# Decoherence of electron spin qubit during transfer between two semiconductor
quantum dots at low magnetic fields

Jan A. Krzywda<EMAIL_ADDRESS>$\langle aQa^{L}\rangle$
Applied Quantum Algorithms, Lorentz Institute and Leiden Institute of Advanced
Computer Science, Leiden University, P.O. Box 9506, 2300 RA Leiden, The
Netherlands  Łukasz Cywiński  Institute of Physics, Polish Academy of
Sciences, al. Lotników 32/46, PL 02-668 Warsaw, Poland

###### Abstract

Electron shuttling is one of the currently pursued avenues towards the
scalability of semiconductor quantum dot-based spin qubits. We theoretically
analyze the dephasing of a spin qubit adiabatically transferred between two
tunnel-coupled quantum dots. We focus on the regime where the Zeeman splitting
is lower than the tunnel coupling, at which interdot tunneling with spin flip
is absent, and analyze the sources of errors in spin-coherent electron
transfer for Si- and GaAs-based quantum dots. Apart from the obvious effect of
fluctuations in spin splitting in each dot (e.g., due to nuclear Overhauser
fields) leading to finite $T_{2}^{*}$ of the stationary spin qubit, we
consider effects activated by detuning sweeps aimed at adiabatic qubit
transfer between the dots: failure of charge transfer caused by charge noise
and phonons, spin relaxation due to enhancement of spin-orbit mixing of
levels, and spin dephasing caused by low- and high-frequency noise coupling to
the electron’s charge in the presence of differences in Zeeman splittings
between the two dots. Our results indicate that achieving coherent transfer of
electron spin in a $10\,\mu$m long dot array necessitates a large and uniform
tunnel coupling, with a typical value of $2t_{c}\gtrsim 60\,\mu$eV.

##  I Introduction

Coherent coupling of qubit registers separated by a few microns is necessary
for the development of scalable quantum computers based on semiconductor
quantum dots [1, 2, 3]. Moving a spin qubit across such a distance, i.e.,
coherent shuttling of a single electron or hole spin, is one of the possible
solutions. Such shuttling can be realized by placing the electron inside a
moving potential generated using surface acoustic waves [4, 5] or metallic
gates [6, 7, 8, 9, 10]. The latter approach has recently allowed for long-
distance charge shuttling [8] and coherent spin shuttling on distances of a
few hundred nanometers [9, 10] in Si/SiGe structures.

Another approach, possible when one-dimensional chains of many tunnel-coupled
quantum dots are connecting the registers, relies on sequential adiabatic
transfer between neighboring quantum dots [11, 12, 13, 14, 15]: the detuning
between two tunnel-coupled quantum dots is changed slowly enough for the
electron to move adiabatically from one dot to another, see Fig. 1(a) and (b).
This method of charge transfer was realized up to four dots in GaAs, in which,
however, significant dephasing was reported [16]. More recently, in Si/SiGe
quantum dots, a successful transfer of charge across 8 dots [11] was shown,
followed by the transfer of a spin eigenstate across a few dots [14], and the
transfer of a qubit in a superposition state between the two [13] and three
[15] quantum dots.

In this paper, we analyze the sources of error in coherent spin qubit transfer
between two tunnel-coupled quantum dots, focusing on Si- and GaAs-based
systems. We consider the spin relaxation and dephasing caused by electron-
phonon interaction, high- and low-frequency charge noise, and hyperfine
interaction with nuclei, with special attention devoted to mechanisms that are
activated by driving the systems through an anticrossing of energy levels
corresponding to electron states localized in the two dots. We work in the
regime of low magnetic fields, defined by requiring the qubit Zeeman
splitting, $E_{Z}$, to be smaller than the gap at the tunneling-induced
anticrossing, $E_{Z}<2t_{c}$, so that there are only two anticrossings (one
per spin direction) in the spectrum, see Fig. 1(b). For higher $E_{Z}$,
additional anticrossings between states with opposite spin appear due to spin-
orbit interaction, making the spin-flip tunneling coupling finite [17, 18, 19,
20, 21, 22]. As it is very hard to achieve perfectly adiabatic (i.e., without
spin-flip) qubit transfer through these anticrossings, interference effects
associated with traversing multiple anticrossings are expected to strongly
affect both the probability of charge transfer [23] and the spin coherence of
the transferred qubit [20]. Optimizing the driving of the two-dot system can
counteract these effects [20], but performing such an optimization for a chain
of many QDs is incompatible with the scalability of such an approach to long-
distance quantum information transfer [7].

Figure 1: Schematic illustration of the paper. a) The mechanics of dot-to-dot
transition, with the relevant energy scales. The arrows indicate the order of
the transition from negative detuning, through avoided crossing at
$\epsilon\\!=\\!0$ up to the point where $\epsilon\gg 2t_{c}$. b) Energy
spectrum of the four-level dot-to-dot transition. Orbital ground states are
denoted using solid, and orbital excited states are denoted using dashed line
for spin-up (red) and spin-down (blue). b) A schematic illustration of the
considered mechanisms contributing to coherent transfer error $Q$: (1) the
Landau-Zener mechanism of leaving the electron behind in the left dot due to
finite sweep rate $v$, (2) charge transfer error, the process of excitation
near the anticrossing due to interaction with environmental noise, followed by
relaxation (inelastic tunneling) at large $\epsilon$, (3) spin relaxation
enhanced by spin-orbital mixing of states near the anticrossing, (4) spin
dephasing due to process (2) activated by difference of Zeeman splittings in
the two dots, $\Delta E_{Z}$, and (5) dephasing caused by low-frequency noise
in Zeeman splitting in each dot, and noise in $\epsilon$ when $\Delta E_{Z}$
is non-zero. c) Scaling of these contributions to error $Q$ with the detuning
sweep rate $v$, shown in log-log scale. At $v$, total error $Q$ is dominated
by the infidelity of charge transfer resulting from the interplay between the
Landau-Zener process (orange line and Eq. 9) and incoherent charge transition
to higher energy state, followed by possible relaxation (blue line and Eq.
20). At low $v$ the error is dominated by the spin relaxation (red line and
Eq. (22)), and loss of coherence between spin eigenstates due to low-frequency
noise (pink line and Eqs (29) and (30)), or random time of inelastic tunneling
from excited to the ground level (gray line and Eq. (24)). In the regime of
small error $Q\ll 1$ the total error (green line in c)) is a sum of all
contributions.

We thus focus on a more robust case of low magnetic fields, while taking into
account the influence of high- and low-frequency noises (phonons and charge
noise) on the dynamics of the transferred qubit. The fact that some of the
error mechanisms discussed here become suppressed with lowering of the $B$
field provides additional motivation for focusing on low $B$. For analogous
reasons, we assume that the valley splitting in each of the Si-based quantum
dots, $E_{VS}$, is sufficiently large [24], i.e., fulfills
$E_{VS}>2t_{c}>E_{Z}$, so that the transfer of an electron initialized in the
valley ground state can be treated, to a very good approximation, as if the
excited valley state was absent.

Let us stress that since the long-term motivation for the research presented
here is achieving coherent qubit transfer across approximately 100 quantum
dots, we use rather moderate values of tunnel coupling, $2t_{c}<100$ $\mu$eV.
For larger values, coherent dot-to-dot transfer was shown experimentally [13,
12], but having such strong and uniform couplings might be hard to maintain
for all pairs of dots in a longer array of them [7].

We analyze in detail five different mechanisms of errors in coherent spin
transfer, which are depicted in Fig. 1(b) and (c). Besides the Landau-Zener
mechanism (1), and the charge transfer error due to interaction with the
environment [23, 25] (2), we discuss mechanisms activated by the finite
coupling between orbital and spin degrees of freedom, in the form of spin
relaxation (3) [26] and spin dephasing (4, 5). The latter is activated by a
nonzero difference in Zeeman spin splitting between the dots, $\Delta E_{Z}$,
due to the gradient of the magnetic field, or differences in dot $g$-factors,
the presence of which introduces dephasing when time spent in each of the dots
becomes random [27, 7]. As we show, this process can take place as a result of
inelastic transfer to an excited state, followed by relaxation (mechanism 4 in
Fig. 1(b,c)), but also during fully adiabatic transfer due to the presence of
low-frequency 1/f noise in detuning [28, 29] (mechanism 5). The dependence of
all the contributions to the phase error of the shuttled qubit, $Q$, on the
detuning sweep rate, $v$, is presented schematically in Fig. 1(c). One can see
that depending on their relative importance, $Q(v)$ can be quite nontrivial:
there can be two local minima, and both $Q\propto 1/v$ and $Q\propto 1/v^{2}$
scalings with lowering $v$ are possible.

The paper is structured as follows: after using Sec. II to introduce the
description of the four-level adiabatic transition, the adiabatic master
equation used throughout the paper, and the model of DQD dots, in Sec. III, we
briefly introduce the most relevant sources of coherent transfer error: charge
transfer error (Sec. III.1), spin relaxation (Sec. III.2), and spin dephasing
due to charge noise and fluctuating spin splittings (Sec. III.3). This is
followed by Sec. IV, in which we compare the results for Si- and GaAs-based
double quantum dots. The paper is concluded by Sec. V, in which we discuss the
implications of the results for mid-range coherent transfer.

##  II The Model

###  II.1 Electron spin qubit in a double quantum dot

We consider ground orbital states of two tunnel coupled quantum dots,
$\ket{L}$ and $\ket{R}$. In presence of magnetic field, each of them splits
into a spin doublet with dot-dependent Zeeman splittings $E_{Z,L}$ and
$E_{Z,R}$, see Fig. 1(a). Without the spin-orbit interaction the double-dot
Hamiltonian is spin-diagonal:

$\hat{H}_{0}(t)=t_{c}\hat{\tau}_{x}+\frac{\epsilon(t)}{2}\hat{\tau}_{z}+\frac{E_{Z}}{2}\hat{\sigma}_{z}+\frac{\Delta
E_{Z}}{4}\hat{\tau}_{z}\hat{\sigma}_{z}.$ (1)

where $\hat{\tau}_{z}=\outerproduct{L}{L}-\outerproduct{R}{R}$,
$\hat{\tau}_{x}=\outerproduct{L}{R}+\outerproduct{R}{L}$, $\Delta
E_{Z}=E_{Z,L}-E_{Z,R}$, $E_{Z}=\tfrac{1}{2}(E_{Z,L}+E_{Z,R})$, and
$\epsilon(t)$ is the time-dependent interdot energy detuning. The
eigenenergies can be written as:

$E_{sq}(t)=\frac{\sigma_{s}}{2}E_{Z}+\frac{q}{2}\Omega_{s}(t),$ (2)

where $\sigma_{s}=\pm 1$ for spin eigenstates $s=\uparrow,\downarrow$, and
$q=\pm\,1$ for excited and ground orbital states respectively. The spin-
dependent orbital gap is given by

$\Omega_{s}(t)=\sqrt{(\epsilon(t)+\sigma_{s}\Delta
E_{Z}/2)^{2}+(2t_{c})^{2}}.$ (3)

Next, we add the coupling between spin and orbital degrees of freedom due
synthetic or intrinsic spin-orbit interaction. In the dot basis used above it
can be written as [20]:

$\hat{V}_{\text{so}}=\frac{b_{\perp}}{2}\hat{\tau}_{z}\hat{\sigma}_{x}+\frac{\alpha_{\text{im}}}{2}\hat{\tau}_{y}\hat{\sigma}_{x}+\frac{\alpha_{r}}{2}\hat{\tau}_{y}\hat{\sigma}_{y},$
(4)

where the $b_{\perp}$ is caused by the difference of transverse magnetic
fields between the QDs due to a magnetic field gradient or a g-tensor
difference (synthetic spin-orbit interaction), and remaining terms are due to
intrinsic spin-orbit coupling.

We assume here $2t_{c}>E_{Z}$, in which states with opposite spins are always
separated in energy. The above spin-orbital coupling introduces the only a
weak mixing of the eigenstates $\ket{\psi_{sq}(t)}$ of Hamiltonian (1), such
that the instantaneous eigenstates of total Hamiltonian
${\hat{H}_{0}(t)+\hat{V}_{\mathrm{so}}}$ can be approximated as:

$|\psi_{sq}^{\prime}(t)\rangle=\ket{\psi_{sq}(t)}+\sum_{s^{\prime}q^{\prime}}\frac{\bra{\psi_{s^{\prime}q^{\prime}}(t)}\hat{V}_{\text{so}}\ket{\psi_{sq}(t)}}{E_{sq}(t)-E_{s^{\prime}q^{\prime}}(t)}\ket{\psi_{s^{\prime}q^{\prime}}(t)},$
(5)

where $\ket{\psi_{sq}(t)}$ are the eigenstates of $\hat{H}_{0}(t)$ with energy
$E_{sq}(t)$ given in Eq. (2). Note that in considered regime of
$2t_{c}>E_{Z}$, denominator is non-zero for any time instant. This allows us
to neglect there modification of the $\hat{H}_{0}$ spectrum due to presence of
$\hat{V}_{\text{so}}$. Since
$\bra{\psi_{s^{\prime}q^{\prime}}}\hat{V}_{\text{so}}\ket{\psi_{sq}}\ll
2t_{c}-E_{Z}$, i.e. the contribution from $\hat{V}_{\text{so}}$ is relatively
small, we use it only to dress the spin-diagonal states
$|\psi_{sq}^{\prime}(t)\rangle$ in Eq. (5), for which now the prime will be
skipped from now.

