A Basis States

# Phonon-mediated decay of singlet-triplet qubits in double quantum dots

## Abstract

We study theoretically the phonon-induced relaxation () and decoherence times () of singlet-triplet qubits in lateral GaAs double quantum dots (DQDs). When the DQD is biased, Pauli exclusion enables strong dephasing via two-phonon processes. This mechanism requires neither hyperfine nor spin-orbit interaction and yields , in contrast to previous calculations of phonon-limited lifetimes. When the DQD is unbiased, we find and much longer lifetimes than in the biased DQD. For typical setups, the decoherence and relaxation rates due to one-phonon processes are proportional to the temperature , whereas the rates due to two-phonon processes reveal a transition from to higher powers as is decreased. Remarkably, both and exhibit a maximum when the external magnetic field is applied along a certain axis within the plane of the two-dimensional electron gas. We compare our results with recent experiments and analyze the dependence of and on system properties such as the detuning, the spin-orbit parameters, the hyperfine coupling, and the orientation of the DQD and the applied magnetic field with respect to the main crystallographic axes.

###### pacs:
73.21.La, 71.70.Ej, 03.67.Lx, 71.38.-k

## I Introduction

The spin states of quantum dots (QDs) are promising platforms for quantum computation.(1); (2) In particular, remarkable progress has been made with - qubits in lateral GaAs double quantum dots (DQDs),(4); (5); (6); (3); (7) where a qubit is based on the spin singlet () and triplet () state of two electrons in the DQD. In this encoding scheme, rotations around the axis of the Bloch sphere can be performed on a subnanosecond timescale (4) through the exchange interaction, and rotations around the axis are enabled by magnetic field gradients across the QDs.(5)

The lifetimes of - qubits have been studied with great efforts. When the qubit state precesses around the axis, dephasing mainly results from Overhauser field fluctuations, leading to short dephasing times .(8); (9); (10); (4); (11); (12) This low-frequency noise can be dynamically decoupled with echo pulses,(4); (13); (14); (15) and long decoherence times have already been measured.(14) In contrast to -rotations, precessions around the axis dephase predominantly due to charge noise.(16); (17) Rather surprisingly, however, recent Hahn echo experiments by Dial et al. (16) revealed a relatively short and a power-law dependence of on the temperature . The origin of the observed decoherence is so far unknown, although the dependence on suggests that lattice vibrations (phonons) may play an important role.

In this work, we calculate the phonon-induced lifetimes of a - qubit in a lateral GaAs DQD. Taking into account the spin-orbit interaction (SOI) and the hyperfine coupling, we show that one- and two-phonon processes can become the dominant decay channels in these systems and may lead to qubit lifetimes on the order of microseconds only. While the decoherence and relaxation rates due to one-phonon processes scale with for the parameter range considered here, the rates due to two-phonon processes scale with at rather high temperatures and obey power laws with higher powers of as the temperature decreases. Among other things, the qubit lifetimes depend strongly on the applied magnetic field, the interdot distance, and the detuning between the QDs. Based on the developed theory, we discuss how the lifetimes can be significantly prolonged.

The paper is organized as follows. In Sec. II we present the Hamiltonian and the basis states of our model. In the main part, Sec. III, we discuss the calculation of the lifetimes in a biased DQD and investigate the results in detail. In particular, we show that two-phonon processes lead to short dephasing times and identify the magnetic field direction at which the lifetimes peak. The results for unbiased DQDs are discussed in Sec. IV, followed by our conclusions in Sec. V. Details and further information are appended.

## Ii System, Hamiltonian, and Basis States

We consider a lateral GaAs DQD within the two-dimensional electron gas (2DEG) of an AlGaAs/GaAs heterostructure that is grown along the [001] direction, referred to as the axis. Confinement in the --plane is generated by electric gates on the sample surface, and the magnetic field is applied in-plane to avoid orbital effects. When the DQD is occupied by two electrons, the Hamiltonian of the system reads

 H = ∑j=1,2(H(j)0+H(j)Z+H(j)SOI+H(j)hyp+H(j)el−ph) (1) +HC+Hph,

where the index labels the electrons, comprises the kinetic and potential energy of an electron in the DQD potential, is the Zeeman coupling, is the SOI, is the hyperfine coupling to the nuclear spins, is the electron-phonon coupling, is the Coulomb repulsion, and describes the phonon bath.

