Spin-orbit coupling and electric-dipole spin resonance in a nanowire double quantum dot

We study the electric-dipole transitions for a single electron in a double quantum dot located in a semiconductor nanowire. Enabled by spin-orbit coupling (SOC), electric-dipole spin resonance (EDSR) for such an electron can be generated via two mechanisms: the SOC-induced intradot pseudospin states mixing and the interdot spin-flipped tunneling. The EDSR frequency and strength are determined by these mechanisms together. For both mechanisms the electric-dipole transition rates are strongly dependent on the external magnetic field. Their competition can be revealed by increasing the magnetic field and/or the interdot distance for the double dot. To clarify whether the strong SOC significantly impact the electron state coherence, we also calculate relaxations from excited levels via phonon emission. We show that spin-flip relaxations can be effectively suppressed by the phonon bottleneck effect even at relatively low magnetic fields because of the very large g-factor of strong SOC materials such as InSb.

. One is the Dresselhaus SOC due to bulk inversion asymmetry 45 . The other is the Rashba SOC generated by structure inversion asymmetry 46 . In general, the SOC strengths depend on system parameters and spatial distributions of the electron wave function. By averaging over the transverse directions y and z, we obtain an effective linear SOC Hamiltonian H x so along the x direction (see Methods) gives the external magnetic field direction, and â (cos , sin ,0) φ φ = is the SOC-induced effective field direction, with φ = arctan(α R /α D ) ∈ [0, π/2] characterizing the relative strength between the Rashba and Dresselhaus SOCs. ϕ is the angle between vectors â and n , i.e., ˆφ = 〈 ⋅ 〉 a n arccos .
When an external magnetic field is applied in the direction n (cos , 0, sin ) θ θ = with strength B, the single-electron Hamiltonian becomes α σ μ σ x x e x a e B n DQD 2 with the vector potential A = (A x , A y , 0), where A x = −By sin θ, A y = −Bz cos θ, g e is the Landé factor, μ B is the Bohr magneton, and σ σ = ⋅ n n . With our assumption of an asymmetric double dot, it follows naturally that the specific value of the Landé factor g e in the left dot is different from that in the right dot, g el ≠ g er 20 . Due to the strong confinements along the transverse directions, 〈y〉 ~ 0, the effects of the magnetic vector potential on the electron orbital dynamics is negligible (detailed calculations are given in Methods), so that the Hamiltonian for the DQD can be simplified as Traditionally SOC is treated as a perturbation in theoretical calculations for semiconductors. However, such a perturbative approach becomes problematic when SOC is strong, in materials such as InSb 41 . For a comprehensive study of the effect of a strong SOC on the electric-dipole transition in a nanowire DQD, in the following calculations we take the SOC term into consideration precisely while treating the Zeeman term perturbatively.
Energy spectrum of the DQD. The energy spectrum of the DQD is calculated by adopting the linear combination of atomic orbitals (LCAO) method. The localized electron wavefunctions are derived by solving the eigenstates of the individual quantum dots. The orthonormal bases used to project the DQD Hamiltonian are obtained by the Schmidt orthogonalization of the local wavefunctions.
Near each of the minima of the DQD potential well along the nanowire axis, V(x) can be approximated as Including the SOC effect, the local Hamiltonian for each single quantum dot can be written as where ψ κn (x) represents an eigenstate of a harmonic oscillator with eigenvalue (n + 1/2)ħω κ , x so is the effective SOC length x so = ħ/(m e α), and |↑ a 〉 and |↓ a 〉 denote the eigenstates of σ a : σ a |↑ a 〉 = |↑ a 〉 and σ a |↓ a 〉 = −|↓ a 〉. |Φ κn↑ 〉 and |Φ κn↓ 〉 are degenerate (Kramers degeneracy), with the eigenvalue given by ε κn The energy levels of H′ κ are thus evenly spaced, with an energy splitting Δ κS = ħω κ .