###  II.2 Interdot transfer of the spin qubit

We assume the initial spin state is an equal spin superposition in a ground
orbital state of the left dot, i.e. the lowest energy orbital eigenstate of
$\hat{H}$ at $-|\epsilon_{0}|\\!\ll\\!-|t_{c}|$, which is well approximated by

$\ket{\Psi(0)}=\frac{|\psi_{\uparrow-}^{\prime}(0)\rangle+|\psi_{\downarrow-}^{\prime}(0)\rangle}{\sqrt{2}}\approx\ket{L}\otimes\frac{\ket{\uparrow}+\ket{\downarrow}}{\sqrt{2}}\,\,.$
(6)

The transfer of the qubit from L to R dot is driven by a linear sweep of
detuning with sweep rate $v$, i.e. $\epsilon(t)=-\Delta\epsilon/2+vt$, from
$t\\!=\\!0$ to $t\\!=\\!t_{f}\\!=\\!\Delta\epsilon/v$, where $\Delta\epsilon$
is the detuning swing that causes the interdot chagre transfer. It is
important to note that the detuning sweep is symmetric with respect to
$\epsilon\\!=\\!0$, i.e. changed from $-\Delta\epsilon/2$ to
$\Delta\epsilon/2$.

We concentrate on a single figure of merit: the coherence between two lowest
in energy instantaneous states at the final time, $t_{f}$, which for
$2t_{c}\\!<\\!E_{Z}$ is simple the spin coherence of an electron in the ground
orbital state localized in the $R$ dot:

$W(t)=2\langle\psi_{\uparrow-}^{\prime}(t_{f})|\hat{\varrho}(t_{f})|\psi_{\downarrow-}^{\prime}(t_{f})\rangle\,\,,$
(7)

where $\ket{\psi_{s-}^{\prime}(t_{f})}$, with $s=\uparrow,\downarrow$ are two
lowest-energy instantaneous eigenstates of $\hat{H}(t_{f})$, previously
defined in Eq. (5). For $\epsilon(t_{f})\\!\gg\\!|t_{c}|$ they are very well
approximated by $\ket{R}\otimes\ket{s}$, i.e. the Zeeman doublet of states of
qubit located in the $R$ dot. In the above equation $\hat{\rho}(t_{f})$ is the
density matrix of the qubit in the double quantum dot. For a closed four-level
system, it is given simply by
$\hat{\rho}(t_{f})=\ket{\Psi(t_{f})}\bra{\Psi(t_{f})}$, where
$\ket{\Psi(t_{f})}$ is the state that evolves from $\ket{\Psi(0)}$ from Eq.
(6) under influence of the time-dependent Hamiltonian, $\hat{H}(t)$. Note that
the factor of $2$ in Eq. (7) makes $W$ normalized in such a way that
$W\\!=\\!1$ corresponds to maximal overlap of the final state with
$\ket{R}\otimes(\ket{\uparrow}+\ket{\downarrow})/\sqrt{2}$, i.e. a perfect
transfer of spin qubit from one dot to another.

We define the transfer error $Q$ as the probability of all the events that do
not lead to a transfer of the electron between two quantum dots with its spin
coherence preserved:

$Q=1-|W(t_{f})|\,\,.$ (8)

According to this definition, failure of charge transfer (i.e. occupation of
$\ket{L}$ orbital state smaller than $1$ at the end of detuning sweep), spin
relaxation, and spin dephasing, all contribute to $Q$. Note that the modulus
in the above definition removes the deterministic phase acquired by the spin
superposition state during the interdot transfer of the qubit.

For a closed four-level system in regime of low magnetic fields
($E_{Z}\\!<\\!2t_{c}$), the spin dynamics during charge transfer can be
neglected, and the only source of transfer error $Q$ is simply the failure of
interdot charge transfer. We focus here on the adiabatic charge transfer
regime, $v\ll(2t_{c})^{2}$, in which the charge transfer error is
exponentially suppressed, as predicted by the Landau-Zener formula [30]:

$Q_{\text{LZ}}=\exp(-2\pi\frac{t_{c}^{2}}{v})\,\,,$ (9)

and shown as a yellow line in Fig. 1c. Note that we use unit in which
$\hbar\\!=\\!1$.

###  II.3 Dynamics in presence of coupling to environment

The physics of coherent transfer error $Q$ is thus rather trivial for a closed
system at low magnetic fields. However, a realistic description of interdot
transfer of a spin qubit should take into account the interaction with
environment relevant for semiconductor quantum dots: a bath of lattice
vibrations (phonons) causing transitions between electronic energy levels
[26], nuclear spins coupling directly to the spin [31], and an environment
containing sources of charge noise: Fermi sea of electrons in nearby metallic
gates (giving rise to Johnson-Nyquist noise), and an ensemble of two-level
systems with finite electric dipole moments causing the $1/f$ noise [32, 33,
34, 35].

Electric field noise leads to fluctuations of mean position of electron
localized in each of the dots. In presence of finite gradient of $z$ component
of the magnetic field, these lead to fluctuations in Zeeman splittings
$E_{Z,L(R)}$ in both QDs [28, 29]. Similar fluctuations are caused by hyperine
interaction with nuclei in each of the dots. We will include these processes
later in the analysis, and now we will focus on processes that are activated
by interdot tranfer of the electron caused by detuning sweep. These requite
the coupling of the environment to the charge states only:

$\hat{V}_{e}=\frac{\hat{V}_{\epsilon}}{2}\hat{\tau}_{z}+\frac{\hat{V}_{t_{c}}}{2}\hat{\tau}_{x},$
(10)

where the $\hat{V}_{\epsilon},\hat{V}_{t_{c}}$ operators acts on the
environmental degrees of freedom and represent quantum fluctuations in
detuning and tunnel coupling respectively. Such environment is typically
assumed to be in thermal equilibrium at low, but finite electron temperature
of $T\approx 100\,$mK such that $2t_{c}/k_{\text{B}}T>1$ [1]. The high-
frequency environmental noise causes inelastic transitions between adiabatic
states of Eq. (1) , which we model by Adiabatic Master Equation (AME) [36, 37,
38] of the form:

$\displaystyle\partial_{t}$
$\displaystyle{{\tilde{\varrho}}}(t)=-i\commutator{{\tilde{H}_{0}}(t)+\partial_{t}\hat{A}^{\dagger}(t)\hat{A}(t)}{{\tilde{\varrho}}(t)}+$
(11)
$\displaystyle-\sum_{q=\pm}\Gamma_{q,\text{spin}}(t)\bigg{(}{\tilde{\sigma}}_{q}{\tilde{\varrho}}(t){\tilde{\sigma}}_{q}^{\dagger}-\frac{1}{2}\anticommutator{{\tilde{\sigma}}_{q}^{\dagger}{\tilde{\sigma}}_{q}}{{\tilde{\varrho}}(t)}\bigg{)}$
$\displaystyle-\sum_{q=\pm,s=\uparrow\downarrow}\Gamma_{q,\text{charge}}^{(s)}(t)\bigg{(}{\tilde{\tau}}_{sq}{\tilde{\varrho}}(t){\tilde{\tau}}_{sq}^{\dagger}-\frac{1}{2}\anticommutator{{\tilde{\tau}}_{sq}^{\dagger}{\tilde{\tau}}_{sq}}{{\tilde{\varrho}}(t)}\bigg{)}\,\,,$

where all of the operators are written in the adiabatic frame,
${\tilde{O}}=\hat{A}(t)\hat{O}(t)\hat{A}^{\dagger}(t)$, i.e. in the basis
associated with the instantaneous states of $\hat{H}_{0}(t)$ from Eq. (1). By
definition, the transformation associated with an operator $\hat{A}(t)$
diagonalizes the Hamiltonian at every time instant:

${\tilde{H}}_{0}(t)=\hat{A}(t)\hat{H}_{0}(t)\hat{A}^{\dagger}(t)=\sum_{s=\uparrow,\downarrow}\frac{1}{2}\bigg{(}E_{Z}\sigma_{s}+\Omega_{s}(t){\tilde{\tau}}_{z}\bigg{)}\outerproduct{s}{s}.$
(12)

Its time-dependence generates the coherent coupling,
$\partial_{t}\hat{A}^{\dagger}(t)\hat{A}(t)$ between the orbital eigenstates,
which is responsible for non-adiabatic evolution that gives $Q_{\text{LZ}}$
from Eq. (9). The remainder of the error $Q$ is caused by the interaction with
the environment.

Firstly, we consider spin-diagonal transitions between the orbital states.
After transformation the operators
${\tilde{\tau}}_{s\pm}=|\tilde{\psi}_{s\pm}(t)\rangle\langle\tilde{\psi}_{s\mp}(t)|$
are responsible for such transitions between the eigenstate of energy
${E_{s\pm}=\sigma_{s}E_{Z}/2\pm\Omega_{s}/2}$ and the eigenstate with energy
${E_{s\mp}=\sigma_{s}E_{Z}/2\mp\Omega_{s}/2}$, respectively. With this process
we associate corresponding time-dependent rates
$\Gamma_{\pm,\text{charge}}^{(s)}(t)$ of excitation (relaxation) for spin-s.
Secondly we add transitions between the eigenstates that are adiabatically
linked to the opposite spins, i.e.
${\tilde{\sigma}}_{+}=|\tilde{\psi}_{\uparrow\pm}(t)\rangle\langle\tilde{\psi}_{\downarrow\pm}(t)|$,
${\tilde{\sigma}}_{-}=|\tilde{\psi}_{\downarrow\pm}(t)\rangle\langle\tilde{\psi}_{\uparrow\pm}(t)|$,
with associated time-depedent transition rates $\Gamma_{+,\text{spin}}(t)$,
$\Gamma_{-,\text{spin}}(t)$. A more detailed derivation of AME can be found in
Appendix. A, while the charge and spin relaxation rates are computed in
Appendix. B.

Finally, in most experimentally relevant scenarios, spin shuttling is
performed in the presence of a low-frequency (slower than the timescale of a
single dot-to-dot charge transfer) fluctuations of electric fields and spin
splittings in both dots that translate to quasistatic noise in the parameters
of Hamiltonian $\hat{H}(t)=\hat{H}_{0}(t)+\delta\hat{H}$, where

$\delta\hat{H}=\delta
t_{c}\hat{\tau}_{x}+\frac{\delta\epsilon}{2}\hat{\tau}_{z}+\frac{1}{2}\bigg{(}\frac{\delta
E_{Z,L}}{2}\outerproduct{L}{L}+\frac{\delta
E_{Z,R}}{2}\outerproduct{R}{R}\bigg{)}\hat{\sigma}_{z}.$ (13)

We compute their influence on the qubit coherence by averaging results of Eq.
(11) over quasistatic and independent fluctuations of detuning
$\delta\epsilon$, tunnel coupling $\delta t_{c}$, and spin-splittings in the
left $\delta E_{Z,L}$ and in the right dot $\delta E_{Z,R}$. Together, the
total contribution to the error from both high-frequency and low-frequency
noise can be written as:

$Q=1-2\overline{\big{|}\langle\tilde{\psi}_{\uparrow-}(t_{f})|{\tilde{\varrho}}(t_{f})|\tilde{\psi}_{\downarrow-}(t_{f})\rangle\big{|}},$
(14)

where the horizontal line represents classical averaging over distribution of
$\delta t_{c}$, $\delta\epsilon$ and $\delta E_{Z,L/R}$.

###  II.4 Parameters for GaAs and Si quantum dots

Quantity | Si | GaAs
---|---|---
Tunnel coupling $2t_{c}$ | 40, 100 $\mu$eV
Range of detuning sweep $\Delta\epsilon$ | 0.5 meV
Temperature $T$ |  $100$ mK
Spin splitting $E_{Z}$ |  $30\,\mu$eV
Spin splitting difference $\Delta E_{Z}$ |  $0.001,1\,\mu$eV |  $0.1\,\mu$eV
Charge relaxation time $\Gamma_{-}(2t_{c})$ | $0.1\,\text{ns}^{-1}$ | $10\,\text{ns}^{-1}$
Dephasing time $T_{2}^{*}$ | 20 $\mu$s | 2 $\mu$s
Intrinsic SOC $|a|$ |  $0.1\,\mu$eV |  $1\,\mu$eV
Synthetic SOC $b_{\perp}$ |  $0,\,1\,\mu$eV
Table 1: Parameters used in the paper for isotopically purified Si devices and
GaAs spin qubit with nuclear spin noise estimation

In this paper, we take into consideration two contrasting and experimentally
relevant cases of isotopically purified Si and GaAs DQD devices. For both of
them the relaxation rates $\Gamma_{-,\text{charge}}^{(s)}(t)$,
$\Gamma_{-,\text{spin}}(t)$ are computed in Appendices B and C. In the polar
GaAs the dominant role in the charge transfer is played by relatively strong
coupling to piezoelectric phonons, which can improve charge transfer at low
sweep rates [25]. However due to inevitable presence of nuclear spins,
coherent transfer that takes longer than $T_{2}^{*}\approx 10$ns is impossible
without additional control. For this reason we consider here GaAs device with
a narrowed nuclear field [39]. In such case the effective dephasing time can
be extended by a few orders of magnitude, i.e. $T_{2,\text{GaAs}}^{*}\approx
2\mu$s, at the cost of continuous estimation of the nuclear field [40]. We
compare such a case against isotopically purified Si (with 800 ppm), with an
order of magnitude longer $T_{2,\text{Si}}^{*}=20\mu$s [41].