The electron-phonon interaction has the form

 Hel−ph=∑q,sWs(q)aqseiq⋅r+h.c., (2)

where is the position of the electron, is a phonon wave vector within the first Brillouin zone, stands for the longitudinal () and the two transverse () phonon modes, and “h.c.” is the hermitian conjugate. The coefficient depends strongly on and , and is determined by material properties such as the relative permittivity , the density , the speed () of a longitudinal (transverse) sound wave, and the constants and for the deformation potential and piezoelectric coupling, respectively. The annihilation operator for a phonon of wave vector and mode is denoted by . The Hamiltonian

 HSOI=α(px′σy′−py′σx′)+β(py′σy′−px′σx′) (3)

contains both Rashba and Dresselhaus SOI. Here and are the momentum operators for the and axes, respectively. The latter coincide with the crystallographic axes [100] and [010], respectively, and and are the corresponding Pauli operators for the electron spin. We take into account the coupling to states of higher energy by performing a Schrieffer-Wolff transformation that removes in lowest order.(18); (19); (20); (23); (21); (22); (24) The resulting Hamiltonian is equivalent to , except that is replaced by

 ˜HSOI≃gμB(rSOI×B)⋅σ, (4)

where is the in-plane factor, is the vector of Pauli matrices, and

 Missing or unrecognized delimiter for \right (5)

Here and are the coordinates of the electron along the main crystallographic axes, whose orientation is provided by the unit vectors and , respectively. The spin-orbit lengths are defined as and , where is the effective electron mass in GaAs and () is the Rashba (Dresselhaus) coefficient. For our analysis, the most relevant effect of the nuclear spins is the generation of an effective magnetic field gradient between the QDs, which is accounted for by . We note that this magnetic field gradient may also result from a nearby positioned micromagnet.(25); (26); (27) For details of and , see Appendix B.

The - qubit in this work is formed by the basis states and , where the notation means that () electrons occupy the left (right) QD. In first approximation, these states read

 |(1,1)S⟩ = |Ψ+⟩|S⟩, (6) |(1,1)T0⟩ = |Ψ−⟩|T0⟩, (7)

with

 |Ψ±⟩=|Φ(1)LΦ(2)R⟩±|Φ(1)RΦ(2)L⟩√2, (8)

where the are orthonormalized single-electron wave functions for the left and right QD, respectively (see also Appendix A).(28); (29) The spin singlet is

 |S⟩=|↑↓⟩−|↓↑⟩√2, (9)

whereas

 |T0⟩=|↑↓⟩+|↓↑⟩√2, (10)

with the quantization axis of the spins along . Analogously, one can define the states and , which are energetically split from the qubit by . For our analysis of the phonon-induced lifetimes, a simple projection of onto this 4D subspace of lowest energy is not sufficient, because

 ∑j(⟨Ψ+|H(j)el−ph|Ψ+⟩−⟨Ψ−|H(j)el−ph|Ψ−⟩)=0. (11)

That is, corrections from higher states must be taken into account in order to obtain finite lifetimes.(23); (30) The spectrum that results from the states considered in our model is plotted in Fig. 1. Depending on the detuning between the QDs, the lifetimes of the qubit are determined by admixtures from , , or states with excited orbital parts.