In the presence of an applied magnetic field, the single-dot Hamiltonian becomes where Δ κZ corresponds to the Zeeman splitting in κ dot. The Zeeman term can be regarded as a perturbation if the ratio

Z S
i.e. the Zeeman splitting Δ κZ is much smaller than the orbital splitting Δ κS , dictating a relatively small magnetic field (see the estimate in ref. 41 ). Within first-order perturbation theory, the two lowest-energy eigenstates of H κ are Here ˆφ = 〈 ⋅ 〉 a n arccos is the angle between unit vectors â and n (i.e., the angle between the effective field from SOC and the applied magnetic field), and It is a ratio between the effective dot size x κ and SOC length x so , therefore is a measure of the SOC strength relative to the confinement energy. For a nanowire quantum dot, η κ is generally small, , even for materials with strong SOC. According to Eq. (9), an applied magnetic field generally leads to hybridization of different spin-orbit states in |Ψ 〉 κ ± , with the degree of orbital mixing proportional to ξ η The orbital states localized in different quantum dots are not orthogonal in general. Nevertheless, from the four lowest-energy localized states |Ψ 〉 κ ± (κ = l, r) and using Schmidt orthogonalization method, we can construct local orthonormal basis states l |Ψ 〉, |Ψ 〉 l , r |Ψ 〉, and r |Ψ 〉. Here and refer to the two pseudo-spin states, whose compositions have been modified by the applied magnetic field as compared to the zero-field Kramers degenerate pair. The analytical expressions for the bases are given in Methods.
Projecting the Hamiltonian H DQD onto this orthonormal basis, the low-energy part of the Hamiltonian H DQD can be written as Here t σσ (σ = , ) is spin-conserved tunnel coupling, σσ t is spin-flipped tunnel coupling and ε κσ is the corresponding single-dot energy. These matrix elements can be obtained by dividing the original Hamiltonian H DQD in Eq. (4) into two parts H DQD = H κ + Δ κ V(x), with H κ either one of the single-dot Hamiltonian, and Δ κ V(x) = V(x) − V κ (x) the double dot correction on H κ . Due to the orthogonality of |Ψ κσ 〉, the tunnelings can be calculated as t σσ′ = 〈Ψ lσ |Δ l V(x)|Ψ rσ′ 〉, with its magnitude proportional to the interdot wave function overlap, . The eigenstates of the nanowire DQD can be obtained numerically by the direct diagonalization of the Hamiltonian H DQD in Eq. (11). We denote these states |Φ i 〉 (i = 1 − 4), with eigenvalues E 1 ≤ E 2 ≤ E 3 ≤ E 4 . In Fig. 2 we give an example energy spectrum of an InSb nanowire DQD, with the corresponding system parameters taken from the experimental data in ref. 20 : ħω l = 5.0 meV, ħω r = 7.5 meV, g el = −32.2, g er = −29.7,  x 200nm so , and d = 40 nm. The effects of the anisotropic g-factors are neglected for simplicity. Except for the interdot distance and the magnetic field strength and orientation, the parameters of the typical InSb nanowire are used in the following calculations for convenience and consistency.
Equation (9) indicates that when , which is satisfied with the parameters used in Fig. 2, the intradot orbital states hybridization is negligible compared with the interdot states mixing. For weaker SOC or strongly coupled DQD,  x d 2 so , the interdot spin-flipped tunneling σσ t is much smaller than the spin-conserved tunneling t σσ . Nevertheless, spin-flipped tunneling leads to a high degree of pseudospin hybridization in states |Φ 2 〉 and |Φ 3 〉 around the anti-crossing point B 0 , as shown in Fig. 2.
