Approach

leads with momentum k and spin σ are created and annihilated by the operators ˆ

c^†_kσ and ˆc_kσ. The Hamiltonian describing the tunneling between the dot and the superconducting leads is written as ˆHT α = ^P_kασdˆ^†_σVdσ,kαˆckασ + ˆc^†_kασV_d^†_σ,kαdˆσ. The hopping V_dσ,kα describes the coupling between the dot and the leads and is assumed to be independent of energy. The operators acting on the dot correspond to ˆd^†_σ and ˆd_σ. The Hamiltonian of the dot ˆH_D and the Zeeman energy are written as HˆD =^P_σ(E0+σµBBz) ˆd^†_σdˆσ and HS(t) = −µ_M(t)B, with the magnetic field B = (0, 0,B_z) and the magnetic moment of the molecular magnet µ_M(t). The magnetic moment is related to the spin of the molecular magnet via µ_M(t) = −γS(t) with the gyromagnetic ratioγ =gMe/(2m). The Landé factor of the molecular magnet is denoted byg_M and has in principle to be determined by comparison with experiments.

In the following, we assume that the Landé factor of the molecular magnet equals that of free electrons andgM = 2 [85]. Due to the magnetic field, the magnetization of the molecular magnet precesses with the Larmor frequency. We assume that the motion of the spin is undamped which can be achieved by dc and rf fields [86,87]. The equation of motion is then given by∂S/∂t=−γS×B. The solution of this equation is S(t) = S(cos(ω_Lt)sin(ϑ)e_x+ sin(ω_Lt)sin(ϑ)e_y+ cos(ϑ)e_z), with the magnitude of the spin |S| = S and the Larmor frequency ω_L = γB_z. The exchange interaction between the molecular magnet and the quantum dot is described by

Hˆ_SD(t) = 1 2

X

σσ⁰

V_sdˆ^†_σ(S(t)σ)σσ⁰dˆ_σ⁰, (3.2) with the coupling V_s between the spin and the quantum dot and the Pauli matrices σ= (σx,σy,σz). The exchange interaction occurs due to the Pauli principle and the electrostatic interaction between the electrons on the quantum dot and the molecular magnet. Eq. (3.2) can be transformed into

HˆSD(t) =^X

σ

σvscos(ϑ) ˆd^†_σdˆσ +vssin(ϑ)e^−iωLtdˆ^†_↑dˆ↓+vssin(ϑ)e^iωLtdˆ^†_↓dˆ↑, (3.3) with v_s= SV_s/2. The first term induces a spin-dependent shift of the energy levels of the dot while the second and third terms account for the spin flip of the electrons occupying the dot. Since the magnetic field enters in the Hamiltonian of the dot and the exchange interaction, we can rewrite the Hamiltonian of the dot in terms of the Larmor frequency of the molecular magnet as ˆH_D =^P_σE₀+σ^~ω₂^Ldˆ^†_σdˆ_σ.

3.4 Approach

The transport properties of the system are described by a nonequilibrium Green’s function approach [88–90]. In order to simplify the evaluation of the Green’s functions, we perform a unitary transformation to the rotating frame of the molecular magnet’s spin, since in this frame the Hamiltonian is time independent. The state vector transforms according to|˜Ψi=U^†|˜Ψi and the Hamiltonian in the rotating frame can

be written as ¯H=U^†HU +i~(∂_tU^†)U with the unitary transformation operator In the rotating frame, the Hamilton operator (3.1) is given by

H¯ =^X

α

( ¯Hα+ ¯H_{T α}) + ¯H_D+ ¯H_S+ ¯H_SD. (3.5) The transformation results in a spin-dependent shift of the quasiparticles’ energies in the leads and the quantum dot. The Hamiltonian of the leads reduces to

H¯_α =^X Due to the spin-dependent energy shift, the Zeeman energy in the Hamiltonian of the dot is canceled by the transformation such that the Hamiltonian of the dot is given by ¯HD =^PσE0dˆ^†_σdˆσ. The spin is fixed in the rotating frame and the exchange Hamiltonian is written as

H¯_SD=^X

σ

σvscos(ϑ) ˆd^†_σdˆσ+vssin(ϑ) ˆd^†_↑dˆ↓+vssin(ϑ) ˆd^†_↓dˆ↑. (3.7) The remaining terms of the Hamiltonian (3.1) are not affected by the transformation (3.4). Figure3.1(b) depicts the system in the frame of the rotating spin. The splitting of the spin-up and spin-down energy levels of the quantum dot appears because of the exchange interaction. According to the first term in Eq. (3.7) the energy shift of the spin-up (spin-down) quasiparticles in the rotating frame is given by +(−)vscos(ϑ).