## Iii Regime of Large Detuning

### iii.1 Effective Hamiltonian and Bloch-Redfield theory

We first consider the case of a large, positive detuning at which the energy gap between and the qubit states is smaller than the orbital level spacing . In this regime, contributions from states with excited orbital parts are negligible, and projection of onto the basis states , , , , , and yields

 ˜H=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝PTδbB20000δbB2V+−V−+PTΩ√2−Ω√2−√2t+P†S−√2t+PS0Ω√2EZ+PT0000−Ω√20−EZ+PT000−√2t+PS00−ϵ+U−V−+PSR00−√2t+P†S000ϵ+U−V−+PSL⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠+Hph. (12)

Here , , , , and are the matrix elements of the electron-phonon interaction, is the tunnel coupling, is the on-site repulsion, , ,

 Ω=gμB ( ⟨ΦL|(rSOI×B)z|ΦL⟩ (13) −⟨ΦR|(rSOI×B)z|ΦR⟩),

and (see also Appendix B.5). We note that the energy in Eq. (12) was globally shifted by . Furthermore, we mention that the state is very well decoupled when is large and positive. In Eq. (12), is mainly included for illustration purposes, allowing also for large and negative and for an estimate of the exchange energy at .

In order to decouple the qubit subspace , we first apply a unitary transformation to that diagonalizes exactly. Then we perform a third-order Schrieffer-Wolff transformation that provides corrections up to the third power in the electron-phonon coupling, which is sufficient for the analysis of one- and two-phonon processes. The resulting effective Hamiltonian can be written as in the interaction representation, where the time is denoted by to avoid confusion with the tunnel coupling. Introducing the effective magnetic fields and and defining as the vector of Pauli matrices for the - qubit,

 Hq=12gμBBeff⋅σ′ (14)

describes the qubit and

 Hq−ph(τ)=12gμBδB(τ)⋅σ′ (15)

describes the interaction between the qubit and the phonons. The time dependence results from

 Hq−ph(τ)=eiHphτ/ℏHq−phe−iHphτ/ℏ. (16)

For convenience, we define the basis of such that . Following Refs. (20); (31), the decoherence time (), the relaxation time (), and the dephasing contribution () to of the qubit can then be calculated via the Bloch-Redfield theory (see also Appendix E), which yields

 1T2=12T1+1Tφ, (17) 1T1=J+xx(ωZ)+J+yy(ωZ), (18) 1Tφ=J+zz(0), (19)

where and

 J+ii(ω)=g2μ2B2ℏ2∫∞−∞cos(ωτ)⟨δBi(0)δBi(τ)⟩dτ. (20)

The correlator is evaluated for a phonon bath in thermal equilibrium and depends strongly on the temperature .

### iii.2 Input parameters

The material properties of GaAs are , , , , and (see also Appendix B.6.1),(32); (33); (34) ,(33); (35); (34) and .(36); (37) In agreement with ,(16) we set , which is the confinement length of the QDs due to harmonic confining potential in the - plane. For all basis states, the orbital part along the axis is described by a Fang-Howard wave function (38) of width (see Appendix A). Unless stated otherwise, we set and ,(40); (39); (41) where is consistent with the assumed (see also Appendix I).(41) We note, however, that adapting to is not required, because changing the width of the 2DEG by several nanometers turns out not to affect our results. All calculations are done for ,(12); (6) , in good agreement with, e.g., Refs. (12); (16), and an interdot distance of . For Figs. 15 (large ), we use , , and .(29) We choose here such that the resulting energy splitting between the qubit states is mostly determined by the hyperfine coupling at , as commonly realized experimentally.(4); (16) The detuning is then set such that and , and we note that this splitting is within the range studied in Ref. (16).

### iii.3 Temperature dependence

Figures 13 consider applied along the axis that connects the two QDs, assuming that the axis coincides with the crystallographic direction. The geometry is realized in most experiments,(13); (15); (17) particularly because GaAs cleaves nicely along [110]. In stark contrast to previous theoretical studies of phonon-limited lifetimes, where ,(20); (42); (43); (44); (45) Fig. 2(a) reveals at considered here, which implies . In the discussion below we therefore focus on the details of the temperature dependence of . We note, however, that the contributions to and from one-phonon processes scale similarly with , and analogously for two-phonon processes. Defining () as the decoherence rate due to one-phonon (two-phonon) processes, Fig. 2(b) illustrates , and so . In the considered range of temperatures, we find . This behavior results from the fact that for our parameters, where is the Boltzmann constant. Therefore, the dominant terms in the formula for are proportional to Bose-Einstein distributions defined as