To model experimental scenario we assume the detuning and tunnel coupling are
affected by the charge noise of 1/f-type with a value of spectral density at
$f=1$Hz, i.e. $S(1\text{Hz})\approx(0.5)^{2}\mu$eV2/Hz [34, 28], that
translates to standard deviation of $\langle\delta\epsilon^{2}\rangle=5\mu$eV
and $\langle\delta 2t_{c}\rangle=0.5\mu$eV. The assumed here, significantly
weaker noise in tunnel coupling is consistent with a typical difference
between the lever arm of plunger and barrier gates [42]. We add the second
source of charge noise in form of Johonson noise from $R=50\Omega$ resistor.
For spin-splitting noise we assume fluctuations in the dots $\delta E_{Z,L}$
and $\delta E_{Z,R}$ are independent and their standard deviation can be
related to typically measured $T_{2}^{*}$, i.e. $\langle\delta
E_{Z,L}^{2}\rangle=\langle\delta E_{Z,R}^{2}\rangle=\sqrt{2}/T_{2}^{*}$. We
assume the $T_{2}^{*}$ is by the presence of the nuclei, or charge noise
affecting the Zeeman splitting in presence of B field gradient [43, 44].

In both devices we use two values of tunnel coupling, $2t_{c}=40$ and $100$
$\mu$eV, and set $E_{Z}=30\mu$eV, such that assumed condition $E_{Z}<2t_{c}$
is met. In Si we use two contrasting values of a difference in Zeeman
splitting between the dots i.e. $\Delta E_{Z}=1,10^{-3}\mu$eV, where the first
of them is more typical in presence of intentional gradient [45], while the
latter can be associated with a typical difference in g-factors $\delta
g/g<10^{-3}$ [46]. For the GaAs we keep typical $\Delta E_{Z}=0.1\mu$eV, but
compare intrinsic spin-orbit coupling $|a|=1\mu$eV, against more optimistic
$|a|=0.1\mu$eV. In Si we keep $|a|=0.1\mu$eV. To avoid interference between
two, intrinsic spin-orbit terms we assume $\Re{a}=0$. For both devices, we
compare intrinsic spin orbit coupling against synthetic spin-orbit coupling
$b_{\perp}$, produced by the longitudinal gradient of magnetic field, for
instance in vicinity of manipulation region. We use value of
$b_{\perp}\\!=\\!1\,\mu$eV, which corresponds to the gradient of the order of
$0.1\,$mT/nm and dots separation $d\approx 100\,$nm.

##  III Sources of coherent transfer error

We concentrate here on small errors, i.e. $Q\ll 1$ for which in the leading
order we have

$Q\approx Q_{\text{charge}}+Q_{\text{relax}}+Q_{\varphi}\,\,,$ (15)

with the total error separated into contributions due to charge transfer error

$Q_{\text{charge}}=(Q_{\uparrow}+Q_{\downarrow})/2,$ (16)

where $Q_{s}$ is the loss of occupation of s-spin eigenstate, i.e.
$Q_{s}=1-2|\langle\tilde{\psi}_{s-}(t_{f})|{\tilde{\varrho}}(t_{f})|\tilde{\psi}_{s-}(t_{f})\rangle|^{2}$,
the spin relaxation

$Q_{\text{relax}}\approx\frac{1}{2}\int_{0}^{t_{f}}\Gamma_{-\text{spin}}(t)\text{d}t,$
(17)

where $\Gamma_{-,\text{spin}}(t)$ is time-dependent spin-relaxation and pure
dephasing contribution

$Q_{\varphi}\approx\frac{1}{2}\langle\delta\varphi^{2}\rangle,$ (18)

where $\langle\delta\varphi^{2}\rangle$ is the square of random phase
$\delta\varphi$ averaged over realisations of low-frequency noise and random
relaxation times. Each of them will be analyzed separately in the remainder of
Sec. III and combined with the parameters corresponding to Si and GaAs devices
in Sec. IV

###  III.1 Charge transfer error $Q_{\text{charge}}$

The necessary condition for coherent communication is the high-fidelity charge
transfer, i.e. deterministic motion of the electron between the dots. With
sufficiently weak spin-orbit interaction and in the relevant regime of
$2t_{c}>E_{Z}$, the spin components can be treated as uncoupled, and hence in
the adiabatic regime (i.e. where $Q_{\text{LZ}}\ll 1$), the charge transfer
error follows the result of [25]:

$Q_{\text{charge}}\approx\int_{0}^{t_{f}}\,\text{d}t\,\Gamma_{+}(\Omega[t])e^{-\int_{0}^{t}\Gamma_{+}(\Omega[t^{\prime}])+\Gamma_{-}(\Omega[t^{\prime}])\text{d}t^{\prime}}.$
(19)

The above equation can be derived from the AME from Eq. (11)), with the
unitary part neglected, i.e. in the limit of $(2t_{c})^{2}\gg v$. At thermal
equilibrium excitation and relaxation rates are related by the detailed
balance condition
$\Gamma_{+}(\Omega[t])=\Gamma_{-}(\Omega[t])e^{-\beta\Omega[t]}$ [47], where
$\beta=1/k_{\text{B}}T$, so $\Gamma_{+}\\!\ll\\!\Gamma_{-}$ in most of the
parameter regime of interest.

When coupling to environment is sufficiently weak, i.e.
$\Gamma_{+}(2t_{c})t_{c}/v\ll 1$, one can consider only a single transition
from ground to excited state, that is followed by a possible relaxation at
finite detuning. This approach, termed Healed Excitation Approximation Limit
(HEAL) in Ref. [25], results in a approximate formula:

$Q_{\text{HEAL}}\approx\Gamma_{+}(2t_{c})\frac{\sqrt{4\pi
k_{\text{B}}Tt_{c}}}{v}e^{-\chi_{\text{HEAL}}}.$ (20)

which shows that if healing factor
${\chi_{\text{HEAL}}=\int_{0}^{t_{f}}\Gamma_{-}(t^{\prime})\text{d}t^{\prime}\ll
1}$, the charge transfer error can be reduced by increasing $v$ since
$Q\propto 1/v$, as larger $v$ reduces the time around avoided crossings,
limiting excitations. On the other hand, slower sweeps, for which
$\chi_{\text{HEAL}}>1$, enable correction of excitations through subsequent
relaxation. This implies that $Q_{\text{charge}}$ can start decreasing below a
certain $v$, as electrons in the excited state relax back to the ground state,
reaching the target dot through inelastic tunneling, see Fig. 2). The
subsequent sections will explore the extent to which such slow transfer of
charge is accompanied by spin dephasing and spin relaxation.

Figure 2: Charge transfer error $Q_{\text{charge}}$ as a function of sweep
rate $v$ for the GaAs (red) and Si (green) DQD devices described in Section
II.4 for two values of tunnel couplings $2t_{c}=40\,\mu$eV (filled dots) and
$2t_{c}=\,100\mu$eV (hollow dots). Dots correspond to the numerical solution
of Adiabatic Master Equation (11), and lines depict analytic formula for
$Q_{\text{HEAL}}$ (20) plus the contribution from Landau-Zener mechanism, i.e.
$Q_{\text{charge}}\approx Q_{\text{LZ}}+Q_{\text{HEAL}}$. We used the
relaxation rates computed in Appendix C.

###  III.2 Spin relaxation error $Q_{\text{relax}}$

In presence of finite spin-orbit interaction Eq. (4), the instantaneous
eigenstates are no longer spin diagonal, and even purely orbital character of
environmental coupling (See Eq. (10)), can induce transition between the two
lowest instantaneous eigenstates, which define the spin qubit.

In this limit, the most relevant contribution comes from the mixing between
$\ket{\psi_{\downarrow+}(t)}$ and $\ket{\psi_{\uparrow-}(t)}$ states around
the avoided crossing. Since $E_{Z}>k_{\text{B}}T$ we concentrate on spin
relaxation, i.e. inelastic transition from $|\psi_{\uparrow-}(t)\rangle$ to
$|\psi_{\downarrow-}(t)\rangle$, which has dominant contribution from:

$\displaystyle\Gamma_{\text{spin}}(\Omega[t])\approx\frac{|\langle\psi_{\uparrow+}(t)|\hat{V}_{\text{so}}|\psi_{\downarrow-}(t)\rangle|^{2}}{(\Omega(t)-E_{Z})^{2}}\Gamma_{-,\text{charge}}(E_{Z})$
(21)

see Appendix B for derivation.

The relaxation is upper bounded by its value at the avoided crossing. To
compute probability of the spin-flip around avoided crossing
$Q_{\text{relax}}$ from Eq. (17), we assume the relaxation rate is constant
and non-negligible only around avoided crossing. Since the electron spends
there a time period of the order of $2t_{c}/v$, we can estimate

$Q_{\text{relax}}\approx\left|\frac{aE_{Z}+2b_{\perp}t_{c}}{(2t_{c})^{2}-E_{Z}^{2}}\right|^{2}\frac{2t_{c}}{v}\,\Gamma_{-,\text{charge}}(E_{Z}).$
(22)

Note that while the contribution from intrinsic spin-orbit coupling, $\propto
aE_{Z}$, vanishes as $E_{Z}$ goes to zero due to Van Vleck cancellation [48,
49], the contribution from the synthetic spin-orbit coupling, $\propto
b_{\perp}t_{c}$, does not exhibit this effect, as discussed in [50]. In Fig. 3
we compare the above simple formula against solution of AME in the limit of
effectively zero temperature, i.e. when $\Gamma_{+}=0$, such that the dots
represents numerically simulated probability of spin relaxation during the
transfer. On the x-axis we mark two tunnel couplings considered in this paper
$2t_{c}=40,100\mu$eV.

The figure indicates a significantly higher predicted probability of spin-
relaxation in GaAs, compared to Si, attributed to larger intrinsic spin-orbit
coupling (approximately an order of magnitude, $a=1\mu$eV) and a much faster
relaxation rate at avoided crossings due to piezoelectric coupling. Even
outside the manipulation region (red, with $b_{\perp}=0$), the resulting
$Q_{\text{relax}}\geq 10^{-3}$ at $v=5\mu$eV/ns poses a serious threat to
coherent transfer in GaAs. Conversely, the intrinsic spin-orbit interaction in
Si (green) induces an effectively negligible error, $Q_{\text{relax}}<10^{-5}$
at $2t_{c}>40\mu$eV. Furthermore, Eq. (22) indicates that the intrinsic SOC
contribution scales as $Q_{\text{relax}}\propto E_{Z}^{2}/t_{c}^{3}$,
suggesting potential reduction with increased tunnel coupling or decreased
magnetic field (Van Vleck cancellation).

However, if the dominant spin-orbit coupling contribution arises from the
transverse gradient of the magnetic field $b_{\perp}$, with $b_{\perp}=1\mu$eV
as an example corresponding to a gradient of $\sim 0.1$mT/nm and dots
separation of $d\approx 100$nm. In such a case the scaling becomes
$Q_{\text{relax}}\propto 1/t_{c}$. In GaAs (black), this contribution becomes
significantly different from the intrinsic case only at large $2t_{c}$ (due to
the lack of Van Vleck cancellation [48]). In Si (orange), it increases the
relaxation rate by about two orders of magnitude, resulting in
$Q_{\text{relax}}\approx 10^{-3}$ at $2t_{c}=40\,\mu$eV and $v=5\,\mu$eV/ns,
potentially becoming a significant contribution to transfer error. It’s
important to note that the error is expected to further increase as the tunnel
coupling approaches $E_{Z}$, eventually becoming sensitive to random
interference at $E_{Z}>2t_{c}$ due to additional avoided crossings.

Figure 3: Probability of spin-relaxation as a function of tunnel coupling for
two different sweep rates $v=5,50\,\mu$eV/ns (solid and hollow dots). We
compare numerical simulations of Adiabatic Master Equation (dots) against
theoretical predictions from Eq. (17) (lines) for GaAs (red) and Si (green)
devices, that are characterized by the relaxation rates from Appendix C. For
each devices we compare contribution from intrinsic and synthetic spin-oribt
coupling with the parameters given in Table 1. For the adiabatic master
equation, we initialize the system in the equal superposition and assume
effectively zero temperature, such that $\Gamma_{+}=0$. We confirm a good
agreement between numerical and analytical methods in the relevant region of
$2t_{c}>E_{Z}=30\,\mu$eV.

###  III.3 Spin dephasing $\langle\delta\varphi^{2}\rangle$

As a last source of error during coherent transfer we discuss pure dephasing
effects, caused by the uncontrolled, random evolution of the phase of electron
qubit during the transition. Main source of such randomness is non-zero dot-
dependent Zeeman splitting combined with a random time spend in the dots. It
is caused by a presence of both low- and high- frequency fluctuations of
electric field as well as the phonon induced transitions, and can be
parameterized by a random excess of time spent in one of the dots $\delta t$.
It translates into stochastic contribution to a relative phase between the
spin eigenstates:

$\delta\varphi\approx\Delta E_{Z}\delta t.$ (23)

For a symmetric sweep the electron on average spends the same amount of time
in each dot, i.e. $\langle\delta t\rangle=0$ and dephasing is related to non-
zero $\langle\delta t^{2}\rangle$. Note that even if a given mechanism leads
to finite $\langle\delta t\rangle$, this simply leads to renormalization of
the deterministic phase acquired by the spin during the shuttling, while the
presence of nonzero $\langle\delta t^{2}\rangle-\langle\delta t\rangle^{2}$
leads to dephasing. The gradient of Zeeman splittings $\Delta E_{Z}$ often
originates from non-zero difference in g-factors between the two dots, i.e.
$\Delta E_{Z}=(\delta g/g)E_{Z}$, or alternatively the intentional gradient of
magnetic field. In both cases we expect $\Delta E_{Z}\ll 2t_{c}$, which allows
to treat $\Delta E_{Z}$ as small perturbation.