In calculating the energy spectrum of the DQD, it is necessary to establish the validity of the perturbation expansion in Eq. (9) and the approximation to neglect orbital effects of the magnetic vector potential in our calculations. The perturbation expansion in Eq. (9) can be justified by the specific values of ξ κ (κ = l, r) at the upper limit of the magnetic field range we consider. With our chosen parameters, when B = 2 T, ξ r < ξ l < 0.75, which still (barely) satisfy the perturbation condition in Eq. (8). As for the orbital effect of the vector potential, we compare the effective magnetic length involve spin flip, leading to EDSR [11][12][13][14][15][16][17][18] . In a DQD with strong SOC, the pseudo-spin composition of the eigenstates vary with magnetic field and interdot distance/tunneling. Moreover, under certain circumstances, intradot spin mixing in the DQD can also affect EDSR. In this Section we investigate how EDSR transition rates depend on different system parameters.
When an AC electric field is applied in the x direction, the Hamiltonian describing the single electron in the DQD reads e d DQD with E and υ representing the amplitude and frequency of the electric field, respectively. The electric-dipole interaction can be treated as a perturbation if j , and the resonant electric-dipole transition rate can be calculated as i j i j where h is the Plank constant. Due to the spinless e-d interaction, the compositions of the pseudospin states |Φ i 〉 and |Φ j 〉 are a crucial factor in determining the magnitude of Ω i↔j . With the transitions involving state |Φ 4 〉 symmetric with respect to those involving state |Φ 1 〉, for simplicity we only consider the electric-dipole transitions involving |Φ 1 〉 in the following calculations.
Magnetic field dependence. In Sec. 2 we have shown that there are two mechanisms leading to different spin states hybridization in the eigenstates of DQD: the SOC-induced intradot states mixing and the interdot spin-flipped tunneling. Because all the mechanisms show strong dependences on the external magnetic field, both the transition rates Ω 2↔1 and Ω 3↔1 will definitely change when varying the magnetic field. As is clearly illustrated in Fig. 3, the variations of Ω 2↔1 and Ω 3↔1 with the magnetic field strength B and orientation ϕ are shown.
For an InSb nanowire DQD with , the interdot tunneling dominates the orbital mixing in the eigenstates of DQD, and the effect of the intradot orbital states mixing can be negligible (the effect of the intradot orbital states mixing is investigated later in the next subsection). In a weak magnetic field, B B 0  , the major pseudospin components of the state |Φ 1 〉 are the same as that of |Φ 3 〉 and different from that of |Φ 2 〉. It follows naturally that Ω Ω . For a fixed ϕ (the angle between the applied magnetic field and the SOC-induced effective field), increasing the magnetic field strength enhances the degree of the interdot pseudospin hybridization in |Φ 2 〉 and |Φ 3 〉, which in turn leads to the rising (falling) of Ω 2↔1 (Ω 3↔1 ), as shown in Fig. 3(c). |Φ 1 〉 ↔ |Φ 2 〉 corresponds to the electric-dipole spin transition for B < B 0 , with Ω 2↔1 representing the EDSR frequency when the AC electric field is on resonance with Δ 12 . As demonstrated in refs 16,17 , the magnitude of the EDSR frequency depends on the effective SOC strength, which can be controlled by changing the magnetic field direction. In Fig. 3(d), for a fixed magnetic field strength, the magnitude of the EDSR frequency Ω 2↔1 as a function of the field orientation ϕ is shown. In particular, when the magnetic field is perpendicular to the SOC field direction, the effect of the SOC-induced mixing reaches its maximum, and the EDSR frequency reaches its peak value. Similarly, Ω 3↔1 also has a strong ϕ dependence.
As B increases beyond B 0 , the major pseudospin components of |Φ 2 〉 and |Φ 3 〉 are swapped. At this point, |Φ 1 〉 ↔ |Φ 3 〉 is the spin-flip transition, with Ω 3↔1 the corresponding EDSR frequency. When increasing magnetic field, the larger energy splitting between |Φ 2 〉 and |Φ 3 〉 weakens the interdot pseudospin hybridization in these levels. As a result, the EDSR frequency Ω 3↔1 decreases, and the orbital transition rate Ω 2↔1 saturates, as shown in Fig. 3(c). in our considered range of magnetic field, so that intradot orbital mixing becomes an important factor in determining the overall spin-flip transition rates. Here we examine the competition between the interdot and intradot mechanisms for spin flip transitions.