 nB(ω)=1eℏω/(kBT)−1 (21)

and may all be expanded according to , keeping in mind that the contributing to are evaluated at because of energy conservation. The time due to two-phonon processes smoothly changes its behaviour from at to with increasing temperature, where are constants. This transition is explained by the fact that, in the continuum limit, the rate corresponds to an integral over the phonon wave vector , where the convergence of this integral is guaranteed by the combination of the Bose-Einstein distribution and the Gaussian suppression that results from averaging over the electron wave functions. More precisely, the decay rate is obtained by integrating over the wave vectors of the two involved phonons. Due to conservation of the total energy, however, considering only one wave vector is sufficient for this qualitative discussion. For , we find that the dominating terms decay with due to factors of type

 fs(q)=e−(q2x+q2y)l2cnB(ωqs)[nB(ωqs)+1], (22)

where and are the projections of onto the and axis, respectively, and is the phonon energy. Whether the Bose-Einstein part or the Gaussian part from provides the convergence of the integral depends on , , and mainly , as the latter can be changed significantly. When the Gaussian part cuts the integral, due to the expansion that applies in this case. When affects the convergence of the integral, terms with higher powers of occur. The resulting temperature dependence is rather complex, but is usually well described by with for different ranges of [see Fig. 2(b)]. The temperature ranges for the different regimes are determined by the details of the setup and the sample. For the parameters considered here, a power-law approximation for yields mainly because of the dephasing due to two-phonon processes (see Figs. 2 and 3), which agrees well with the experimental data of Ref. (16).

Figure 3 shows the resulting temperature dependence of for different spin-orbit lengths. Remarkably, the calculation yields short even when SOI is completely absent. Keeping fixed by adapting the value of , one finds that decreases further with increasing SOI. As seen in Eq. (12), couples to the triplet states and . An important consequence of the resulting admixtures is that greater detunings are required in order to realize a desired . In Fig. 3, for instance, increases from (no SOI) to (, ). As explained below, increasing decreases the lifetimes because it enhances the effects of through reduction of the energy gap (see also Fig. 1).

### iii.4 Origin of strong dephasing

The results discussed thus far have revealed two special features of the phonon-mediated lifetimes of - qubits in biased DQDs. First, , as seen in Fig. 2(a). Second, the strong decay does not require SOI, as seen in Fig. 3. These features have not been observed in previous calculations for, e.g., spin qubits formed by single-electron(39); (20) or single-hole(42); (43) or two-electron(23) states in GaAs QDs, hole-spin qubits in Ge/Si nanowire QDs,(44) or electron-spin qubits in graphene QDs.(45) Therefore, we discuss the dominant decay mechanism for - qubits in DQDs in further detail and provide an intuitive explanation for our results.

Assuming again a large, positive detuning , with , and setting (no SOI), the states , , and of Eq. (12) are practically decoupled from the qubit. The relevant dynamics are then very well described by

 ˜H=⎛⎜ ⎜ ⎜⎝0δbB20δbB2V+−V−−√2t+P†S0−√2t+PS−ϵ+U−V−+˜P⎞⎟ ⎟ ⎟⎠+Hph, (23)

with , , and as the basis states and

 ˜P=PSR−PT. (24)

In the absence of SOI, the hyperfine interaction () is the only mechanism that couples the spin states and enables relaxation of the - qubit. We note that even when is nonzero the relaxation times are largely determined by the hyperfine coupling instead of the SOI for the parameters considered in this work. At sufficiently large temperatures, where , is negligible in the calculation of , leading to pure dephasing, . In addition, the matrix element turns out to be negligible for our parameters. Following Appendix G, we finally obtain

 1T2=1Tφ=2t4ℏ2(Δ′S)6∫∞−∞⟨˜P2(0)˜P2(τ)⟩dτ (25)

from this simple model, where

 Δ′S=√(U−V+−ϵ)2+8t2 (26)

corresponds to the energy difference between the eigenstates of type and (using ). We note that terms of type and must be removed from in Eq. (25), as the Bloch-Redfield theory requires to vanish (see also Appendix G).(46) In Fig. 4, we compare from Eq. (25) with derived from Eq. (12) for (see also Fig. 3), and find excellent agreement at where relaxation is negligible.

