˆHt are given by Eq. (5.2) and (5.3) respectively. The superconducting lead is described by the BCS Hamiltonian [see Eq. (2.25)] ˆHs = P
kσ[εkˆa†kσˆakσ+ ∆ˆa†−k↓ˆa†k↑+ ∆†ˆak↑aˆ−k↓] with the annihilation operators with spin σ and momentum kof the superconductor given by ˆakσ. The quantum dot is modeled by the Hamiltonian ˆHd = ε0ˆnd with the bare energy level of the dot ε0. The last
term in Eq. (5.23) is given by Eq. (5.5) and corresponds to the flexural mode and the coupling between the quantum dot and the flexural modes of the resonator.
5.3.2 Results for a single mode
The damping and phonon occupation are calculated in the same way as in Sec.5.2.2 with Eqs. (5.15) and (5.16), respectively.
Before we give the expression for the damping, we briefly recall the results obtained in Sec. 2.3.1for a normal-superconducting contact. An electron incoming below the superconducting gap can be either Andreev reflected or normal reflected. Above the gap, AR and NR is still possible but also quasiparticles contribute to transport.
In the case of a normal-superconducting junction we can divide the quasiparticle contributions into direct tunneling (DT) and branch crossing (BC). Sketches of the corresponding processes of AR, NR, DT and BC are shown in Fig.2.2.
Similar to the normal-superconducting contact, we can divide the total mechanical damping due to the electron-vibration interaction into a contribution from below and above the superconducting gap. Below the gap, inelastic Andreev reflection (γAR) and inelastic normal reflection (γNR) are possible. Above the gap inelastic direct quasiparticle tunneling (γDT), inelastic branch crossing (γBC) and inelastic tunneling via an intermediate state (γIS) start to contribute to transport [210]. In all these processes the electrons or holes interact with the resonator and absorb or emit a vibrational energy quantum. The total mechanical damping can then be written as
γtot=γAR+γNR+γDT+γBC+γIS. (5.24) We remark that for the discussion in this section, it is sufficient to separate between transport processes below and above the superconducting gap.
In the following, we give the exact result of the damping to infinite order in the tunneling t between the quantum dot and the leads. All damping coefficients in Eq. (5.24) can be expressed in terms of the transmission amplitudes
Tνsν¯(ε) =Gν(ε)F∗(ε+sω)−F(ε)G∗¯ν(ε+sω) , (5.25) Tννs (ε) =Gν(ε)G∗ν(ε+sω)−F(ε)F∗(ε+sω) . (5.26) The elements Gν(ε) and F(ε) correspond to the exact Green’s function of a dot coupled to a normal and a superconducting lead. A calculation of the retarded Green’s function in Nambu space gives (see appendixC.1)
GˆR(ε) = G+(ε) F(ε) F(ε) G−(ε)
!
= 1 D
−(ε+ε0+iΓn+εΓs/Ω(ε)) ∆Γs/Ω(ε)
∆Γs/Ω(ε) −(ε−ε0+iΓn+εΓs/Ω(ε))
!
(5.27) withD= ∆2Γ2s/Ω2(ε)−(ε+ε0+iΓn+εΓs/Ω(ε)) (ε−ε0+iΓn+εΓs/Ω(ε)), Ω(ε) = p∆2−(ε+iη)2 and an infinitesimal small real part η.
5.3 Effects of quasiparticles The rates γAR,γDT,γBC and γIS are then given by
γAR=X
ν,s
sγνsν¯, (5.28)
γDT=X
sν
s(γDT,nssν +γDT,snsν ) , (5.29) γBC=X
sν
s(γBC,nssν +γBC,snsν ) , (5.30) γIS=X
sν
s(γIS,nssν +γIS,snsν ) , (5.31)
with ν = (e,h) = ±, ν0 = (e,h) = ± and the notation ¯ν = −ν. The indexes n andslabel a particle in the normal and superconducting lead, respectively. The first (second) index thereby refers to the incoming (outgoing) particle. Then, the first term in the Eqs. (5.29)-(5.31) describes an incoming particle in the normal lead and an outgoing particle in the superconducting lead. Similarly, the second term in the Eqs. (5.29)-(5.31) corresponds to an incoming particle in the superconducting lead and an outgoing particle in the normal metal.
The reflection inside the normal lead (NR) and the superconducting lead (SR) are
γNR=X
ν,s
sγννs , (5.32)
γSR=X
ν,s
s(γsνDT,SR+γBC,SRsν +γIS,SRsν ) . (5.33)
Reflection for an incoming particle inside the superconductor can be separated into DT, BC and IS.
The damping ratesγννs 0 in Eqs. (5.28) and (5.32) can be written as
γννs 0(ω) =λ2Γ2n 2
Z dε
2πfν(ε)[1−fν0(ε+sω)]|Tννs 0(ε)|2 , (5.34)
in which we have defined the Fermi functionfν(ε) = 1/[1 + exp((ε−ν eV))/T].
Introducing the density of states of the superconductor with
ρ(ε) =θ(ε2−∆2)ε/pε2−∆2, (5.35)
we can write the rates as
The processes of DT and BC can be understood in the following picture. We consider an incoming electron from the normal metal above the gap. The electron is transmitted to the quantum dot and then to the superconductor. In this way the charge of an electron is transfered form the normal to the superconductor. To the lowest order in the coupling (Γn= Γs = Γ), the process of DT is proportional to Γ2. Instead of being transmitted, the electron can also be reflected at the superconductor and again at the quantum dot. In the process of a BC, the electron is then transmitted as a hole above the gap. Therefore, to the lowest order in the coupling, BC is proportional to Γ4.
Similar to the inelastic Andreev reflection discussed in Sec. 5.2, the electron can interact with the resonator before or after an Andreev reflection which is described by the two amplitudes in the transmission.
A calculation of the phonon occupation as in Sec.4.3.4, gives the result
n= 1 From Eq. (5.37), one expects that the processes above the gap can drive the res-onator to a nonequilibrium state. A strong modification of the phonon occupation