In a low magnetic field, the effect of the intradot orbital states mixing on the EDSR, compared with the interdot mechanism, is negligible as long as ξ κ is small. As the Zeeman splitting increases, the rising value of ξ κ enhances the strength of the intradot orbital states mixing, see Eq. (9). Meanwhile, the interdot pseudospin states mixing weakens with the increase of B for B > B 0 . There thus exists a turning magnetic field B t : for B < B t , EDSR is dominated by the interdot state hybridization; for high fields the state mixing is dominated by the intradot mechanism. This change is also reflected in the variation of the EDSR frequency Ω 3↔1 around the turning field B t , as shown in the inset of Fig. 4. As the B field increases and approaches B t , the magnitude of the EDSR frequency Ω 3↔1 decreases with the growth of B as the interdot state mixing mechanism becomes less efficient, so it reverts that trend when B > B t as the intradot mechanism becomes more effective.
The turning field B t is a symbol for the competition between these two different mechanisms. Its magnitude mainly depends on the interdot distance. For the InSb nanowire DQD with ϕ = π/2, B t as a function of d is shown in Fig. 4. The downward trend of B t with the increase of d can be explained by the decline of the interdot state mixing, which requires a smaller magnetic field to counteract.
At large interdot distances, the magnitude of B t tends to be stable. This is because at a large interdot distance intradot orbital mixing dominates over interdot pseudospin hybridization, even for smaller magnetic field B ≤ B 0 . Now B 0 mainly depends on the orbital energy difference between the QDs Δ o ≡ ħω r − ħω l , and nearly independent of d, The dependence on the interdot distance. The underlying dependence of the interdot barrier on the interdot distance means that spin tunneling, and single-electron energy spectrum of the nanowire DQD in general, depend on d 48 . In Fig. 5 we show the energy spectrum of the nanowire InSb DQD as a function of d, with B = 0.3 T and ϕ = π/2. Because of the asymmetry in the confinement potential along the wire axis, we limit ourselves to consider the case with a nonzero finite interdot distance exclusively. ) is much smaller than the orbital energy difference Δ o between the dots, so that the effect of the interdot states mixing on the energy spectrum is negligible. The energy spectrum of the DQD is essentially the sum of the energy spectrums of the single QDs in this case, with |Φ 〉 |Ψ 〉 When the interdot distance decreases, the interdot tunnel coupling increases exponentially, and the energy spectrum of the DQD changes correspondingly. When d ~ d 0 , the energy scale of the interdot tunnel coupling becomes comparable to the orbital energy difference between the QDs, so that the eigenstates of the DQD are delocalized pseudospin states. If the interdot distance further decreases, the two dots start to merge. The "interdot tunneling" will be of the same magnitude as the orbital excitation energy in the individual QDs. At this limit, the character of the electronic states shifts back from molecular-like to atomic-like again like the case of large interdot distance 48 , although the composition of the orbital states are dramatically different. The energy splitting between |Φ 2 〉 and |Φ 3 〉 is now dominated by the single-particle single-dot excitation energy (as compared to tunnel splitting in the case of a double dot), which results in a sharp rise in this energy gap, as shown in Fig. 5.