The above analysis provides further insight and gives explanations for the results observed in this work. First, Eq. (25) illustrates that dephasing requires two-phonon processes and cannot be achieved with a single phonon only. As dephasing leaves the energy of the electrons and the phonon bath unchanged, the single phonon would have to fulfill . However, phonons with infinite wavelengths do not affect the lifetimes, which can be explained both via [see Eq. (2)] and via the vanishing density of states at for acoustic phonons in bulk. Thus, in all our calculations, where is the relaxation rate due to one-phonon processes. Second, as discussed above, we find that the hyperfine interaction in combination with electron-phonon coupling presents an important source of relaxation in this system.(24) Third, the strong dephasing at large detuning results from two-phonon processes between states of type and . This mechanism is very effective because the spin state remains unchanged. Therefore, the dephasing requires neither SOI nor hyperfine coupling, and we note that Eq. (25) reveals a strong dependence of on the tunnel coupling and the splitting . Hence, the short in the biased DQD can be interpreted as a consequence of the Pauli exclusion principle. When the energy of the right QD is lowered (), the singlet state of lowest energy changes from toward , since the symmetric orbital part of the wave function allows double-occupancy of the orbital ground state in the right QD. The triplet states, however, remain in the (1,1) charge configuration. While this feature allows tuning of the exchange energy and readout via spin-to-charge conversion on the one hand,(4) it enables strong dephasing via electron-phonon coupling on the other hand: effectively, phonons lead to small fluctuations in ; due to Pauli exclusion, these result in fluctuations of the exchange energy and, thus, in dephasing. This mechanism is highly efficient in biased DQDs, but strongly suppressed in unbiased ones, as we show in Sec. IV and Appendix H.

### iii.5 Angular dependence

We also calculate the dependence of and on the angle between and the axis, assuming that . The results for and are plotted in Fig. 5. Remarkably, the phonon-induced lifetimes of the qubit are maximal when and minimal when . The difference between minimum and maximum increases strongly with the SOI, and for and we already expect improvements by almost two orders of magnitude. These features can be understood via the matrix elements of the effective SOI,(22); (23); (24)

 Ω=FSOI(a,lc)EZlDcos(θB−θ)+lRcos(θB+θ)lDlR, (27)

where () is the angle between (the axis) and the crystallographic axis [110], and is a function of and . From this result, we conclude that there always exists an optimal orientation for the in-plane magnetic field for which the effective SOI is suppressed and, thus, for which the phonon-mediated decay of the qubit state is minimal (comparing the lifetimes at fixed ). Remarkably, one finds for () that this suppression always occurs when (), independent of and . In the case where , the finite in our model results from admixtures with , as explained in Sec. III.4. Due to the hyperfine interaction, these admixtures also lead to finite . We wish to emphasize, however, that suppression of the effective SOI only results in a substantial prolongation of the lifetimes when the spin-orbit lengths are rather short, as the dominant decay mechanism in biased DQDs is very effective even at .

## Iv Regime of Small Detuning

All previous results were calculated for a large detuning . Now we consider an unbiased DQD, i.e., the region of very small . The dominant decay mechanism in the biased DQD is strongly suppressed at , where the basis states and are both split from by a large energy . Adapting the simple model behind Eq. (25) to an unbiased DQD yields

 8t4ℏ2(U−V+)6∫∞−∞⟨˜P2(0)˜P2(τ)⟩dτ (28)

as the associated dephasing time (see Appendix H for details). Comparing the prefactor with that of Eq. (25) results in a remarkable suppression factor below for the parameters in this work. As explained in Appendix H, this suppression factor may also be estimated via for fixed , where is the splitting between the eigenstates of type and at large and is the above-mentioned splitting at .

