A quantum dot (QD) is an artificially low-dimensional structure that can be filled with electrons (or holes). Two or more QDs can be coupled to form multiple-QD systems (i.e., artificial molecules). Because the degrees of freedom of the QDs are well controllable, it is possible to add or remove the electrons in the QDs, and the QD system can be coupled via tunnel barriers to electrodes, in which electrons can be exchanged. Accordingly, the artificial molecule provides an excellent model system in which the thorough investigation of quantum many-body properties in a confined geometry can be performed 1 2 3 4 5 6 . Among the various multiple-QD systems, an Aharonov-Bohm (AB) interferometer containing double QDs (DQDs) is of particular interest and importance, in which two QDs are embedded in the opposite arms of the AB ring, respectively, and they are coupled to each other via barrier tunneling. As a tunable two-level system, the parallel DQD system that can become one of the promising candidates for the quantum bit in quantum computation has received more attention 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . However, in an actual DQD system, the coherent indirect coupling between two QDs via a reservoir is very essential. Kubo et al. introduced the parameter ₁ αcharacterizing the indirect coupling strength, and Gurvitz also indicated the fundamentality of the sign of the coherent indirect coupling parameter 21 22 . Kubo et al. investigated the pseudospin Kondo effect in a lateral DQD system using the slave-boson mean-field method and found that the exotic pseudospin Kondo effect occurs when a coherent indirect coupling is presented through the common reservoirs 23 . Recently, Kubo and co-workers calculated the shot noise and Kondo effect in a DQD structure with the coherent indirect coupling. Their results demonstrate that the coherent indirect coupling can generate a novel antiferromagnetic exchange phenomenon 24 . Trocha and Barnaś studied theoretically the spin-dependent transport through a DQD coupled to ferromagnetic leads. They observed that the Fano antiresonance of the linear conductance relies on the sign of the indirect coupling in the nondiagonal coupling elements 8 . Furthermore, the transport properties of a DQD system has been considered in the orbital Kondo regime. That the Kondo temperature and Kondo resonances are susceptible to the coherent indirect coupling parameter is also revealed 25 . In addition, if a QD is formed in a semiconductor two-dimensional electron gas structure without the inversion symmetry in the growth direction, the Rashba spin-orbit interaction (RSOI) will emerge, and the RSOI can induce the spin-related phase factor in the tunneling matrix elements and the spin-flip effect. The RSOI results from a relativistic effect at the low speed limit, and it can couple the electron spin to its orbital motion, thus providing a possible way to control the spin degree of freedom by means of an external electric field. As a consequence, the coherent indirect coupling and the RSOI make the quantum transport through the QD systems rich and varied 26 27 28 29 30 .

On the other side, the subgap transport through heterostructures with nano-objects (such as QDs, molecules, nanowires, etc.) coupled to one conductor and another superconducting lead has attracted a great deal of attention over the past years due to the fundamental physics and its potential applications 31 32 33 34 35 . Andreev reflection (AR) usually occurs in the hybrid systems, in which two electrons with opposite spins enter the superconductor from the normal metal region, leading to the formation of a Cooper pair in the superconducting region 36 37 38 . In comparison with the standard mechanism of normal AR, the crossed AR (CAR) is a nonlocal dynamics process which occurs at the contact between a superconductor and two normal leads, where two subgap electrons from different metals enter into the superconductor and generate a Cooper pair there 39 40 41 42 . AR (or CAR) in nanoscopic heterostructures gives rise to a rich subgap structure in the current-voltage characteristics. Accordingly, understanding the AR and CAR has attracted theoretical and experimental attention mainly because the AR (or CAR) may create the entangled electrons in a solid-state device, and CAR can be readily probed by spin selection using ferromagnetic electrodes. This approach is almost unrealized for entangler devices, since projecting the spin will cause the destruction of entanglement 43 . Based on the CAR, the controlled Cooper pair splitting has been realized in terms of a two-quantum dot Y-junction 44 , which opens a possible route towards a test of the Einstein-Podolsky-Rosen (EPR) paradox and Bell inequalities in solid-state systems. Herrmann et al. used carbon nanotube DQD as Cooper pair beam splitters and realized the quantum optic-like experiments with spin-entangled electrons 45 . These results show that the CAR has an important application in testing a fundamental property of quantum mechanics.