Since the compositions of the DQD eigenstates vary with the interdot distance, particularly near d 0 , the electric-dipole transition rates change quite dramatically as well. In the inset of Fig. 5 we plot the transition rates Ω 2↔1 and Ω 3↔1 as a function of d. At a large interdot distance  d d 0 , the DQD eigenstates can be approximated as the eigenstates of the individual QDs, as explained above. Thus, |Φ 2 〉 ↔ |Φ 1 〉 is an intradot spin-flip transition while |Φ 3 〉 ↔ |Φ 1 〉 is an interdot transition. Because of the vanishingly small interdot state mixing, the magnitude of the interdot transition rate will be smaller than that of the intradot spin-flip transition rate, Ω 3↔1 < Ω 2↔1 . As the interdot distance decreases, the rapidly rising interdot coupling means both transition rates increase quickly as the states become mixed. When  d ~ d 0 , the eigenstates of the DQD are delocalized, and the electric-dipole spin transition |Φ 2 〉 ↔ |Φ 1 〉 is dominated by the interdot pseudospin tunneling. As d decreases further, the magnitudes of Ω 2↔1 and Ω 3↔1 become stable because the DQD merges into a single QD. The electric-dipole transition in a single nanowire QD was investigated in ref. 41 . In this limit, Ω 3↔1 approaches eEx h ( 2/2) / , while Ω 2↔1 can be approximated by ξ η η − eEx h exp( ) / l l l l 2 . Thus, when  d d 0 the main mechanism of the EDSR turns back to the intradot orbital states mixing again. In short, in the parameter range we have considered, the electric-dipole transition rates depend sensitively on the interdot tunneling/distance.

Phonon-induced relaxation between the energy levels. Electron-phonon (e-ph) interaction,
together with spin-orbit coupling, is the main cause of spin relaxation in a quantum dot [49][50][51][52][53][54][55][56][57][58][59][60][61] . Accurately determining the relaxation rates is thus a necessary condition for quantitatively assessing the fidelity of the electric-dipole transitions. Recall that B 0 is the field at which |Φ 2 〉 and |Φ 3 〉 cross in the absence of SOC. Based on the major pseudospin components of the eigenstates involved in relaxation, Γ 2→1 corresponds to phonon-induced spin relaxation for B B 0  , while Γ 3→1 is the phonon-induced spin relaxation rate when  B B 0 . For relaxation between energetically close levels, we only consider the e-ph interaction with acoustic phonons and ignore the optical phonons. For acoustic phonons, there are two types of e-ph interaction: the piezoelectric and deformation potential interactions 62 . Including the e-ph interaction, the complete Hamiltonian describing the DQD reads where the e-ph interaction is given by 55,56  3ˆˆ. Here q = (q x , q y , q z ) is the phonon wave vector, with q representing its magnitude, r = (x, y, z) denotes the electron position, and λ is the polarization of the phonon, with ê and c λ being the polarization vector and sound velocity of the phonon mode. The phonon annihilation (creation) operator is denoted by b (b † ). ρ and V are the mass density and the volume of the sample, respectively. Using the Fermi golden rule, the phonon-induced relaxation rate between the eigenstates |Φ i 〉 and |Φ j 〉 (i > j) can be calculated as where Δ ij is the energy difference between the eigenstates |Φ i 〉 and |Φ j 〉, n th is the thermal occupation of the phonon mode with ħω λ = Δ ij ≡ E i − E j . At low temperatures when k T ij B  Δ , n th ≈ 0. The matrix element M ij depends on the spatial distribution of the electron wave functions in three dimensions (see Methods). Taking the phonon mode density into consideration, the phonon-induced relaxation rate can be rewritten as i j ij q q 2 2 , df , pe 2  with the integral region satisfying the energy conservation condition ħc λ q = Δ ij . The electron-phonon interaction Hamiltonian in Eq. (16) is spin independent. As such the spin composition of |Φ i 〉 and |Φ j 〉 plays a key role in determining Γ i→j : a mostly spin-conserved relaxation would be much faster than a relaxation involving spin flip. Since the degree of spin mixing depends on the interplay between the external field and the spin-orbit coupling, the relaxations between different eigenstates can generally be regulated by varying the magnetic field strength and direction [54][55][56][57][58][59][60][61] . Here we focus on the dependence of the relaxation rates on the magnetic field strength for a fixed magnetic field direction. The numerical results are shown in Fig. 6, where we plot Γ 2→1 and Γ 3→1 as functions of the magnetic field strength, with the system parameters taking the values as in Fig. 2.