####  III.3.1 Non-adiabatic dephasing

As a first relevant mechanism leading to $\langle\delta t^{2}\rangle\neq 0$,
we consider the dephasing during charge relaxation between the eigenstates of
Fig. 1 b). They are caused by the energy exchange between the electron and the
environment, that is in resonance with the instantaneous orbital gap
$\Omega_{s}(t)$ from Eq. (3). In the limit of a single excitation at avoided
crossing, the contribution $\langle\delta\varphi^{2}\rangle$ corresponds to a
random period of time spent in the excited orbital state. This time period is
set by the time between the excitation and subsequent relaxation, which brings
the electron charge back to the ground state. To illustrate this mechanism, we
set the excitation probability to $P_{e}$ and the relaxation rate to a
constant $\Gamma_{-}$. In such a case the waiting time is given by the
distribution $p(\delta t)=\Gamma_{-}e^{-\Gamma_{-}\delta t}$, with a variance
$\langle\delta t^{2}\rangle=\Gamma_{-}^{-2}$. Directly from the above we have
[27]:

$\langle\delta\varphi^{2}\rangle_{\text{non-adiab}}=P_{e}\left(\frac{\Delta
E_{Z}}{\Gamma_{-}}\right)^{2},$ (24)

where $P_{e}$ is the probability of excitation from ground to excited state
around avoided crossing. In absence of coherent excitations (L-Z), it can be
estimated using Eq. (20) with $\chi_{\text{HEAL}}=0$ (before the relaxation),
such that $P_{e}\approx\Gamma_{+}(2t_{c})\sqrt{4\pi k_{\text{B}}Tt_{c}}/v$
[25]. This results in phase error that scales as

$\langle\delta\varphi^{2}\rangle_{\text{non-
adiab}}\propto\sqrt{2t_{c}k_{B}T}\frac{(\Delta
E_{Z})^{2}}{v\Gamma_{-}(2t_{c})}e^{-\beta 2t_{c}}$ (25)

with the two most easily tunable parameters.

In Fig. (4), we use dots to compare the results of AME (Eq.(11)) with
approximate expression (Eq. (25) for GaAs (a) and Si (b) at different $\Delta
E_{Z}$ values. The simple model assumes constant relaxation rate and spin-
splitting, likely overestimating AME due to relaxation and effectively lower
$\Delta E_{Z}$ around $\epsilon\ll 2t_{c}$. In GaAs, negligible dephasing at
$\Delta E_{Z}\\!\leqslant\\!0.1$ $\mu$eV is attributed to fast relaxation with
strong phonon coupling. Si exhibits lower relaxation rates, leading to longer
times in the excited state. For $\Delta E_{Z}\\!=\\!0.1\,\mu$eV (orange), the
error combined with losing charge ($Q_{\text{charge}})$ has a single minimum,
indicating significant phase loss during charge relaxation. Fig. 4 reveals a
region where dephasing during relaxation is effective inactive, and the error
is dominated by $Q_{\text{charge}}$. This is no longer true at slower sweeps,
where the dashed lines align quantitatively with AME at $v\\!<\\!50$
$\mu$eV/ns for GaAs and $v\\!<\\!10$ $\mu$eV/ns for Si.

Figure 4: Dephasing during inelastic tunneling due to random time spent in the
excited state as a function of sweep rate $v$ for GaAs (a) and SiGe (b)
devices, with relaxation rates from Appendix C. We show results for the tunnel
coupling of $2t_{c}=40\mu$eV and for different values of Zeeman splittings
difference $\Delta E_{Z}=10^{-1}\,\mu$eV, $1\,\mu$eV for GaAs and $\Delta
E_{Z}=10^{-3}\,\mu$eV, $10^{-2}\,\mu$eV amd $10^{-1}\,\mu$eV in Si. Dots
corresponds to solutions of Adiabatic Master Equation without averaging over
slow noise, while the dashed lines lines represents approximate formula Eq.
(24). For the case of Si and $\Delta E_{Z}=10^{-1}$ (green solid lines), the
analytical formula overestimates the error since $\Delta E_{Z}/\Gamma_{-}\geq
1$.

####  III.3.2 Adiabatic dephasing

Second mechanism leading to spin dephasing, takes place even if the evolution
is fully adiabatic, i.e. the electron stays in the ground orbital level during
dot-to-dot transfer. The dephasing is then caused by low-frequency noise in
the Hamiltonian parameters, which includes charge noise of a typical in a form
of 1/f noise contributing to small fluctuations of detuning
$\delta\epsilon(t)$ and tunnel coupling $\delta t_{c}(t)$, as well as the
fluctuating magnetic field due to presence of nuclear spins or charge noise
induced modifications of g-factor, which we model by a dot-dependent spin-
splitting noise: $\delta E_{Z,L},\delta E_{Z,R}$. Together, the above
fluctuations modify the energy splitting between the lowest energy eigenstates
$\omega_{\text{spin}}(t)=E_{\uparrow-}(t)-E_{\downarrow-}(t)$, that defines
the spin qubit energy splitting

$\omega_{\text{spin}}(t)=E_{Z}+\tfrac{1}{2}[\delta E_{Z,L}+\delta
E_{Z,R}-\Delta\Omega(t)],$ (26)

where $\Delta\Omega(t)=\Omega_{\uparrow}(t)-\Omega_{\downarrow}(t)$ is a
difference in spin-depedent orbital splitting, written explicitly as

$\displaystyle\Omega_{s}(t)=\Big{\\{}\big{[}\epsilon+\delta\epsilon+\tfrac{\sigma_{s}}{2}(\Delta
E_{Z}+$ $\displaystyle\delta E_{Z,L}-\delta E_{Z,R})\big{]}^{2}$
$\displaystyle+[2t_{c}+2\delta t_{c}]^{2}\Big{\\}}^{1/2}$ (27)

where $\sigma_{s}=\pm 1$ for $s=\uparrow,\downarrow$ respectively.

The random fluctuations of spin splitting, contribute to a random phase
$\delta\varphi$, since

$\varphi=\int_{0}^{t_{f}}\omega_{\text{spin}}(t)\,\text{d}s=\varphi_{0}+\delta\varphi,$
(28)

which after averaging over realisations of all four quasistatic processes
leads to dephasing $Q\approx\tfrac{1}{2}\langle\delta\varphi^{2}\rangle$.

Figure 5: Total coherent transfer error $Q$ as a function of sweep rate $v$
for two values of tunnel couplings $2t_{c}=40\,\mu$eV (filled dots) and
$2t_{c}=100\,\mu$eV (hollow dots), with results for a GaAs DQD in panel (a),
and for a Si DQD in panel (b). Dots connected by lines represent solutions of
the Adiabatic Master Equation averaged over quasistatic noises in spin
splitting and detuning, while the dashed and dotted lines are the analytical
predictions due to separate mechanisms. The pink lines are the depahsing error
from Eq. (29) due to $1/f$ noise in detuning having an amplitude of
$S_{1}=0.5$ $\mu$eV/$\sqrt{\mathrm{Hz}}$, marked by the corresponding $\Delta
E_{Z}$, and the blue lines are phase error due to spin-splitting fluctuations
in two QDs marked by corresponding $T_{2}^{*}$ values, Eq. (30). For Si we
also plot the phase error due to charge transfer error caused by high-
frequency noise, Eq. (24), as gray dashed lines, corresponding to two values
of $2t_{c}$. In both panels the colors of solid lines correspond to distinct
values of spin-orbit couplings (SOCs). For GaAs in (a) we plot the results for
intrinsic SOC $|a|=1\,\mu$eV in red, and for synthetic SOC $b=1\,\mu$eV ($a\ll
1\mu$eV) in black. In the numerical simulation of the full error we use use
$T_{2}^{*}\\!=\\!2$ $\mu$s and $\Delta E_{Z}=10^{-1}\mu$eV. For Si (b) we have
the intrinsic SOC $|a|=0.1\mu$eV, and we compare the AME results for synthetic
SOC $b\\!=\\!0$ (green) versus $b=1\,\mu$eV (yellow), $T_{2}^{*}=20\,\mu$eV
and $\Delta E_{Z}=10^{-3}\,\mu$eV. Influence of dephasing due to $1/f$ noise
in detuning for larger values of $\Delta E_{Z}$ on the total error can be
estimated by taking a maximum of solid line and the pink dashed line
corresponding to the relevant $\Delta E_{Z}$. The remaining parameters can be
found in Table 1.

Calculation of $\delta\varphi$ to the lowest order in fluctuating quantitites
$\delta\epsilon$, $\delta E_{Z,L(R)}$, and $\delta t_{c}$ is given in Appendix
D. Let us now discuss the results for $\langle\delta\varphi^{2}\rangle$ in two
limits, in which it is has a simple and physically transparent form. For
sufficiently large $\Delta E_{Z}$ the most relevant mechanism leading to non-
zero $\langle\delta\varphi^{2}\rangle$ are the fluctuations of dots energy
detuning $\delta\epsilon$, which are directly related to the electrical noise
and can be seen as a uncertainty in the exact position of the avoided
crossing. In such a case similarity to the previous section, a time period
spent in each dot is random, and hence $\langle\delta t^{2}\rangle$ becomes
non-zero (See Eq. 23). The value of $\delta\epsilon$ shifts initial and final
values of the detunings $\pm vt_{f}+\delta\epsilon$, which directly translates
into $\delta t\approx\delta\epsilon/v$, and hence:

$\langle\delta\varphi^{2}\rangle=\Delta
E_{Z}^{2}\frac{\langle\delta\epsilon^{2}\rangle}{v^{2}}.$ (29)

Note that due to symmetry reasons slow, quasistatic fluctuations in tunnel
coupling $\delta t_{c}$ are not expected to introduce any significant random
spin phase $\delta\varphi$. For more quantitative derivation of Eq. (29) and
effects of tunnel coupling noise see Appendix D. We highlight that for typical
amplitude of those fluctuations $\delta t_{c}\ll\delta\epsilon$, and in the
limit of $v\ll t_{c}^{2}$ significant modification of the coherent charge
excitations $Q_{\text{LZ}}$ are not seen in the numerical results (See [51]
for this effect).

In the limit of sufficiently low $\Delta E_{Z}$ the dephasing is dominated by
a standard inhomogenous broadening, caused by the fluctuations of spin-
splitting. Assuming fluctuations in the dots are independent $\langle\delta
E_{Z,L}\delta E_{Z,R}\rangle\\!=\\!0$ and $\langle
E_{Z,L}^{2}\rangle=\langle\delta E_{Z,R}^{2}\rangle=\sqrt{2}/T_{2}^{*}$, the
contribution to a random phase reads:

$\langle\delta\varphi^{2}\rangle=\bigg{(}\frac{\Delta\epsilon}{vT_{2}^{*}}\bigg{)}^{2},$
(30)

which measures the ratio of transfer time $\Delta\epsilon/v$ and typically
measured dephasing time $T_{2}^{*}$. The above result holds for the nuclar
spin dominated $T_{2}^{*}$, for which independence of $\delta E_{Z,L}$ and
$\delta E_{Z,R}$ is expected. In the alternative scenario, when $T_{2}^{*}$ is
dominated by the charge noise, e.g. g-factor fluctuations, the
$\langle\delta\varphi^{2}\rangle$ is expected to be slightly smaller (larger)
for typically weakly correlated (anti-correlated) spin-splitting noise [52,
53, 54].

For numerical test of approximate analytical formulas Eqs. (29) and (30) see
Appendix D.

##  IV Case study: GaAs and Si DQD devices

We conclude this paper with the case study of the coherent double-dot
transition in isotopically purified Si ($T_{2}^{*}=20\mu$s) and compare it
against GaAs with continuous estimation of the Overhauser field [39, 40]
($T_{2}^{*}=2\mu$s). The remaining parameters of the devices are discussed in
Section II.4. The main result of the paper is shown in Fig. 5, where we the
dots connected with solid lines mark the coherent transfer error of equal spin
superposition computed using the Adiabatic Master Equation and averaged over
low-frequency fluctuations (see Sec. II). For the two DQD devices, we use
transition rates from Appendix C and compute the transfer error $Q$ for the
combination of parameters $2t_{c}=40,100\,\mu$eV (filled and hollow dots
respectively), and a selection of relevant parameters: single-dot spin
coherence time $T_{2}^{*}=2\,\mu$s (GaAs with field narrowing),
$T_{2}^{*}=20\,\mu$s (Si), spin splitting gradient $\Delta E_{Z}$ between
$10^{-3}$ and $1$ $\mu$eV and various spin-orbit couplings. For GaAs, we
compare the relatively strong intrinsic $|a|=1\,\mu$eV (red) against spin-
orbit coupling originating from the magnetic field gradient close to the
manipulation region $b_{\perp}=1\mu$eV (black). For Si, we use the same value
of synthetic SOC (yellow) but use a smaller intrinsic $|a|=0.1\,\mu$eV
(green). We additionally add lines corresponding to analytical predictions,
dephasing noise related to spin splitting (blue, dashed), dephasing due to
detuning fluctuations (violet dashed), and charge excitation using single
excitation approximation limit (gray dashed). For comparison, we also add the
line corresponding to $T_{2}^{*}=20$ns without field narrowing in GaAs (dotted
blue line in a).

As expected, in the limit of large sweep rates, in both devices transfer error
$Q$ is limited by charge excitation due to the Landau-Zener mechanism.
Starting from the largest considered sweep rates, decreasing $v$ is initially
improving the transfer until another mechanism becomes dominant. Those
mechanisms, contrary to LZ, typically worsen the transfer as the transfer time
increases, i.e. as $v$ decrease. This gives rise to an interesting trade-off
mechanism we discuss below.