To our knowledge, the AR in the DQD with a maximum coupling |α| = 1 has been studied widely. However, the quantum transport through a four-terminal AB interferometer including a DQD in the presence of the AR, the coherent indirect coupling, and RSOI is less explored. Motivated by recent theoretical and experimental advances in the DQD systems 7 8 9 10 13 15 16 19 21 22 23 24 25 44 45 , one may expect that the interplay of the coherent indirect coupling and the RSOI in the presence of the AR will add new physics to hybrid quantum systems, which may have practical applications for future spintronics. Consequently, we investigate the AR in the above-mentioned system in this paper. It is found that the RSOI and a nonzero coherent indirect coupling cause the spin-polarized AR when the polarizations of two ferromagnetic leads are parallel, but for antiparallel (AP) arrangement of the polarizations of two ferromagnetic leads, the CAR can contribute the spin-polarized AR conductance. We note that the convex shape of the Andreev-reflected tunneling magnetoresistance (ARTMR) versus the magnetic flux relies on the sign of the coherent indirect coupling parameter, and there are extreme values in the plot of the ARTMR versus the coherent indirect coupling parameter. Even the negative ARTMR also occurs. This is a spin valve effect in the AR process. It is interesting to note that the sign of the coherent indirect coupling parameter leads to the swap effect in the spin-polarized current plot, and the pure spin current can be produced when two ferromagnetic leads are fully polarized. The spin-dependent AR current can be controlled by means of the gate voltage, RSOI, magnetic flux, and so on. These results provide the ways to manipulate the spin-dependent transport by means of the system parameters.

We consider a hybrid four-terminal AB interferometer including a parallel DQD coupled to two ferromagnetic reservoirs and two superconductors as shown in Figure 1. The system is described by the following Hamiltonian:

H = H F + H S + H DQD + H T ,

where H _F is the Hamiltonian of the left and right ferromagnetic electrodes

H F = ∑ ν = L , R ∑ k σ ε ν , k σ c ν , k σ † c ν , k σ .

Here, c ν , k σ † ( c ν , k σ ) is the creation (annihilation) operator in the lead νwith energy ε _ν,kσ . H _S represents two superconducting reservoirs with chemical potential μ _s = 0 and the energy gap Δ,

H S = ∑ γ = 1 , 2 ∑ k σ ε γ , k σ c γ , k σ † c γ , k σ + ∑ k ( Δ c γ , k ↑ † c γ , − k ↓ † + h.c. ) .

H _DQD in Equation 1 denotes the DQD Hamiltonian

H DQD = ∑ i σ ε i d i σ † d i σ + t c ∑ σ ( d 1 σ † d 2 σ + h.c. ) ,

in which d i σ † ( d i σ , i = 1 , 2 ) represents the creation (annihilation) operator of the electron with energy ε _i in the dot i; t _c is the coupling strength taken as a real parameter. The last term, H _T, in Equation 1 corresponds to the tunneling Hamiltonian between the DQD and four leads,

H T = ∑ ki ν σ ( T ν , k ( i ) c ν , k σ † d i σ + h.c. ) + ∑ ki ν σ ( T γ , k ( i ) c γ , k σ † d i σ + h.c. ) ,

where the tunneling matrix elements between the DQD and two ferromagnetic leads are T L , k ( 1 ) = | T L 1 | e i ϕ / 4 , T L , k ( 2 ) = | T L 2 | e − i ϕ / 4 , T R , k ( 1 ) = | T R 1 | e − i ϕ / 4 e − i φ R 1 / 4 , and T R , k ( 2 ) = | T R 2 | e i ϕ / 4 e i φ R 2 / 4 . The phase shift due to the total magnetic flux threading into the AB ring is assumed to be ϕ = 2Π(Φ_L + Φ_R)/ϕ ₀ with the flux quantum ϕ ₀ = h/e. The phase factor φ _Ri comes from the RSOI in dot i, which is tunable in the experiments 46 47 . T γ , k ( i ) = T S 1 ( T S 2 ) as the tunneling coupling between the DQD and two superconductors is also assumed to be independent of k and σ.