In a low magnetic field with  B B 0 , the major pseudospin components of |Φ i 〉 (i = 1 − 4) dictate that |Φ 3 〉 → |Φ 1 〉 is a charge transition, while |Φ 2 〉 → |Φ 1 〉 is a spin-flip transition, thus 3 1 As B increases, spin-flipped tunneling results in the hybridization between |Φ 〉 2 0 and |Φ 〉 3 0 , so that Γ 3→1 decreases while Γ 2→1 increases. The slight oscillations in both relaxation rates are most likely due to the matching between DQD charge density and phonon wave vector (recall that the e-ph interaction Hamiltonian contains a e iq·r factor). Furthermore, with the large g-factor for InSb, Zeeman splitting reaches 1 meV when the magnetic field is only a fraction of 1 Tesla. The corresponding phonon wave length is in the order of 10 nm, already below the quantum dot size, so that phonon bottleneck effect starts to become apparent for spin-flip relaxation 63 .
When B = B 0 the spin states of |Φ 2 〉 and |Φ 3 〉 are equally mixed. The energy gap between |Φ 2 〉 and |Φ 3 〉 means that the relaxation rates are generally not identical at B 0 . The rates are determined by a competition mostly between phonon density of states consideration and the phonon bottleneck effect: the former favors the larger-energy 3 → 1 transition, while the latter favors the smaller-energy 2 → 1 transition.
When the magnetic field strength exceeds B 0 , |Φ 2 〉 (|Φ 3 〉) becomes the pseudospin up (down) state. As illustrated in Fig. 2, for B > B 0 the energy splitting Δ 21 (proportional to the tunnel coupling in the DQD) tends to be stable with the growth of B, while Δ 31 , now the Zeeman splitting, keeps increasing. Thus Γ 2→1 approaches a constant value when  B B 0 . Γ 3→1 , on the other hand, keeps decreasing due to the reduction in spin mixing and the increasing influence of the phonon bottleneck effect. The relaxation calculation here is done using bulk phonons. In a suspended nanowire, confined phonons on the nanowire should be used, and we expect the relaxation rates to be further suppressed because of the much smaller phonon density of states and stronger anisotropy due to the nanowire geometry 23,64 .

Discussion and Conclusion
In this paper, we study the electronic properties of a nanowire DQD within the frame of effective mass approximation (EMA). For a thin nanowire, the energy scale of the electronic dynamics along the axis direction is much smaller than the energy scale of the excitations in the transverse directions. As such in our consideration the confined electron always stays in the transverse ground state 18,22,23,28 .
Our calculations are based on a truncated double harmonic potential. Within EMA, the DQD confinement potential is usually approximated by a quartic function, a biquadratic function, or a Gaussian function 7,[38][39][40]48 . All these model potentials give rise to results consistent with the experimental results at a qualitative level 7,40,55,58 . Therefore, the simplicity associated with the truncated double harmonic well (biquadratic function) model potential becomes the deciding factor for our choice. The relatively concise expressions within this model allows us to get to the basic physics more easily.
The system parameters used in our calculations are taken from the experimental data in ref. 20 . The low-energy spectrum of a single electron in the DQD is obtained using the LCAO method. In the calculation, the SOC is taken into account precisely, while the applied magnetic field is treated as a perturbation.
Our calculations show that in a DQD, there exist two different mechanisms that lead to EDSR: the intradot pseudospin state mixing and the interdot spin-flipped tunneling. The EDSR frequency is determined by the combined effect of these two mechanisms, in which the dominant role can be varied by changing the system parameters. When the EDSR is dominated by the interdot spin-flipped tunneling, we show that the electric-dipole transition rates depend sensitively on the magnitude and orientation of the applied field. The intradot orbital mixing becomes more important when we reduce the tunnel coupling, so the two dots become independent, or when we increase tunneling to the degree when the double dot merges into a single dot. In the intermediate regime the interdot spin mixing is more effective. For a fixed tunnel coupling/interdot distance, the electric-dipole driven transition rates experience a dip as the magnetic field increases, when the DQD transitions from the interdot-mixing dominated low-field region to the intradot-mixing dominated high-field region.