In GaAs, the subsequent mechanism following the reduction of $v$ is spin
relaxation induced by spin-orbit mixing near avoided crossings. Despite using
the same value of intrinsic and synthetic spin-orbit coupling (when the latter
is finite), a difference between the influence of the two couplings is
visible, in particular at large $2t_{c}=100\,\mu$eV, for which the intrinsic
one suffers from Van Vleck cancellation since $2t_{c}\gg E_{Z}=30\mu$eV (Red,
hollow dots in the figure). The best achievable in this scenario $Q=10^{-5}$
takes place at $v=10^{3}\,\mu$eV/ns and increases by the order of magnitude if
the SOC is of the intrinsic character. For smaller $2t_{c}=40\,\mu$eV one can
achieve $Q=5\times 10^{-3}$ around $v=10^{3}\,\mu$eV, with no visible
difference between the two SOCs. At slower sweeps the $Q$ is further
increasing, until the error becomes dominated by the charge noise (dashed
violet line), which changes the scaling to $Q\propto 1/v^{2}$.

Alternatively, at the slowest sweeps the error can be dominated by the spin
splitting fluctuations in the dots. The position at which it happens depends
on $T_{2}^{*}$ (see the dashed blue line for $T_{2}^{*}=2\,\mu$s). It means
that for GaAs without field monitoring ($T_{2}^{*}=20$ns) this mechanism can
become the dominant (see dotted blue line), and effectively create a single
minimum of $Q=10^{-4}$ at $2t_{c}=100\,\mu$eV and $Q=10^{-2}$ at
$2t_{c}=40\,\mu$eV.

In Si, a similar behaviour of a trade-off, between Landau-Zener excitations
and a mechanism of dephasing due to low-frequency noise, is expected at
$\Delta E_{Z}=1\mu$eV (pink dotted line in b). It means that close to large
longitudinal gradients (manipulation regions) the error might be not smaller
than $Q=10^{-5}$ at $2t_{c}=100\,\mu$eV and $Q=10^{-4}$ at $2t_{c}=40\,\mu$eV.
Away from that regions, where order of magnitude-wise $\Delta E_{Z}=10^{-3}$,
the shape of $Q$ is more complex.

The spin relaxation mechanism in Si becomes relevant only in regions with a
large transverse gradient (depicted by yellow lines with $b_{\perp}=1,\mu$eV).
In its presence, especially at larger $2t_{c}=100,\mu$eV, the shape of the $Q$
curve (the yellow line with hollow dots) resembles the GaAs case, exhibiting a
transition from Landau-Zener (L-Z) dominance at large $v$ to spin relaxation
dominance at small $v$. For all the other parameter sets, a transition occurs
to regime dominated by incoherent charge transfer error that is well described
in the SEAL approximation, marked by gray dashed lines. The interplay between
these two charge transfer errors results in a minimum, reaching as low as
$Q=10^{-7}$ at $v=10^{3},\mu$eV/ns with a larger tunnel coupling of
$2t_{c}=100\,\mu$eV, and $Q=10^{-4}$ at $v=200,\mu$eV/ns with smaller
$2t_{c}=40\,\mu$eV.

In contrast to GaAs, as $v$ is further decreased, $Q(v)$ curve can turn
downwards again. This occurs particularly in the absence of strong synthetic
SOC (green lines), and manifests at $v$ small enough to allow for relaxation
from the excited to the target state, i.e. the HEAL process from Sec. III.1.
This process can temporarily decrease the error for smaller $v$, until the
dephasing due to low-frequency noise in $\epsilon$ or spin splitting become
dominant at lowest $v$ due to $Q=1/v^{2}$ scaling. Depending on the exact
values of $T_{2}^{*}$ (blue line), charge noise amplitude and $\Delta E_{Z}$
(black dashed line), it can give rise to a second minimum of $Q$ at low $v$,
which happened in case of $2t_{c}=40\,\mu$eV (green line with hollow dots).
However, obtained minimum of $Q=5\times\,10^{-4}$ was slightly larger than the
one at large $v$, and is hardly visible at larger $2t_{c}=100\,\mu$eV.

##  V Discussion and summary

We now discuss implications of our results for coherent spin transfer through
a long ($\sim 1-10$ $\mu$m) quantum dot array. For the mechanism of charge
transfer error and spin relaxation, $Q$ is additive: the error accumulated
during transfer is a sum of the contributions from each dot-to-dot transfers,
i.e.

$Q_{N}=\sum_{n=1}^{N}Q_{\text{charge}}^{(n)}+Q_{\text{relax}}^{(n)},$ (31)

where $Q^{(n)}$ is the error accumulated during transition between $(n-1)$th
and $n$th dot. In the present context, single charge-transfer error even means
that the electron gets left behind in one of the dots when it should transfer
to the next one. However, note that in the simplest version of a scalable
shuttling scheme, in which a few global voltages are used to simultaneously
trigger the appropriate swings of interdot detunings in a long chain of QDs
[7], an electron that got left behind will start traveling in the opposite
direction in subsequent driving cycles. In a more complicated scheme it would
resume its motion in the correct direction along the QD chain. However, even
in the latter scenario, the delay associated with staying in a random dot
(with a random deviation of Zeeman splitting from the acerage value) for one
period of driving, will lead to appreciable dephasing if the delayed qubit.
For simplicity, we assume here that every charge transfer failure means an
irreversible loss of coherence of the shuttled qubit.

For a QD chain with uniform parameters, for which each pair of adjacent QDs
has $2t_{c}^{(n)}\approx 2t_{c}^{(0)}$, the overall transfer error can be
estimated as $Q_{N}=NQ^{(1)}$. Based on the result from Fig. 5, the largest
considered $2t_{c}=100\mu$eV would allow for coherent transfer, defined as
$Q^{(N)}<10^{-3}$, across $N=1000$ and $N=10-100$ dots in Si and GaAs
($|a|=1\,\mu$eV) respectively. With a distance between the dots of $d=100$nm
this corresponds to $L=100\mu$m and $L=1-10\mu$m respectively, which is
consistent with requirement posed by [2]. From our results for smaller
$2t_{c}$, it is clear that $2t_{c}=40\mu$eV would not allow for a transfer
across $L\\!=\\!10\mu$m even in a perfectly uniform QD chain. We predict that
the minimal tunnel coupling required for such a transfer is $2t_{c}\approx
60\mu$eV, for which high precision in choosing $v$ is required, while the
resulting fidelity depends on the details of experimental device, i.e.
temperature, $T_{2}^{*}$, and the amplitude and spatial correlations of charge
noise.

In a more realistic situation of QD chain with tunnel couplings distributed
with appreciable standard deviation around a mean value that is “large enough”
for low coherent transfer error, the transfer can be limited by a few weak
links, or even dominated by a single weakly coupled pair, i.e.
$Q_{N}\approx\text{max}(Q)$. Again, based on results from Fig. 5 this puts
constrains on the weakest tunnel coupling along the chain to $2t_{c}\approx
40\,\mu$eV as achievable error was slightly above $Q^{(1)}=10^{-3}$ in GaAs
and slightly below in Si. In such a case, the presence of second minimum at
low $v$ in Si can be in principle used to improve the transfer error by
adjusting the sweep rate for the whole transfer, or only when the qubit is
passing through the weak link. However the latter would pose a challenge for
the electronic control, and also require a characterization of the chain,
while the former would significantly slow down the electron transfer, exposing
the qubit to stronger dephasing due to local low-frequency spin splitting
fluctuations, i.e. errors described by Eq. (30).

The additivity of errors might no longer hold for dephasing caused by
spatiotemporally correlated low-frequency noise, due to possibly coherent sum
of random phases:

$\delta\varphi_{N}=\sum_{n=1}^{N}\delta\varphi^{(n)},$ (32)

which contribution to an error after averaging depends on the correlations
$\langle\delta\varphi^{2}\rangle=\sum_{nm}\langle\delta\varphi^{(n)}\delta\varphi^{(m)}\rangle$.
However for a linear chain, each dot is visited only once, which together with
only weak correlation of the noise in nearest neighbours
$\langle\delta\varphi^{(n)}\delta\varphi^{(n+1)}\rangle\ll\langle[\delta\varphi^{(n)}]^{2}\rangle$
is expected to introduce a small modification to linear scaling, such that for
uniform case of $\langle[\delta\varphi^{(n)}]^{2}\rangle\approx Q_{\varphi,1}$
we can estimate

$\langle\delta\varphi_{N}^{2}\rangle\approx\xi NQ_{\varphi,1}$ (33)

where we introduced constant numerical factor $\xi\gtrsim 1$ encapsulating
spatial correlation of charge noise between nearest and next-nearest
neighbours [53]. The above is the manifestation of motional narrowing process,
which in comparison to stationary system, decreases coherent error by the
virtue of interacting with many uncorrelated environments [55]. Such motional
narrowing leading to decreased dephasing of moving qubit compared to the
stationary one was observed in recent shuttling experiment with a moving QD in
Si/SiGe [9].

In view of latest experimental results in Ge holes [15], we highlight that
correlations of low-frequency noise are expected to be more relevant for
sequential transition between few quantum dots. Since in [15] tunnel coupling
$2t_{c}\approx 100\mu$eV, allowed for $v\approx 10^{3}\,\mu$eV/ns, no charge
excitations are expected there, and the error should come from dephasing noise
caused by very large $\Delta E_{Z}$ specific to holes in Ge [56, 57, 58].
However, in such a setup of shuttling back and forth between two or three
dots, the same dot is visited multiple times, meaning that even without
spatial correlations of the noise, slow fluctuations $\delta\varphi$ can add
coherently, which can result in $\langle\delta\varphi^{2}\rangle>N^{\alpha}$,
with $\alpha>1$, scaling. Moreover, repetition of a shuttling process could
create a noise filter that commensurate with shuttling frequency. Thus,
predicting exact relation between spectral density of the noise, electron path
and dephasing would require a separate, careful analysis outside of the scope
of this paper.

In summary, we have analyzed possibility of coherent electron transfer between
two tunnel-coupled semiconductor quantum dots. We analyzed contrasting
examples of spin qubits realised in GaAs and Si nanostructures. We have shown
that maintaining qubit coherence would benefit from using large detuning sweep
rates, that according to the Landau-Zener formula are enabled by maximising
the tunnel coupling. Slower sweeps activates a number of error mechanisms,
leading to charge excitation, spin relaxation, and spin dephasing. In the
process we have identified the origins of such errors in a form of inelastic
transition between the instantaneous levels, slow fluctuations of detuning
both activated by the dot-dependent Zeeman splitting, and spin relaxation
induced by the spin-orbit interaction. To provide qualitative result we have
modelled environment for the relevant case of Si and GaAs quantum dots. In
general, in both devices an optimal sweep rate was found, as a result of the
non-monotonic dependence of $Q$ on sweep rate $v$. Typically, the trade-off
resulting in a local minimum of $Q(v)$ is between the increasing probability
of losing the electron charge at high $v$, and spin relaxation or spin
dephasing, becoming more dangerous at low $v$.

In GaAs, longer transfer time expose the electron spin to the inhomogenous
dephasing associated with the $T_{2}^{*}$. In principle this issue can be
fixed when $T_{2}^{*}$ is enhanced, for instance by continuous estimation of
the nuclear field [39, 40, 59]. However, as we have shown in this paper, this
improvement is limited by another source of error: the spin-relaxation due to
non-negligible spin-orbit coupling, which dominates the error at lower $v$. We
have speculated that even for optimized DQD geometry, i.e. weaker spin-orbit
interaction, the error is unlikely to allow for coherent transfer at $\mu$m
distance.

In contrast, we have found that in Si-based devices the transfer error as a
function of $v$ exhibits more interesting structure, with up to two local
minima. The first of them has been already identified with the interplay
between the coherent and incoherent charge excitation, that result in leaving
the electron in the initial dot [23, 25]. Here we have found an increase of
the phase error due to charge noise activated in the presence of interdot
difference in Zeeman splittings, which has to be taken into account in case of
spin shuttling in vicinity of the manipulation region. We have reported that
away from this region, where the values of the spin splitting difference are
not enhanced, similar contribution is shifted to lower $v$, at which its
influence becomes comparable to that of low-frequency fluctuations of the
local spin splitting. For Si, this takes place at low sweep rates, at which
the charge transfer becomes inelastic and decreasing with $v$. As a result a
second minimum of an error at much lower $v$ can be created, however as we
have shown no significant improvement in comparison to the other minimum is
expected.

Concluding, we have analyzed the most relevant mechanism leading to error in
coherent electron spin qubit transfer between two Si- and GaAs- quantum dots.
We have found a trade-off behaviour between the mechanisms affecting charge
transfer and electron spin phase, as typically the latter one is lost for the
slowest sweeps. We have identified local minima of the error in coherent
transfer, and analyzed their implication for the coherent transfer across few
$\mu$m. We have proven that coherent shuttling using pre-defined quantum dots
in isotopically purified Si is more promising in comparison to GaAs, even in
presence of nuclear field narrowing in the latter. We have found that in
silicon, coherent transfer across few $\mu$m would require $2t_{c}>40\,\mu$eV
at the weakest link and $2t_{c}>60\,\mu$eV for majority of dots pair. This
requirement for hundred-dot array makes this method of transfer challenging
for future designs.

##  Appendix A Lindblad form of the Adiabatic Master Equation

In this appendix we derive Lindblad form of the Adiabatic Master Equation
provided in Eq. (11).

###  A.1 Instantaneous states, energies and adiabatic frame

We start by considering the spin-diagonal Hamiltonian

$\hat{H}(t)=t_{c}\hat{\tau}_{x}+\frac{\epsilon(t)}{2}\hat{\tau}_{z}+\frac{E_{Z}}{2}\hat{\sigma}_{z}+\frac{\Delta
E_{Z}}{4}\hat{\tau}_{z}\hat{\sigma}_{z}.$ (34)

where $2t_{c}$ is the tunnel coupling, $\epsilon(t)$ is the dots deutning,
$\Delta E_{Z}=E_{Z,L}-E_{Z,R}$ and $E_{Z}=\tfrac{1}{2}(E_{Z,L}+E_{Z,R})$ are
related to dot-dependent spin-splittings $E_{Z,L},E_{Z.R}$.