Using the nonequilibrium Green’s function technique, the spin-dependent current through the left ferromagnetic reservoir can be expressed as 48 49

I L σ = ie ℏ ∫ d ε 2 Π Tr ( σ ̂ z Γ L { G < ( ε ) + F L [ G r ( ε ) − G a ( ε ) ] } ) ,

where Tr is the trace in the spin space; σ ̂ z is a 4 × 4 matrix with Pauli matrix σ _z as its diagonal components; G ^r,a,<(ε) are retarded, advanced, and lesser Green’s functions in the generalized 4 × 4 Nambu notation.

G r ( t , t ′ ) = − i θ ( t − t ′ ) 〈 { Ψ ( t ) , Ψ † ( t ′ ) } 〉 ,

G < ( t , t ′ ) = i 〈 Ψ † ( t ′ ) Ψ ( t ) 〉 ,

with the vector Ψ † = ( d 1 ↑ † , d 1 ↓ , d 2 ↑ † , d 2 ↓ ) .

After some algebraic manipulations, the spin-dependent current can be derived from Equation 6:

I L ↑ = e h ∫ d ε [ T ↑ AR ( ε ) ( f L − f ¯ L ) + T ↑ CAR ( ε ) ( f L − f ¯ R ) + T ↑ LR ( ε ) ( f L − f R ) + T ↑ QS ( ε ) ( f L − f S ) ] ,

I L ↓ = e h ∫ d ε [ T ↓ AR ( ε ) ( f L − f ¯ L ) + T ↓ CAR ( ε ) ( f R − f ¯ L ) + T ↓ LR ( ε ) ( f ¯ R − f ¯ L ) + T ↓ QS ( ε ) ( f S − f ¯ L ) ] ,

in which T σ AR and T σ CAR are the spin-dependent AR and CAR coefficients, respectively. T σ LR represents the single-particle tunneling through FL-DQD-FR or FR-DQD-FL. T σ QS corresponds to the probability of the quasiparticle tunneling among two superconductors and the left ferromagnetic lead. f L ( f ¯ L ) , f R ( f ¯ R ) , and f _S are Fermi-Dirac distribution functions. The derivation of the spin-dependent current is minutely given in the Appendix.

Since we mainly focus on the AR process at zero temperature limit and set |e V _L| = |e V _R| < Δ, T σ QS will vanish. In the case of e V _L = e V _R, the current from the quasiparticle tunneling through FL-DQD-FR or FR-DQD-FL becomes zero; as a consequence, the AR dominates the transport through the four-terminal AB interferometer.

In the following numerical calculations, we mainly elucidate the spin-dependent AR process in the four-terminal AB interferometer with the coherent indirect coupling and the RSOI. We take e = h = k _B = 1, and set Δ = 1 as the energy unit. Throughout the paper, the symmetric couplings with Γ 1 ν = Γ 2 ν = Γ s = Γ = 0 . 2 Δ and |P _L| = |P _R| are considered as a typical case.

In this paper, we have analyzed the AR of a four-terminal AB interferometer containing a coupled DQD with with the RSOI and the coherent indirect coupling via two ferromagnetic leads. The formulas of the transmission coefficients are derived based on the framework of the nonequilibrium Green’s function technique. For P configuration, the spin-polarized AR can occur, stemming from the RSOI and a nonzero coherent indirect coupling. On the contrary, for AP configuration, the spin-polarized AR always happens because of the CAR mechanism. Under the introduction of the ARTMR, we find that the sign of the ARTMR versus the magnetic flux keeps invariable for different parameter α, but the convex shape of the ARTMR depends distinctly on the sign of the parameter α. With the increase of the RSOI strength, the ARTMR versus the parameter α exhibits the more significant nonmonotonic features, and there exist the extreme values in the ARTMR plot, even the negative ARTMR also emerges. Since the energy levels of the DQD can be manipulated via the gate voltage, we can obtain the optimal spin-polarized current. A pure spin current can be generated via the CAR and two half-metal leads. Moreover, the strong RSOI and the reduction of the destructive interference (α = 1) favor the enhancement of the spin current. Thus, this device may become an effective spin-current generator, and the pure spin current is tuned in terms of the magnetic flux, the RSOI strength, and so forth. These results offer the ways to manipulate the spin-dependent transport via the four-terminal AB setup.