Finally, we have calculated phonon-induced relaxation rates among the DQD energy levels. The very large g-factors for strong SOC materials, such as InSb that we consider, mean that phonon bottleneck effect kicks in at much lower magnetic field for spin-flip transitions compared to materials such as GaAs. Overall, our results on low-energy spectrum, controllable electric-dipole transitions, and relaxations should provide useful input for experimental studies of quantum coherent manipulations in a nanowire DQD.

Methods
Derivation of the effective Hamiltonian. The nanowire DQD Hamiltonian in Eq. (4) is derived within the effective mass approximation. We choose our coordinate system according to the geometry of the nanowire and the applied field. Specifically, we choose the x-axis along the axis of the nanowire, as illustrated in Fig. 1(a). When an external magnetic field is applied, we choose it to lie in the xz-plane, so that the field can be expressed as B = B(cos θ, 0, sin θ). The complete Hamiltonian describing an electron in a nanowire DQD is where the first term is the kinetic energy, with the kinetic momentum P = p + eA and the vector potential A = B(−y sin θ, −z cos θ, 0), U(r) is the confinement potential in three dimensions, H so (r) represents the spin-orbit interaction, and the last term denotes the Zeeman term, with g e and μ B being the location-dependent Landé g-factor and Bohr magneton, respectively. Here g e is location-dependent, with the specific value of the g-factor of the left QD being different from that of the right QD g el ≠ g er .
The DQD confinement potential for the electron is modelled by a asymmetric double well harmonic potential along the nanowire axis, V x m where ω l ≠ ω l , and 2d is the interdot distance. In the transverse direction we consider a strong harmonic potential along the y direction, V y m y ( ) (1/2) e y 2 2 ω = , and a large gradient potential along the z direction, V(z) = eE z z for z ≥ 0 and V(z) = ∞ for z < 0. Due to the strong transverse confinements, we assume that the electron is always in the ground state along the y and z, so that the transverse orbital dynamics is frozen: where y m /( ) e y 0  ω = and τ = (2eE z m e /ħ 2 ) 1/3 . The characteristic length scales of the wavefunction along the y and z directions can thus be quantified by y 0 and z z z z dz ∞ , respectively. The lowest-order effective Hamiltonian for an electron moving along the x-axis can be obtained by averaging over the y and z directions, The first term on the right side of Eq. (21) represents the effective kinetic Hamiltonian. Substituting the kinetic momentum expression into Eq. (21), the effective kinetic Hamiltonian can be expanded as with B x = B cos θ and B z = B sin θ. Since ψ(y) is an even function, the second term on the right side of Eq. (22)  where we have used the identity 〈p y 〉 = 0. In the considered range of magnetic field with 1  ξ , the contribution of (eB x ) 2 〈z 2 〉 to H D x is negligible compared with the other two terms in the bracket on the right side of Eq. (25 The total effective SOC Hamiltonian along the nanowire axis is thus given by α σ α σ Substituting Eqs (22) and (28) into Eq. (21), the effective Hamiltonian describing the DQD along the wire axis can be simplified as Eq. (4) in the main text, Construction of the orthonormal basis. The analytic formulas for the orthonormal bases |Ψ 〉 κ and |Ψ κ⇓ 〉 (κ = l, r) are given. Using the perturbation theory, the two lowest-energy eigenstates of the local Hamiltonian H κ can be approximated as the equation (9)  Here ξ κ denotes the ratio between the Zeeman splitting and the orbital splitting in κ dot, ξ κ ≡ Δ κZ /Δ κS , and is much less than one, which ensures the validity of the perturbation theory. η κ corresponds to the ratio between the effective dot size x κ and SOC length x so . In a nanowire quantum dot, η κ is generally a small number , even for materials with strong SOC. Therefore, in order to facilitate the numerical calculations in the main text and account the effect of high orbital states, the summation in Eq. (31) is truncated, and only keep the n = 1 term. Thus, the corresponding normalized local wave functions can be written as