At any point in time, the above Hamiltonian can be diagonalized via the
rotation around the y-axis:
${\hat{A}(t)\\!=\\!\sum_{s=\uparrow,\downarrow}e^{-i\theta_{s}(t)\hat{\sigma}_{y}/2}\outerproduct{s}{s}}$
using spin-dependent orbital angle defined via
$\text{ctan}\,\theta_{s}(t)=[\epsilon(t)+\sigma_{s}\Delta E_{Z}/2]/2t_{c}$
where $\sigma_{s}=\pm 1$ are for spin-up and spin-down respectively. After
diagonalization we have:

${\tilde{H}}_{0}(t)=\hat{A}^{\dagger}(t)\hat{H}(t)\hat{A}(t)=\frac{1}{2}\sum_{s=\uparrow,\downarrow}\bigg{(}\Omega_{s}[t]{\tilde{\tau}}_{z}+\sigma_{s}E_{Z}\bigg{)}\outerproduct{s}{s},$
(35)

where $\Omega_{s}[t]=\sqrt{(2t_{c})^{2}+(\epsilon[t]+\sigma_{s}\Delta
E_{Z}/2)^{2}}$ is the spin-dependent orbital gap, while ${\tilde{\tau}}_{z}$
is the Pauli matrix in the adiabatic frame, in which only coefficients of the
system state vary in time, and the corresponding state vectors are time-
independent. In such an adiabatic frame we define the state
$|\tilde{\psi}(t)\rangle$ via the transformation:

$\displaystyle\ket{\psi(t)}$
$\displaystyle=\sum_{s}a_{s+}(t)\ket{\psi_{s+}(t)}+a_{s-}(t)\ket{\psi_{s-}(t)}$
$\displaystyle=\hat{A}(t)\bigg{[}\sum_{s}\tilde{a}_{s+}(t)\ket{\psi_{s+}(t_{0})}+\tilde{a}_{s-}(t)\ket{\psi_{s-}(t_{0})}\bigg{]}$
$\displaystyle=\hat{A}(t)|\tilde{\psi}(t)\rangle.$ (36)

When $|\tilde{\psi}(t)\rangle=\hat{A}^{\dagger}(t)\ket{\psi(t)}$ is
substituted to Schroedinger equation, we have
$i\partial_{t}\big{(}\hat{A}(t)|\tilde{\psi}(t)\rangle\big{)}=\hat{H}_{0}(t)A(t)|\tilde{\psi}(t)\rangle$,
which multiplied from the left by $\hat{A}^{\dagger}(t)$ produces the
Schroedinger equation in the adiabatic frame:

$i\partial_{t}|\tilde{\psi}(t)\rangle=\left({\tilde{H}}_{0}(t)-i\hat{A}^{\dagger}(t)\partial_{t}\hat{A}(t)\right)|\tilde{\psi}(t)\rangle={\tilde{H}}|\tilde{\psi}(t)\rangle.$
(37)

In the above equation we obtained the coupling between adiabatic levels,

$\hat{A}^{\dagger}(t)\partial_{t}\hat{A}(t)=\frac{1}{2}\sum_{s=\uparrow,\downarrow}\partial_{t}\theta_{s}(t){\tilde{\tau}}_{y}\outerproduct{s}{s},$
(38)

which for the linear drive $\epsilon(t)=vt$ reads
$\partial_{t}\theta_{s}=2vt_{c}/\Omega_{s}^{2}(t)$.

###  A.2 Adiabatic Bloch-Redfield equation

We now add environment in thermal equilibrium with environmental Hamiltonian
$\hat{H}_{e}$, i.e. $\hat{\varrho}_{e}=e^{-\beta\hat{H}_{e}}/\mathcal{Z},$
where $\mathcal{Z}=\text{Tr}\\{e^{-i\hat{H}_{e}}\\}$ and the inverse thermal
energy is $\beta=1/k_{B}T$. The environment couples only to the charge states
of the electron, i.e.

$\hat{V}_{e}=\tfrac{1}{2}\left(\hat{V}_{x}\hat{\tau}_{x}+\hat{V}_{z}\hat{\tau}_{z}\right)$
(39)

where $\hat{V}_{i}$ are the real operators acting on environmental degrees of
freedom only.

The von Neumann equation for the density matrix of the qubit-environment
system in the adiabatic frame reads

$\frac{\partial}{\partial
t}{\tilde{\varrho}}_{qe}(t)=-i\commutator{{\tilde{H}}(t)+{\tilde{V}}_{e}(t)+\hat{H}_{e}}{{\tilde{\varrho}}_{qe}(t)},$
(40)

where we have used the operators in the adaibatic frame i.e.
${\tilde{O}}(t)=\hat{A}^{\dagger}(t)\hat{O}\hat{A}(t)$. We now follow standard
treatment, i.e. perform Born-Markov approximation that leads to the Bloch-
Redfield equation in the interaction picture w.r.t.
${\tilde{H}}(t)+\hat{H}_{e}$, i.e.

$\displaystyle\frac{\partial}{\partial t}{\tilde{\varrho}}_{q,I}(t)=-\Tr_{e}$
$\displaystyle\left\\{\commutator{{\tilde{V}}_{e,I}(t)}{\int_{0}^{\infty}\text{d}s\,{\tilde{V}}_{e,I}(t-s){\hat{\varrho}}_{q,I}^{\prime}(t)\hat{\varrho}_{e}}\right\\}$
$\displaystyle\qquad+(h.c.),$ (41)

in which we have defined

${\tilde{O}_{I}(t)=e^{iH_{e}t}\hat{U}_{q}^{\dagger}(t,t_{0})\tilde{O}U_{q}(t,t_{0})e^{-iH_{e}t}},$

with
${\tilde{U}_{q}}(t,t_{0})=\mathcal{T}\exp{-i\int_{t_{0}}^{t}{\tilde{H}}(t^{\prime})\text{dt'}}$.

The Adiabatic Master Equation is now obtained by unwinding the interaction
picture only with respect to the system, i.e. multiplying from the left by
$\hat{U}_{q}(t,t_{0})$ and from the right by $\hat{U}_{q}^{\dagger}(t,t_{0})$,
which after simplifications gives:

$\displaystyle\frac{\partial}{\partial t}{\tilde{\varrho}}_{q}(t)=$
$\displaystyle-i\commutator{{\tilde{H}}(t)}{{\tilde{\varrho}}_{q}(t)}-\Tr_{e}\bigg{\\{}\bigg{[}{{\tilde{V}}_{e,I_{e}}}(t),\int_{0}^{\infty}\text{d}s$
$\displaystyle\Big{(}{\tilde{U}}(t,t-s){{\tilde{V}}_{e,I_{e}}}(t-s){\tilde{U}}^{\dagger}(t,t-s)\Big{)}\hat{\varrho}_{q}^{\prime}(t)\bigg{]}$
$\displaystyle+(h.c.)\bigg{\\}},$ (42)

where
${\tilde{V}}_{e,I_{e}}(t)=e^{i\hat{H}_{e}t}\hat{A}^{\dagger}(t)\hat{V}_{e}(t)\hat{A}e^{-i\hat{H}_{e}t}$
is the operator in the adiabatic frame and in the interaction picture with
respect to environmental Hamiltonian only.

We now perform the central approximation of adiabatic master equation, which
is related to a free-evolution operator inside the dissipative part, which is
approximated as:

$\tilde{U}(t,t-s)=\mathcal{T}\exp(-i\int_{t-s}^{t}{\tilde{H}}(t^{\prime}))\approx\exp(-i{\tilde{H}}_{0}(t)\,s).$
(43)

That is equivalent of saying that within the short correlation time of the
environment $s<\tau_{c}$, both coherent (adiabatic) coupling between the
adiabatic levels as well as a rotation of the adiabatic frame can be
neglected. The latter also means that we can write:

$\displaystyle\hat{U}(t,t-s)$
$\displaystyle{\tilde{V}}_{e,I_{e}}(t-s)\hat{U}^{\dagger}(t,t-s)$ (44)
$\displaystyle\approx
e^{-i{\tilde{H}}(t)\,s}\hat{A}^{\dagger}(t)\hat{V}_{e,I_{e}}(t-s)\hat{A}(t)e^{i{\tilde{H}}(t)\,s}$

where $\hat{V}_{e,I_{e}}(t)=e^{i\hat{H}_{e}t}\hat{V}_{e}e^{-i\hat{H}_{e}t}$.
It is now convenient to introduce the resolution of the unity
${\sum_{n}\outerproduct{n(t_{0})}{n(t_{0})}\approx\sum_{n}\tilde{\Pi}_{n}}$ in
terms of time-independent states in the adiabatic picture, use
$\hat{A}(t)\ket{n(t_{0})}=\ket{n(t)}$,
$\tilde{H}_{0}(t)\ket{n(t_{0})}=E_{n}\ket{n(t_{0})}$ and arrive at:

$\displaystyle\sum_{nm}$ $\displaystyle
e^{-i{\tilde{H}}(t)\,s}\tilde{\Pi}_{n}\hat{A}^{\dagger}(t)\hat{V}_{e,I_{e}}(t-s)\hat{A}(t)\tilde{\Pi}_{m}e^{i{\tilde{H}}(t)\,s}$
(45)
$\displaystyle=\sum_{nm}\bra{n(t)}V_{e,I_{e}}(t-s)\ket{m(t)}e^{-iE_{nm}s}\tilde{\Pi}_{nm}$

where $E_{nm}=E_{n}(t)-E_{m}(t)$ is the difference in the energies of the
adiabatic states $\ket{n(t)},\ket{m(t)}$ and
$\tilde{\Pi}_{nm}=\outerproduct{n(t_{0})}{m(t_{0})}$.

Together the adiabatic Master equation for the density matrix of the electron
state now reads:

$\displaystyle\frac{\partial}{\partial
t}{\tilde{\varrho}}_{q}(t)=-i\commutator{{\tilde{H}}(t)}{{\tilde{\varrho}}_{q}(t)}$
(46)
$\displaystyle-\sum_{nmlk}\commutator{{\tilde{\Pi}}_{nm}}{{\tilde{\Pi}}_{kl}{\tilde{\varrho}}_{q}(t)\int_{0}^{\infty}\text{d}s\,e^{-iE_{kl}[t]s}A_{nmkl}(s)}+(h.c.),$

in which we have defined

$\displaystyle A_{nmkl}(s)$
$\displaystyle=\Tr{\hat{V}_{e,I_{e}}^{nm}(t)\hat{V}_{e,I_{e}}^{kl}(t-s)\hat{\varrho}_{e}}$
$\displaystyle=\Tr{\hat{V}_{e,I_{e}}^{nm}(s)\hat{V}_{e,I_{e}}^{kl}(0)\hat{\varrho}_{e}}$
(47)

with $\hat{V}_{e,I_{e}}^{nm}(t)=\bra{n(t)}\hat{V}_{e,I_{e}}(t)\ket{m(t)}$ for
brevity. We now substitute
$\hat{V}_{e}=\frac{1}{2}\sum_{i=x,z}\hat{\tau}_{i}\hat{V}_{i}$ a which
convenient gives:

$A_{nmkl}(s)=\sum_{i=x,y}\tau_{i,nm}(t)\tau_{i,kl}(t)C_{i}(s),$ (48)

where $\tau_{i,nm}(t)=\bra{n(t)}\hat{\tau}_{i}\ket{m(t)}$ and
$C_{i}(s)=\Tr\\{\hat{V}_{i}(s)\hat{V}_{i}(0)\hat{\varrho}_{e}\\}$ is the
correlation function, in which we have assumed that two environmental
operators are statistically independent, i.e.
${\langle\hat{V}_{z}\hat{V}_{x}\rangle=\Tr_{e}\\{\hat{V}_{z}\hat{V}_{x}\hat{\varrho}_{e}\\}=0}$.
We finally express the correlation function in terms of spectral density of
the noise, i.e. $C_{i}(r)=\int_{-\infty}^{\infty}S_{i}(\omega)e^{-i\omega
r}\frac{\text{d}\omega}{2\pi}$, which substituted to the expression in the
commutator reads:

$\displaystyle\int_{0}^{\infty}\text{d}s\,e^{-iE_{kl}[t]s}C_{nmkl}(r)=\sum_{i=x,z}\frac{\tau_{i,nm}(t)\tau_{n,kl}(t)}{8}S_{i}(E_{lk}[t]),$
(49)

In the above, we have used the identity $\int_{0}^{\infty}e^{i\omega
t}\text{d}t=\pi\delta(\omega)+P(1/\omega)$ and ignored the principle value as
it generates only deterministic shift in the energy. In thermal equilibrium we
have $S(\omega)=S(-\omega)e^{-\beta\omega}$ where $\beta=1/k_{\text{B}}T$.