In this appendix, we present the derivation of the current formulas in detail.

Let g ^r(ε) and G ^r(ε) denote the retarded Green’s function of the DQD without and with the coupling to the external reservoirs. In the Nambu space, g ^r(ε) can be given as

[ g r ( ε ) ] − 1 = ε − ε 1 0 − t c 0 0 ε + ε 1 0 t c − t c 0 ε − ε 2 0 0 t c 0 ε + ε 2 .

Based on the following Dyson equation, the retarded Green’s function of the system can be written as G ^r(ε)]⁻¹ = g ^r(ε)⁻¹−Σ ^r, in which Σ r = Σ L r + Σ R r + Σ S 1 r + Σ S 2 r . The lesser Green’s function G ^<(ε) = G ^r(ε)Σ ^< G ^a(ε), where G ^a(ε) = G ^r(ε)] ^† and Σ < = Σ L < + Σ R < + Σ S 1 < + Σ S 2 < In the wide-band limit approximation, the retarded self-energy can be derived from the definition

Σ L r = − i 2 Γ L = − i 2 Γ 1 ↑ L 0 α Γ 1 ↑ L Γ 2 ↑ L e i ϕ / 2 0 0 Γ 1 ↓ L 0 α Γ 1 ↓ L Γ 2 ↓ L e − i ϕ / 2 α Γ 1 ↑ L Γ 2 ↑ L e − i ϕ / 2 0 Γ 2 ↑ L 0 0 α Γ 1 ↓ L Γ 2 ↓ L e i ϕ / 2 0 Γ 2 ↓ L ,

Σ R r = − i 2 Γ R = − i 2 Γ 1 ↑ R 0 α Γ 1 ↑ R Γ 2 ↑ R e − i ϕ σ / 2 0 0 Γ 1 ↓ R 0 α Γ 1 ↓ R Γ 2 ↓ R e i ϕ σ / 2 α Γ 1 ↑ R Γ 2 ↑ R e i ϕ σ / 2 0 Γ 2 ↑ R 0 0 α Γ 1 ↓ R Γ 2 ↓ R e − i ϕ σ / 2 0 Γ 2 ↓ R ,

Σ S 1 r = − i 2 Γ S 1 = − i 2 Γ 1 ρ 1 ( ε ) 1 − Δ ε 0 0 − Δ ε 1 0 0 0 0 0 0 0 0 0 0 ,

Σ S 2 r = − i 2 Γ S 2 = − i 2 Γ 2 ρ 2 ( ε ) 0 0 0 0 0 0 0 0 0 0 1 − Δ ε 0 0 − Δ ε 1 ,