Consequently, the lifetimes and in the unbiased DQD are no longer limited by or , but by states with an excited orbital part (see Fig. 1). We therefore extend the subspace by the basis states , , , and , and proceed analogously to the case of large detuning (see Appendixes A and C for details). The asterisk denotes that the electron is in the first excited state, leading to an energy gap of compared to the states without asterisk. Setting , the orbital excitation is taken along the axis, because states with the excitation along turn out to have negligible effects on the qubit lifetimes. From symmetry considerations, states with the excited electron in the right QD should only provide quantitative corrections of the lifetimes by factors on the order of 2 and are therefore neglected in this analysis. The resulting temperature dependence of , and is shown in Fig. 6. The plotted example illustrates that two-phonon processes affect only at rather high temperatures when is small, leading to for a wide range of due to single-phonon processes. In stark contrast to the biased DQD, we find . Remarkably, the absolute value of is of the order of milliseconds, which exceeds the at large by 2–3 orders of magnitude. For , , and typical sample temperatures , we find that the lifetimes can be enhanced even further.

## V Conclusions and outlook

In conclusion, we showed that one- and two-phonon processes can be major sources of relaxation and decoherence for - qubits in DQDs. Our theory provides a possible explanation for the experimental data of Ref. (16), and we predict that the phonon-induced lifetimes are prolonged by orders of magnitude at small detunings and, when the SOI is strong, at certain orientations of the magnetic field. Our results may also allow substantial prolongation of the relaxation time recently measured in resonant exchange qubits.(47)

While the model developed in this work applies to a wide range of host materials, the resulting lifetimes depend on the input parameters and, thus, on the setup and the heterostructure. By separately neglecting the deformation potential coupling () and the piezoelectric coupling (), we find that the qubit lifetimes of Figs. 26 for GaAs DQDs are limited by the piezoelectric electron-phonon interaction, the latter providing much greater decay rates than the deformation potential coupling. Consequently, the phonon-limited lifetimes of singlet-triplet qubits may be long in group-IV materials such as Ge or Si,(48); (49); (50) where the piezoelectric effect is absent due to bulk inversion symmetry.

Essentially, there are two different schemes for manipulating singlet-triplet qubits in DQDs electrically. The first and commonly realized approach is based on biased DQDs and uses the detuning to control the exchange energy.(4) Alternatively, the exchange energy can be controlled by tuning the tunnel barrier(1) rather than the detuning. Our results suggest that the second approach is advantageous, as it applies to unbiased DQDs for which the phonon-mediated decay of the qubit state is strongly suppressed. In addition, one finds at very small detunings ,(28) which implies that not only but also at , where now stands for the average over some random fluctuations of . Therefore, singlet-triplet qubits in unbiased DQDs are also protected against electrical noise. The latter, for instance, turned out to be a major obstacle for the implementation of high-fidelity controlled-phase gates between - qubits.(6) Keeping in mind that two-qubit gates for singlet-triplet qubits may also be realized with unbiased DQDs,(7) we conclude that operation at with a tunable tunnel barrier is a promising alternative to the commonly realized schemes that require nonzero detuning. As single-qubit gates for - qubits correspond to two-qubit gates for single-electron spin qubits, the regime is also beneficial for many other encoding schemes.

###### Acknowledgements.
We thank Peter Stano, Fabio L. Pedrocchi, Mircea Trif, James R. Wootton, Robert Zielke, Hendrik Bluhm, and Amir Yacoby for helpful discussions and acknowledge support from the Swiss NF, NCCR QSIT, SNANO, and IARPA.