###  A.3 Local secular approximation

In the dissipative part of Eq. (46) there is a sum over four different
adiabatic states. However some of their combination include quickly
oscillating terms, which will not contribute strongly to the
$\tilde{\varrho}_{q}$. To see this mechanism we go back to the interaction
picture with respect to $\hat{H}_{0}$, i.e.:

$\displaystyle\frac{\partial}{\partial t}$
$\displaystyle{\tilde{\varrho}}_{q,I}(t)=-i\commutator{\hat{U}_{0}^{\dagger}(t,t_{0})A^{\dagger}(t)\partial_{t}A(t)\hat{U}_{0}(t,t_{0})}{\varrho_{q,I}(t)}$
$\displaystyle-\frac{1}{8}\sum_{nmkl}e^{-i\int_{0}^{t}(E_{nm}[t^{\prime}]+E_{kl}[t^{\prime}])\text{d}t^{\prime}}\commutator{{\tilde{\Pi}}_{nm}}{\tilde{\Pi}_{kl}\hat{\varrho}_{q,I}(t)}$
$\displaystyle\times\sum_{i=x,z}\tau_{i,nm}(t)\tau_{i,kl}(t)S_{i}(E_{lk}[t])+(h.c.).$
(50)

We now attempt to neglect those highly oscillating terms. Neglecting all of
the terms where $E_{nm}[t^{\prime}]\neq-E_{kl}(t)$, i.e. when $n=l$ and $m=k$,
we correspond to a global secular approximation, that leads to a Lindlbad
equation. However such an approximation would not preserve any coherence
between spin-up and spin-down states during their orbital dissipation.

Intuitively the amount of spin coherence lost during the orbital relaxation
depends on the ratio between $\Delta E_{Z}/\Gamma_{\pm}$, i.e. slow relaxation
would randomize time spend in two orbital eigenstates of different spin
splitting. Since typicall $\Delta E_{Z}/\Gamma_{\pm}$ is small, we keep the
terms where $|E_{nm}[t^{\prime}]-E_{kl}[t^{\prime}]|\leq\Delta E_{Z}$, i.e. we
perform a local secular approximation with respect to the orbital gap [60]. As
a result there are only 20 non-vanishing combinations of indices, which are
summarized in Table 2.

| n | m | k | l | $E_{nm}+E_{kl}$ | $E_{kl}$ | SOC needed
---|---|---|---|---|---|---|---
c | $s-$ | $s+$ | $s+$ | $s_{-}$ | 0 | $\Omega_{s}$ | no
c | $s+$ | $s-$ | $s-$ | $s+$ | 0 | $-\Omega_{s}$ | no
c | $s-$ | $s+$ | $\overline{s}+$ | $\overline{s}-$ | $-\sigma_{s}\Delta\Omega/2$ | $\Omega_{\overline{s}}$ | no
c | $s+$ | $s-$ | $\overline{s}-$ | $\overline{s}+$ | $\sigma_{s}\Delta\Omega/2$ | $-\Omega_{\overline{s}}$ | no
s | $\uparrow q$ | $\downarrow q$ | $\downarrow q$ | $\uparrow q$ | 0 | $-E_{Z}-q\Delta\Omega$ | yes
s | $\downarrow q$ | $\uparrow q$ | $\uparrow q$ | $\downarrow q$ | 0 | $E_{Z}+q\Delta\Omega/2$ | yes
s | $\uparrow q$ | $\downarrow q$ | $\downarrow\overline{q}$ | $\uparrow\overline{q}$ | $q\Delta\Omega/2$ | $-E_{Z}+q\Delta\Omega/2$ | yes
s | $\downarrow q$ | $\uparrow q$ | $\uparrow\overline{q}$ | $\downarrow\overline{q}$ | $-q\Delta\Omega/2$ | $E_{Z}-q\Delta\Omega/2$ | yes
cs | $\uparrow q$ | $\downarrow\overline{q}$ | $\downarrow\overline{q}$ | $\uparrow q$ | 0 | $-E_{Z}+q\overline{\Omega}$ | yes
cs | $\downarrow q$ | $\uparrow\overline{q}$ | $\uparrow\overline{q}$ | $\downarrow q$ | 0 | $E_{Z}-q\overline{\Omega}$ | yes
Table 2: Relevant transition rates, that survives local secular approximation.
We use the notation in which $\sigma_{s}=\pm 1$ for spin-up and spin-down
respectively while $q=\pm 1$ for excited and ground orbital state. In the
dissipation part we neglect difference in the relaxation rates that are of the
order of $\Delta\Omega<\Delta E_{Z}\ll\Omega_{0}(t)$.

Additionally since $\Delta E_{Z}$ is also much smaller than any other
timescale, in the dissipative part we neglect the difference between spin-
diagonal orbital splittings
$\Delta\Omega=\Omega_{\uparrow}-\Omega_{\downarrow}\to 0$, and set:

$\Omega_{\uparrow}(t)\approx\Omega_{\downarrow}(t)\approx\Omega_{0}(t)=\sqrt{\epsilon^{2}(t)+(2t_{c})^{2}},$
(51)

which also means that $\overline{\Omega}=\Omega_{0}$. Note that this is
consistent with local secular approximation, as with this assumptions all of
the transition listed in Table 2 have $E_{nm}+E_{kl}=0$. Additionally, what
follows is that all of the charge transitions (c) are associated with the same
energy transfer $\pm\Omega_{0}$, the spin transitions (s) with common energy
transfer of $\pm E_{Z}$, while combined charge-spin transitions with
$\pm(\Omega_{0}-E_{Z})$ (cs1) and $\pm(\Omega_{0}+E_{Z})$ (cs2).

In the same limit of small $\Delta E_{Z}$, the matrix elements can be
approximated as their value at $\Delta E_{Z}=0$, for which the spin-up and
spin-down transitions are alligned and
$\theta_{\uparrow}=\theta_{\downarrow}=\theta_{0}$. This would mean:

$\bra{\psi_{s+}(t)}\hat{\tau}_{i}\ket{\psi_{s-}(t)}\approx\bra{+(t)}\hat{\tau}_{i}\ket{-(t)},$
(52)

where $\bra{+(t)}\hat{\tau}_{z}\ket{-(t)}=-\cos\theta_{0}$,
$\bra{+(t)}\hat{\tau}_{x}\ket{-(t)}=\sin\theta_{0}$ and
$\text{ctan}\theta_{0}=\epsilon(t)/2t_{c}$.

As a result of the assumptions above, the final form of Adiabatic Master
Equation has the Lindblad form, i.e. it can be written as:

$\displaystyle\frac{\partial}{\partial
t}{\tilde{\varrho}}_{q}(t)=-i\commutator{{\tilde{H}}(t)}{{\tilde{\varrho}}_{q}(t)}+\sum_{q=\pm}\bigg{[}$
(53)
$\displaystyle+\bigg{(}{\tilde{L}}_{q,c}(t){\tilde{\varrho}}_{q}(t)\hat{\tilde{L}}_{q,c}^{\dagger}(t)-\frac{1}{2}\anticommutator{{\tilde{L}}_{q,c}^{\dagger}(t){\tilde{L}}_{q,c}(t)}{{\tilde{\varrho}}_{q}(t)}\bigg{)}$
$\displaystyle+\bigg{(}{\tilde{L}}_{q,s}(t){\tilde{\varrho}}_{q}(t)\hat{\tilde{L}}_{q,s}^{\dagger}(t)-\frac{1}{2}\anticommutator{{\tilde{L}}_{q,s}^{\dagger}(t){\tilde{L}}_{q,s}(t)}{{\tilde{\varrho}}_{q}(t)}\bigg{)}$
$\displaystyle+\bigg{(}{\tilde{L}}_{q,cs1}(t){\tilde{\varrho}}_{q}(t)\hat{\tilde{L}}_{q,cs1}^{\dagger}(t)-\frac{1}{2}\anticommutator{{\tilde{L}}_{q,cs1}^{\dagger}(t){\tilde{L}}_{q,cs1}(t)}{{\tilde{\varrho}}_{q}(t)}\bigg{)}$
$\displaystyle+\bigg{(}{\tilde{L}}_{q,cs2}(t){\tilde{\varrho}}_{q}(t)\hat{\tilde{L}}_{q,cs2}^{\dagger}(t)-\frac{1}{2}\anticommutator{{\tilde{L}}_{q,cs2}^{\dagger}(t){\tilde{L}}_{q,cs2}(t)}{{\tilde{\varrho}}_{q}(t)}\bigg{)}\bigg{]},$

in which we defined time-dependent Lindblad operators of explicit form:

$\displaystyle{\tilde{L}}_{\pm,c}(t)$
$\displaystyle=\sqrt{\Gamma_{\pm,c}(t)}\sum_{s=\uparrow,\downarrow}|\tilde{\psi}_{s\pm}\rangle\langle\tilde{\psi}_{s\mp}|$
$\displaystyle{\tilde{L}}_{-,s}(t)$
$\displaystyle=\sqrt{\Gamma_{-,s}(t)}\sum_{q=\pm}|\tilde{\psi}_{\downarrow
q}\rangle\langle\tilde{\psi}_{\uparrow q}|$
$\displaystyle{\tilde{L}}_{+,s}(t)$
$\displaystyle=\sqrt{\Gamma_{+,s}(t)}\sum_{q=\pm}|\tilde{\psi}_{\uparrow
q}\rangle\langle\tilde{\psi}_{\downarrow q}|$
$\displaystyle{\tilde{L}}_{-,cs1}(t)$
$\displaystyle=\sqrt{\Gamma_{-,cs1}(t)}\,|\tilde{\psi}_{\uparrow-}\rangle\langle\tilde{\psi}_{\downarrow+}|$
$\displaystyle{\tilde{L}}_{+,cs1}(t)$
$\displaystyle=\sqrt{\Gamma_{+,cs1}(t)}\,|\tilde{\psi}_{\downarrow+}\rangle\langle\tilde{\psi}_{\uparrow-}|$
$\displaystyle{\tilde{L}}_{-,cs2}(t)$
$\displaystyle=\sqrt{\Gamma_{-,cs2}(t)}\,|\tilde{\psi}_{\downarrow-}\rangle\langle\tilde{\psi}_{\uparrow+}|$
$\displaystyle{\tilde{L}}_{+,cs2}(t)$
$\displaystyle=\sqrt{\Gamma_{+,cs2}(t)}\,|\tilde{\psi}_{\uparrow+}\rangle\langle\tilde{\psi}_{\downarrow-}|.$
(54)

Finally the excitaiton rates are related to the relaxation rates via the
fluctuation-dissipation theorem, which for the environment at thermal
equilibrium gives:

$\displaystyle\Gamma_{+,c}(t)$
$\displaystyle\approx\Gamma_{-,c}(t)e^{-\beta\Omega_{0}(t)}$
$\displaystyle\Gamma_{+,s}(t)$ $\displaystyle\approx\Gamma_{-,s}(t)e^{-\beta
E_{Z}}$ $\displaystyle\Gamma_{+,cs1}(t)$
$\displaystyle\approx\Gamma_{-,cs1}(t)e^{-\beta(\Omega_{0}(t)-E_{Z})}$
$\displaystyle\Gamma_{+,cs2}(t)$
$\displaystyle\approx\Gamma_{-,cs2}(t)e^{-\beta(\Omega_{0}(t)+E_{Z})}$ (55)

So together the AME can be characterized by four relaxation rates
$\Gamma_{-,c}(t)$, $\Gamma_{-,s}(t)$, $\Gamma_{-,cs1}(t)$ and
$\Gamma_{-,cs2}(t)$, that will be computed in Appendix. B

###  A.4 Equations of motion in absence of spin-orbit coupling

We finally use Eq. (53) to write equation of motion for the case where
evolution is adiabatic (slow sweep rates) and the spin-orbit is not present,
or negligibly small. The first conditions corresponds to avoiding dynamical
term in the adiabatic Hamiltonian, such that $\tilde{H}\approx\tilde{H}_{0}$.
The second condition, means that only charge transitions are possible
(transitions c in Table 2). As a result the equation for probability of
occupying ground and excited state
$\dot{P}_{s\pm}(t)=\langle\tilde{\psi}_{s\pm}(t)|\hat{\tilde{\varrho}}_{q,I}(t)|\tilde{\psi}_{s\pm}(t)\rangle$
reads:

$\displaystyle\dot{P}_{s\pm}(t)$
$\displaystyle=P_{s\mp}(t)\Gamma_{s\pm}(t)-P_{s\pm}(t)\Gamma_{s\mp}(t),$ (56)

while the coherence between the spin-like states in the excited
$W_{+}=\langle\tilde{\psi}_{\uparrow+}(t)|\hat{\tilde{\varrho}}_{q,I}(t)|\tilde{\psi}_{\downarrow-}(t)\rangle$
and ground
$W_{-}=\langle\tilde{\psi}_{\uparrow-}(t)|\hat{\tilde{\varrho}}_{q,I}(t)|\tilde{\psi}_{\downarrow-}(t)\rangle$
states reads:

$\dot{W}_{\pm}(t)=-W_{\pm}(t)\Gamma_{\mp}(t)+W_{\mp}(t)\Gamma_{\pm}(t)e^{\mp
i\int_{t_{0}}^{t}\Delta\Omega(t^{\prime})\text{d}t^{\prime}}$ (57)

The above equation shows that the spin coherence can decrease due to inelastic
transitions between the adiabatic states. This can be seen in the language of
perturbation theory, i.e. finding the first order correction $W_{-}^{(1)}(t)$
to unperturbed $W_{+}(0)=1$, $W_{-}(0)=0$, which gives:

$W_{-}^{(1)}(t)=\int_{0}^{t}\text{d}t^{\prime}\Gamma_{-}(t^{\prime})\exp(-\int_{t_{0}}^{t^{\prime\prime}}\Gamma_{-}(t)+i\Delta\Omega(t^{\prime\prime})).$
(58)

For the constant $\Gamma_{-}$ and $\Delta\Omega=\Delta E_{Z}$ we can compute
the phase error associated with orbital relaxation
$Q_{\varphi}=(1-|W_{-}^{(1)}(t)|)$:

$Q_{\varphi}=\frac{1}{2}\left(\frac{\Delta E_{Z}}{\Gamma_{-}}\right)^{2},$
(59)

which reconstructed Eq.(24) for initially occupied excited state $P_{e}=1$. In
the opposite limit of relatively large $\Delta E_{Z}\geq\Gamma_{\pm}$, we
would have $\dot{W}_{\pm}(t)=-W_{\pm(t)}\Gamma_{\mp}(t)$, which effectively
recovers the global secular approximation, i.e. the orbital relaxation does no
longer preserve any spin coherence.