where Γ ij σ ν = 2 Π ∑ k σ T ν , k ( i ) ∗ T ν , k ( j ) ρ ν σ with ρ _νσ being the density of states of the spin σ band in the lead ν. We calculate the tunneling matrix element by means of the Bardeen’s formula, i.e., T ν , k ( i ) = ℏ 2 ∫ S [ ψ ν , k ( r → ) ∇ ψ d ( i ) ∗ ( r → ) − ψ d ( i ) ∗ ( r → ) ∇ ψ ν , k ( r → ) ] dS / 2 m e , where m _e is the effective mass, S is the region of the integration, ψ ν , k ( r → ) is the wave function of evanescent mode of the lead ν, and ψ d ( i ) is the wave function of an electron localized in the QD i. Considering the propagation of electrons in the reservoir ν, this propagation process (the wave number dependence of T ν , k ( i ) ) induces the coherent indirect coupling via the reservoir ν between two QDs, which is characterized with the parameter α ^ν . We assume that (X _i , Y _i , 0) is the center position of the ith QD, X ₁ = X ₂ = X _D and L = |Y ₁−Y ₂|. thus, α ^ν is given by α ν = α ∼ ( 2 X D ) 3 / [ L 2 + ( 2 X D ) 2 ] 3 2 based on 21 . We find |α| ≤ 1 and decreases with L, and |α| = 1 corresponds to L = 0. We define α ≡ Γ 12 σ ν / Γ 11 σ ν Γ 22 σ ν , in which Γ 11 σ ν = Γ 1 σ ν , Γ 22 σ ν = Γ 2 σ ν . With the definition of the spin polarization P ν = ( Γ i ↑ ν − Γ i ↓ ν ) / ( Γ i ↑ ν + Γ i ↓ ν ) in the lead ν, the tunneling matrix element can be written as Γ i ↑ ν = Γ i ν ( 1 + P ν ) and Γ i ↓ ν = Γ i ν ( 1 − P ν ) with Γ i ν = ( Γ i ↑ ν + Γ i ↓ ν ) / 2 . Γ γ = 1 , 2 = 2 Π ∑ k σ | T γ , k ( i ) | 2 N γ σ , N _γσ is the density of states when the superconductor is the normal state, and ρ _γ (ε) is the modified BCS density of states ρ γ ( ε ) ≡ ρ ( ε ) = | ε | θ ( | ε | − Δ ) ε 2 − Δ 2 + ε θ ( Δ − | ε | ) i Δ 2 − ε 2 . With the RSOI phase factor φ _R = φ _R1−φ _R2, the spin-dependent phase factor is given by ϕ _σ = ϕ + 2σ φ _R . We mainly take account of the case of the symmetric coupling between two superconducting electrodes and DQD, that is, Γ _γ = Γ_s. According to the fluctuation-dissipation theorem, the lesser self-energy can be given as Σ ν < = F ν ( Σ ν a − Σ ν r ) and Σ S γ < = i F γ Γ S γ ρ ~ γ , where ρ ~ γ ≡ ρ ~ ( ε ) = ( | ε | / ε 2 − Δ 2 ) θ ( | ε | − Δ ) . F _ν and F _γ are, respectively,

F ν = f ν ( ε − e V ν ) 0 0 0 0 f ν ( ε + e V ν ) 0 0 0 0 f ν ( ε − e V ν ) 0 0 0 0 f ν ( ε + e V ν ) ,

F γ = f s ( ε ) 0 0 0 0 f s ( ε ) 0 0 0 0 f s ( ε ) 0 0 0 0 f s ( ε ) ,

in which f _ν (ε−e V _ν ) = f _ν , f ν ( ε + e V ν ) = f ¯ ν , and f _s (ε) are the Fermi distribution functions. By substituting Equations 15 to 20) into Equation 6, we can obtain the spin-related current as shown in Equations 9 and 10. The AR (CAR) coefficients ( T σ AR and T σ CAR ) and the probability of the quasiparticle tunneling ( T σ LR and T σ QS ) can be calculated as

T ↑ AR ( ε ) = Γ 1 ↑ L Γ 1 ↓ L | G 12 r ( ε ) | 2 + Γ 1 ↑ L Γ 2 ↓ L | G 14 r ( ε ) | 2 + Γ 2 ↑ L Γ 1 ↓ L | G 32 r ( ε ) | 2 + Γ 2 ↑ L Γ 2 ↓ L | G 34 r ( ε ) | 2 + 2 α Γ 1 ↓ L Γ 1 ↑ L Γ 2 ↑ L Re [ e − i ϕ / 2 G 12 r ( ε ) G 23 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↑ L Γ 2 ↓ L Re [ e − i ϕ G 12 r ( ε ) G 43 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↑ L Γ 2 ↓ L Re [ G 14 r ( ε ) G 23 a ( ε ) ] + 2 α Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↓ L Re [ e − i ϕ / 2 G 12 r ( ε ) G 41 a ( ε ) ] + 2 α Γ 2 ↓ L Γ 1 ↑ L Γ 2 ↑ L Re [ e − i ϕ / 2 G 14 r ( ε ) G 43 a ( ε ) ] + 2 α Γ 2 ↑ L Γ 1 ↓ L Γ 2 ↓ L Re [ e − i ϕ / 2 G 32 r ( ε ) G 43 a ( ε ) ] ,