## Appendix A Basis States

We consider a GaAs/AlGaAs heterostructure that contains a two-dimensional electron gas (2DEG). Electric gates on the top of the sample induce a double quantum dot (DQD) potential that confines electrons and enables the implementation of a singlet-triplet qubit. Assuming that this spin qubit is based on low-energy states of two electrons in the DQD, we consider the four states of lowest energy,

 |(1,1)S⟩ = |Ψ+⟩|S⟩, (29) |(1,1)T+⟩ = |Ψ−⟩|T+⟩, (30) |(1,1)T0⟩ = |Ψ−⟩|T0⟩, (31) |(1,1)T−⟩ = |Ψ−⟩|T−⟩, (32)

two states with a doubly occupied quantum dot (QD),

 |(0,2)S⟩ = |ΨR⟩|S⟩, (33) |(2,0)S⟩ = |ΨL⟩|S⟩, (34)

and four additional states that feature one electron in a first excited orbital state,

 |(1∗,1)S⟩ = |Ψe+⟩|S⟩, (35) |(1∗,1)T+⟩ = |Ψe−⟩|T+⟩, (36) |(1∗,1)T0⟩ = |Ψe−⟩|T0⟩, (37) |(1∗,1)T−⟩ = |Ψe−⟩|T−⟩, (38)

as the basis in this problem. In the notation used above, the first and second index in parentheses corresponds to the occupation number of the left and right QD, respectively. The asterisk denotes that the electron in the QD is in the first excited state. The spin part of the wave functions consists of the singlet and the triplets , , and ,

 |S⟩ = |↑↓⟩−|↓↑⟩√2, (39) |T0⟩ = |↑↓⟩+|↓↑⟩√2, (40) |T+⟩ = |↑↑⟩, (41) |T−⟩ = |↓↓⟩, (42)

where () corresponds to an electron spin oriented along (against) the externally applied magnetic field, see Appendix B.

As the two minima in the DQD potential may be approximated by the confining potential of a 2D harmonic oscillator, the one-particle wave functions for ground and first excited states can be constructed from the eigenstates of the harmonic oscillators.(28) Defining the growth axis of the heterostructure as the axis, we consider harmonic confinement potentials around with as the confinement length in the QDs. The axis connects the two QDs, pointing from the left to the right one. The interdot distance is , is the effective mass of electrons in GaAs, and is the orbital level spacing in each QD. With these definitions, the orbital parts of the 2D harmonic oscillator wave functions (ground, excited along , excited along ) can be written as

 ϕL,R(x,y) = 1√πlce−[(x±a)2+y2]/(2l2c), (43) ϕxL,R(x,y) = √2πl4c(x±a)e−[(x±a)2+y2]/(2l2c), (44) ϕyL,R(x,y) = √2πl4cye−[(x±a)2+y2]/(2l2c). (45)

The confining potential along the axis may be considered as a triangular potential of type

 V(z)={∞,z<0,Cz,z>0, (46)

where is a positive constant with units energy/length and corresponds to the interface between AlGaAs () and GaAs (). The ground state in such a potential can be approximated by the Fang-Howard wave function,(38)

 ϕFH(z)=θ(z)z√2a3ze−z/(2az), (47)

with as a positive length and

 θ(z)={0,z<0,1,z>0, (48)

as the Heaviside step function. The Fang-Howard wave function from Eq. (47) is normalized and fulfills

 ⟨ϕFH|z|ϕFH⟩=3az, (49)

which may be interpreted as the width of the 2DEG.

Following Refs. (29); (28); (51) for constructing wave functions in the DQD potential, we define overlaps between the harmonic oscillator wave functions,

 s = ⟨ϕL|ϕR⟩=e−a2l2c, (50) sx = ⟨ϕxL|ϕxR⟩=s(1−2a2l2c), (51) sy = ⟨ϕyL|ϕyR⟩=s, (52)

and

 g = 1−√1−s2s, (53) gx = 1−√1−s2xsx, (54) gy = 1−√1−s2ysy=g. (55)

Then the normalized orbital parts of the one-particle wave functions for the DQD are

 ΦL,R(r) = ϕL,R(x,y)−gϕR,L(x,y)√1−2sg+g2ϕFH(z), (56) Φe,xL,R(r) = ϕxL,R(x,y)−gxϕxR,L(x,y)√1−2sxgx+g2xϕFH(z), (57) Φe,yL,R(r) = ϕyL,R(x,y)−gϕyR,L(x