In order to evaluate the Green’s functions of the system, we divide the structure into three subsystems, which are the left lead (L), the right lead (R), and the quantum dot (D) [91]. The spin dependence and the superconducting state are taken into account by writing the Green’s functions in Nambu-spin space. Since the system is out of equilibrium, we additionally write the Green’s functions in Keldysh space. In Keldysh-Nambu-spin space, the Green’s functions have the structure

Gˇ_ββ⁰(t,t⁰) = Gˆ^R_ββ0 Gˆ^K_ββ0

0 Gˆ^A_ββ0

!

(t,t⁰). (3.8)

The symbols ˇ and ˆ denote a matrix in Keldysh-Nambu-spin and Nambu-spin space, respectively. The labels R, A, and K indicate the retarded, advanced, and Keldysh elements of the Green’s function ˇG_ββ⁰(t,t⁰). The indices β and β⁰ refer to op-erators in one of the three subsystems; e.g., the retarded Green’s function ˆG^R_LD is given by ˆG^R_LD(t,t⁰) = −iθ(t − t⁰)h{ψ_L(t),ψ_D^†(t⁰)}i, where we have introduced the operators of the quantum dot and the leads in Nambu-spin space as ψD = (d↑ d↓ d^†_↑ d^†_↓)^T and ψα = (cα↑ cα↓ c^†_−α↑ c^†_−α↓)^T with the index α referring to the momentum k_α in the left or right lead. The elements of the matrix in (3.8) are re-lated by G^< = (1/2)G^K−G^R+G^A with the lesser Green’s function defined by G^<_DL(t,t⁰) =ihψ^†_L(t⁰)ψ_D(t)i.

3.4 Approach Taking into account allkstates of the leads, we define the matricesG_ββ⁰ andV_ββ⁰ as (G_ββ⁰)ij = ˆG_βi,β⁰

in Nambu-spin space. The indices i and j indicate all k states in the leads. The matrices (3.8) of all subsystems are then combined in an enlarged Hilbert space into one matrix defined by ˜G. The full and the unperturbed Green’s function are written as

The coupling of the quantum dot to the leads and to the molecular magnet is given by con-taining the coupling between the quantum dot and the leads are diagonal in Keldysh space and are in Nambu space given by ˆV_α,d.

Fourier-transforming the Dyson equation to energy space, we obtain

G˜ = ˜G₀+ ˜G₀V˜G˜ , (3.9) and can calculate the retarded and advanced Green’s functions of the dot as

Gˆ^R/A_DD =1−Gˆ^R/A_0D ˆΣ^R/A+ ˆV_DD⁻¹Gˆ^R/A_0D , (3.10) where the self-energy is defined as ˆΣ = ˆVDLˆgLLVˆLD + ˆVDRgˆRRVˆRD = ˆΣL + ˆΣR. These self-energies are obtained by summation over all quasiparticle states in the left and right leads, respectively. This summation can be replaced by the integral P

kα → _(2π)¹ 3

R d³kα→N0R

dξαR dΩ_kα

4π with the normal density of states at the Fermi energy, N0. The integration over the quasiparticle energies in the superconductor leads to the so-called quasiclassical Green’s function [see Sec.2.2.3] which read [90,92]

ˆ

together with the normalization condition gˆα^R/A

2

=−π²ˆ1. In the wide-band limit, the retarded and advanced self-energies are approximated by ˆΣ^R/Aα = Nα|Vˆ|²gˆα^R/A

with the energy E^R/A defined as E^R/A=E±iη and an infinitesimal small real part η →0. In the rotating frame, the self-energies are given by (setting ˜ω=~ω_L/2) spin-up and spin-down unperturbed Green’s function of the dot in Nambu-spin space is given by ( ˆG^R_0D)⁻¹_11/22 =E^R−E0 and ( ˆG^R_0D)⁻¹_33/44 =−(E^R+E0). The unperturbed Green’s function and the self-energy enable us to evaluate the full Green’s function of the dot (3.10).

The average charge current operator in the Nambu-spin space from the left lead to the quantum dot is obtained by using the Heisenberg equation of motion

Jˆ_L=−ie

~

DhHˆ¯, ˆ¯N_L^iE=−ie

~

DhHˆ¯_T, ˆ¯N_L^iE , (3.12) with the Hamilton operator ˆ¯H in the rotating frame, the number operator ˆ¯N_L =

1 The Josephson current (3.12) can then be written in terms of the lesser Green’s function as [93]

J_L= e 2~

Z dE

2π Tr ˆσ₀ G^<_DLV_LD−V_DLG^<_LD . (3.13) By using the Dyson equation (3.9) in the enlarged Hilbert space, we calculate the elements G^<_DL and G^<_LD with the help of the relationG^<= (1/2)G^K−G^R+G^A. Fermi energy in Nambu-spin space is shifted and ˆF is given by

Fˆ =