T ↑ CAR ( ε ) = Γ 1 ↑ L Γ 1 ↓ R | G 12 r ( ε ) | 2 + Γ 1 ↑ L Γ 2 ↓ R | G 14 r ( ε ) | 2 + Γ 1 ↓ R Γ 2 ↑ L | G 32 r ( ε ) | 2 + Γ 2 ↑ L Γ 2 ↓ R | G 34 r ( ε ) | 2 + 2 α Γ 1 ↑ L Γ 1 ↓ R Γ 2 ↓ R Re [ e − i ϕ σ / 2 G 12 r ( ε ) G 21 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↑ R Γ 2 ↓ R Re [ e i ( ϕ − ϕ σ ) / 2 G 34 r ( ε ) G 21 a ( ε ) ] + 2 α Γ 2 ↓ R Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 34 r ( ε ) G 41 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 1 ↓ R Γ 2 ↓ R Re [ e i ( ϕ + ϕ σ ) G 32 r ( ε ) G 41 a ( ε ) ] + 2 α Γ 1 ↓ R Γ 1 ↑ L Γ 2 ↑ L Re [ e − i ϕ / 2 G 12 r ( ε ) G 23 a ( ε ) ] + 2 α Γ 2 ↑ L Γ 1 ↓ R Γ 2 ↓ R Re [ e − i ϕ σ / 2 G 34 r ( ε ) G 23 a ( ε ) ] ,

T ↑ LR ( ε ) = Γ 1 ↑ L Γ 1 ↓ R | G 11 r ( ε ) | 2 + Γ 1 ↑ L Γ 2 ↑ R | G 13 r ( ε ) | 2 + Γ 1 ↑ R Γ 2 ↑ L | G 31 r ( ε ) | 2 + Γ 2 ↑ L Γ 2 ↑ R | G 33 r ( ε ) | 2 + 2 α Γ 1 ↑ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ϕ σ / 2 G 13 r ( ε ) G 11 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 2 ↑ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ( ϕ σ + ϕ ) / 2 G 33 r ( ε ) G 11 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 2 ↑ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ( ϕ σ − ϕ ) / 2 G 13 r ( ε ) G 13 a ( ε ) ] + 2 α Γ 2 ↑ R Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 33 r ( ε ) G 31 a ( ε ) ] + 2 α Γ 2 ↑ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ϕ σ / 2 G 33 r ( ε ) G 13 a ( ε ) ] + 2 α Γ 1 ↑ R Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 31 r ( ε ) G 11 a ( ε ) ] ,

T ↑ QS ( ε ) = Γ s ρ ~ { Γ 1 ↑ L [ | G 11 r ( ε ) | 2 + | G 12 r ( ε ) | 2 + | G 13 r ( ε ) | 2 + | G 14 r ( ε ) | 2 ] + Γ 2 ↑ L [ | G 31 r ( ε ) | 2 + | G 32 r ( ε ) | 2 + | G 33 r ( ε ) | 2 + | G 34 r ( ε ) | 2 ] + 2 α Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 32 r ( ε ) G 21 a ( ε ) ] + 2 α Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 33 r ( ε ) G 31 a ( ε ) ] + 2 α Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 34 r ( ε ) G 41 a ( ε ) ] − 2 Γ 1 ↑ L Δ ε Re [ G 12 r ( ε ) G 11 a ( ε ) + G 14 r ( ε ) G 31 a ( ε ) ] − 2 Γ 2 ↑ L Δ ε Re [ G 32 r ( ε ) G 13 a ( ε ) + G 33 r ( ε ) G 43 a ( ε ) ] − 2 α Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 31 r ( ε ) G 21 a ( ε ) + e i ϕ / 2 G 34 r ( ε ) G 31 a ( ε ) ] − 2 α Γ 1 ↑ L Γ 2 ↑ L Re [ e i ϕ / 2 G 32 r ( ε ) G 11 a ( ε ) + e i ϕ / 2 G 33 r ( ε ) G 41 a ( ε ) ] } ,

T ↓ AR ( ε ) = Γ 1 ↑ L Γ 1 ↓ L | G 21 r ( ε ) | 2 + Γ 2 ↑ L Γ 1 ↓ L | G 23 r ( ε ) | 2 + Γ 1 ↑ L Γ 2 ↓ L | G 41 r ( ε ) | 2 + Γ 2 ↑ L Γ 2 ↓ L | G 43 r ( ε ) | 2 + 2 α Γ 1 ↓ L Γ 1 ↑ L Γ 2 ↑ L Re [ e − i ϕ / 2 G 23 r ( ε ) G 12 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↑ L Γ 2 ↓ L Re [ e − i ϕ G 43 r ( ε ) G 12 a ( ε ) ] + 2 α 2 Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↑ L Γ 2 ↓ L Re [ G 41 r ( ε ) G 32 a ( ε ) ] + 2 α Γ 1 ↑ L Γ 1 ↓ L Γ 2 ↓ L Re [ e − i ϕ / 2 G 41 r ( ε ) G 12 a ( ε ) ] + 2 α Γ 2 ↓ L Γ 1 ↑ L Γ 2 ↑ L Re [ e − i ϕ / 2 G 43 r ( ε ) G 14 a ( ε ) ] + 2 α Γ 2 ↑ L Γ 1 ↓ L Γ 2 ↓ L Re [ e − i ϕ / 2 G 32 r ( ε ) G 43 a ( ε ) ] ,

T ↓ CAR ( ε ) = Γ 1 ↓ L Γ 1 ↑ R | G 21 r ( ε ) | 2 + Γ 1 ↓ L Γ 2 ↑ R | G 23 r ( ε ) | 2 + Γ 1 ↑ R Γ 2 ↓ L | G 41 r ( ε ) | 2 + Γ 2 ↓ L Γ 2 ↑ R | G 43 r ( ε ) | 2 + 2 α Γ 1 ↓ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ϕ σ / 2 G 23 r ( ε ) G 12 a ( ε ) ] + 2 α 2 Γ 1 ↓ L Γ 2 ↓ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ( ϕ σ − ϕ ) / 2 G 43 r ( ε ) G 12 a ( ε ) ] + 2 α 2 Γ 1 ↓ L Γ 2 ↓ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ( ϕ σ + ϕ ) / 2 G 23 r ( ε ) G 14 a ( ε ) ] + 2 α Γ 2 ↑ R Γ 1 ↓ L Γ 2 ↓ L Re [ e i ϕ / 2 G 23 r ( ε ) G 34 a ( ε ) ] + 2 α Γ 1 ↑ R Γ 1 ↓ L Γ 2 ↓ L Re [ e i ϕ / 2 G 21 r ( ε ) G 14 a ( ε ) ] + 2 α Γ 2 ↓ L Γ 1 ↑ R Γ 2 ↑ R Re [ e i ϕ σ / 2 G 43 r ( ε ) G 14 a ( ε ) ]

T ↓ LR ( ε ) = Γ 1 ↓ L Γ 1 ↓ R | G 22 r ( ε ) | 2 + Γ 1 ↓ L Γ 2 ↓ R | G 24 r ( ε ) | 2 + Γ 1 ↓ R Γ 2 ↓ L | G 42 r ( ε ) | 2 + Γ 2 ↓ L Γ 2 ↓ R | G 44 r ( ε ) | 2 + 2 α Γ 1 ↓ R Γ 1 ↓ L Γ 2 ↓ L Re [ e i ϕ σ / 2 G 22 r ( ε ) G 24 a ( ε ) ] + 2 α 2 Γ 1 ↓ L Γ 2 ↓ L Γ 1 ↓ R Γ 2 ↓ R Re [ e − i ( ϕ σ + ϕ ) / 2 G 44 r ( ε ) G 22 a ( ε ) ] + 2 α 2 Γ 1 ↓ L Γ 2 ↓ L Γ 1 ↓ R Γ 2 ↓ R Re [ e i ( ϕ σ − ϕ ) / 2 G 24 r ( ε ) G 24 a ( ε ) ] + 2 α Γ 2 ↓ R Γ 1 ↓ L Γ 2 ↓ L Re [ e i ϕ / 2 G 24 r ( ε ) G 44 a ( ε ) ] + 2 α Γ 2 ↓ L Γ 1 ↓ R Γ 2 ↓ R Re [ e − i ϕ σ / 2 G 44 r ( ε ) G 24 a ( ε ) ] + 2 α Γ 1 ↓ L Γ 1 ↑ R Γ 2 ↑ R Re [ e − i ϕ σ / 2 G 24 r ( ε ) G 22 a ( ε ) ] ,