##  Appendix B Relaxation rates

We now follow [25] and find expressions for the charge and spin relaxation
rate, which will be evaluated for Si and GaAs devices in the next section C.

The relaxation is caused by the interaction with environment. We cast
electron-environment in the form:

$\hat{V}=\frac{1}{2}\hat{V}_{x}\hat{\tau}_{x}+\frac{1}{2}\hat{V}_{z}\hat{\tau}_{z},$
(60)

where $\tau_{i}$ are written in the dot basis $\ket{L/R}$ and the
$\hat{V}_{i}$ are acting on the environmental degrees of freedom only. They
correspond to noise in tunnel coupling $\hat{V}_{x}$ and noise in detuning
$\hat{V}_{z}$, which are assumed to be independent, i.e.
$\langle\hat{V}_{x}\hat{V}_{z}\rangle=0$. Using Fermi Golden rule the
relaxation rate between any of the states can be written as:

$\Gamma_{sq\to
s^{\prime}q^{\prime}}=\frac{1}{4}\sum_{i=x,z}\big{|}\bra{\psi_{sq}}\hat{\tau}_{i}\ket{\psi_{s^{\prime}q^{\prime}}}\big{|}^{2}S_{i}(E_{sq}-E_{s^{\prime}q^{\prime}}),$
(61)

where $S_{i}(\Omega)$ is the spectral density of the environment, while the
time-dependence of the relaxation rate $\Gamma_{sq\to
s^{\prime}q^{\prime}}(t)$, that results from the time-depedence of the states
$\ket{\psi_{sq}(t)}$ and the energies $E_{sq}(t)$ was omitted for brevity.

###  B.1 Charge relaxation $\Gamma_{-.c}(t)$

For the charge relaxation we consider the transition between spin-diagonal
states, i.e. $\ket{\psi_{s+}}\to\ket{\psi_{s-}}$, i.e.

$\Gamma_{s+\to
s-}(t)=\frac{1}{4}\sum_{i=x,z}\big{|}\bra{\psi_{s+}(t)}\hat{\tau}_{i}\ket{\psi_{s-}(t)}\big{|}^{2}S_{i}(\Omega_{s}[t]),$
(62)

Using definition of the instatnanous states Eq. (LABEL:), we have:

$\displaystyle\big{|}\bra{\psi_{s+}(t)}\hat{\tau}_{x}\ket{\psi_{s-}(t)}\big{|}^{2}$
$\displaystyle=\sin^{2}\theta_{s}(t)$
$\displaystyle\big{|}\bra{\psi_{s+}(t)}\hat{\tau}_{z}\ket{\psi_{s-}(t)}\big{|}^{2}$
$\displaystyle=\cos^{2}\theta_{s}(t),$ (63)

where $\sin\theta_{s}(t)=2t_{c}/\Omega_{s}(t)$. Following assumption from A,
in computing relaxation rate we neglect small difference between the orbital
energies, and in this way find the spin-independent rate:

$\Gamma_{-,c}(t)\approx\frac{1}{4}\bigg{(}\cos^{2}\theta_{0}[t]S_{x}(\Omega_{0}[t])+\sin^{2}\theta_{0}[t]S_{z}(\Omega_{0}[t])\bigg{)},$
(64)

where $\tan\theta_{0}=2t_{c}/\epsilon(t)$.

###  B.2 Spin relaxation $\Gamma_{s}(t)$

The spin relaxation is activated by the spin-orbit coupling, which hybridises
spin-up and spin-down states around avoided crossing, i.e. the correction to
states due to presence of spin orbit interaction from Eq. 4 can be written as:

$\ket{\psi_{sq}^{\prime}}=\ket{\psi_{sq}}+\sum_{q^{\prime}s^{\prime}\neq
qs}\frac{\bra{\psi_{s^{\prime}q^{\prime}}}\hat{V}_{so}\ket{\psi_{sq}}}{E_{sq}-E_{s^{\prime}q^{\prime}}}\ket{\psi_{q^{\prime}s^{\prime}}}.$
(65)

As an illustration for the ground state we have:

$\displaystyle\ket{\psi_{-\downarrow}^{\prime}}=\ket{\psi_{-\downarrow}}$
$\displaystyle+\frac{1}{2}\,\frac{a^{*}\sin\tfrac{\Delta\theta}{2}-b\cos\overline{\theta}}{\Delta\Omega-
E_{Z}}\ket{\psi_{-\uparrow}}$
$\displaystyle-\frac{1}{2}\,\frac{a\cos\tfrac{\Delta\theta}{2}-b\sin\overline{\theta}}{\overline{\Omega}+E_{Z}}\ket{\psi_{+\uparrow}},$
(66)

in which $\Delta\theta=\theta_{\uparrow}-\theta_{\downarrow}$,
$\overline{\theta}=(\theta_{\uparrow}+\theta_{\downarrow})/2$ and similarly
$\Delta\Omega=\Omega_{\uparrow}-\Omega_{\downarrow}$,
$\overline{\Omega}=\Omega_{\uparrow}+\Omega_{\downarrow}$. For the second
lowest in energy eigenstate we have:

$\displaystyle\ket{\psi_{-\uparrow}^{\prime}}=\ket{\psi_{-\uparrow}}$
$\displaystyle+\frac{1}{2}\,\frac{a\sin\tfrac{\Delta\theta}{2}-b\cos\overline{\theta}}{E_{Z}-\Delta\Omega}\ket{\psi_{-\downarrow}}$
$\displaystyle+\frac{1}{2}\,\frac{a^{*}\cos\tfrac{\Delta\theta}{2}+b\sin\overline{\theta}}{\overline{\Omega}-E_{Z}}\ket{\psi_{+\downarrow}}.$
(67)

The hybridised states can be substituted for the expression for spin
relaxation rate, i.e.

$\displaystyle\Gamma_{\uparrow q\to\downarrow
q}(t)=\frac{1}{4}\sum_{i}\bigg{|}\bra{\psi_{q\uparrow}^{\prime}(t)}\hat{\tau}_{i}\ket{\psi_{q\downarrow}^{\prime}(t)}\bigg{|}^{2}S_{i}(E_{Z}+q\Delta\Omega[t])$
(68)

Again following assumptions made in the derivation of AME (See Appendix A) we
use the fact that $\Delta E_{Z}$ is much smaller than any other energy scale,
which allows us to neglect a small difference between spin relaxations in
ground and excited state associated with
$\Delta\Omega=\Omega_{\uparrow}-\Omega_{\downarrow}$. This ammounts to setting
$\Delta\theta\to 0$, $\Delta\Omega\to 0$,
$\overline{\theta}\to\theta_{0}=\arcsin(2t_{c}/\Omega_{0})$,
$\overline{\Omega}=\Omega_{0}=\sqrt{(2t_{c})^{2}+\epsilon^{2}(t)}$ and
writing:

$\Gamma_{-,s}(t)=\bigg{|}\frac{aE_{Z}+bt_{c}}{\Omega_{0}^{2}(t)-E_{Z}^{2}}\bigg{|}^{2}\Gamma_{-\text{charge}}(E_{Z})$
(69)

###  B.3 Spin-charge relaxation $\Gamma_{sc}(t)$

We finally compute the relaxation rate associated with simultaneous change of
both orbital and spin. At finite $E_{Z}$ we have two distinct transitions:

$\displaystyle\Gamma_{\uparrow+\,\to\,\downarrow-}(t)=\frac{1}{4}\sum_{i}\big{|}\langle\psi_{\uparrow+}^{\prime}(t)|\hat{\tau}_{i}|\psi_{\downarrow-}^{\prime}(t)\rangle\big{|}^{2}S_{i}(\overline{\Omega}(t)+E_{Z})$
$\displaystyle\Gamma_{\downarrow+\,\to\,\uparrow-}(t)=\frac{1}{4}\sum_{i}\big{|}\langle\psi_{\downarrow+}^{\prime}(t)|\hat{\tau}_{i}|\psi_{\uparrow-}^{\prime}(t)\rangle\big{|}^{2}S_{i}(\overline{\Omega}(t)-E_{Z}),$
(70)

note that now since $E_{Z}$ is not essentially much smaller then
$\overline{\Omega}\approx\Omega_{0}(t)$, the transitions should be associated
with two distinct Lindbladians, meaning that no spin coherence is expected to
survive the relaxation.

To compute the matrix elements we use perturbation theory (see above) and
write:

$\displaystyle\ket{\psi_{\uparrow+}^{\prime}}=\ket{\psi_{\uparrow+}}$
$\displaystyle+\frac{1}{2}\,\frac{a\sin\tfrac{\Delta\theta}{2}+b\cos\overline{\theta}}{\Delta\Omega+E_{Z}}\ket{\psi_{+\downarrow}}$
$\displaystyle-\frac{1}{2}\,\frac{a^{*}\cos\tfrac{\Delta\theta}{2}+b\sin\overline{\theta}}{\overline{\Omega}+E_{Z}}\ket{\psi_{\downarrow-}},$
$\displaystyle\ket{\psi_{\downarrow+}^{\prime}}=\ket{\psi_{\downarrow+}}$
$\displaystyle-\frac{1}{2}\,\frac{a^{*}\sin\tfrac{\Delta\theta}{2}+b\cos\overline{\theta}}{\Delta\Omega+E_{Z}}\ket{\psi_{\uparrow+}}$
$\displaystyle+\frac{1}{2}\,\frac{a\cos\tfrac{\Delta\theta}{2}-b\sin\overline{\theta}}{\overline{\Omega}-E_{Z}}\ket{\psi_{\uparrow-}}.$
(71)

Interestingly, in the limit of small $\Delta E_{Z}$, i.e. $\Delta\theta\to 0$,
and in the leading order, the orbital relaxation with a spin-flip is driven
solely by the synthetic spin-orbit interaction, i.e.

$\displaystyle\Gamma_{-,cs1}(t)$
$\displaystyle=\bigg{|}\frac{E_{Z}b\cos\theta_{0}}{E_{Z}^{2}-\Delta\Omega(t)^{2}}\bigg{|}^{2}\Gamma_{-,c}(\Omega_{0}[t]+E_{Z})\,$
$\displaystyle\approx\frac{b^{2}\epsilon^{2}}{E_{Z}^{2}\Omega_{0}^{2}}\Gamma_{-,c}(\Omega_{0}[t]+E_{Z}).$
(72)

where in the approximate expression we assumed $E_{Z}\gg\Delta\Omega>\Delta
E_{Z}$. Similarly we have:

$\Gamma_{-,cs2}(t)\approx\frac{b^{2}\epsilon^{2}}{E_{Z}^{2}\Omega_{0}^{2}}\Gamma_{-,c}(\Omega_{0}[t]-E_{Z}).$
(73)

In view of results above we now argue that charge transition with spin flip is
the higher order correction to transfer error, and for this reason is
negligibly small to all the other sources of errors considered in this paper.
Firstly, such a charge excitation with a spin flip is highly improbable event,
which can be linked to the Boltzman factor $e^{-\beta[\Omega_{0}(t)\pm
E_{Z}]}$ being negligibly small everywhere apart from the vicinity of avoided
crossing (See charge relaxation). At this region of small detuning
$\Gamma_{+,cs}\propto\epsilon^{2}/\Omega_{0}^{2}$ is small as well. The same
arguments holds for the relaxation, which is unlikely to happen unless
$\epsilon\geq 2t_{c}$. Also a prefactor of $b^{2}/E_{Z}^{2}$ is expected to be
negligibly small unless the transition takes place at manipulation region or
at very low magnetic fields. As a result we conclude this process can be
neglected in the analysis, which we confirm by including it in the numerical
simulation of AME without a change in overall result.

##  Appendix C Relaxation rates in Si and GaAs: Spectral densities

We first consider charge relaxation caused by the charge noise. The noise in
detuning is modeled by extrapolating typically measured power spectra of 1/f
and Johnon noise of the form:

$S_{z}^{1/f}(\Omega)=\frac{S_{1}\omega_{1}}{\Omega},\,\,S_{z}^{J}(\Omega)=\frac{Z}{R_{q}}\frac{\Omega}{1-e^{-\beta\Omega}}$
(74)

where we use $Z=50\Omega$ and $R_{q}=\pi/e^{2}\approx 13k\Omega$.

Figure 6: Spectral densities $S_{i}(\Omega)$ as a function of the energy
transfer $\omega$ for a different mechanisms: Phonons in Si (green), phonons
in GaAs (red), 1/f noise (black), Johnson noise (blue). With solid line we
depict $S_{z}(\Omega)$, i.e. fast noise in detuning, while dashed line
represent $S_{x}(\Omega)$, i.e. fast noise in tunnel coupling.

We assume the noise in tunnel coupling is independent and its power spectrum
is rescaled by a factor of $\alpha=10^{-2}$, i.e. $S_{x}(\Omega)=\alpha
S_{z}(\Omega)$. This can be related to a typically small overlap of the
wavefunction and empirically observed difference between lever arms of barrier
and plunger gates [42].

Figure 7: Charge relaxation rate in GaAs (red) and Si (green) as a function of
deutning for two different tunnel couplings considered in this paper
$2t_{c}=40\mu$eV (solid line) and $2t_{c}=100\mu$eV (dashed line).

As a second source of the relaxation we consider phonons, for which the
spectral densities can be derived from the general form of phonon-electron
interaction:

$\hat{V}_{\text{el}-\text{ph}}=\sum_{j=p,d}\sum_{\\\
\lambda=L,T}\sum_{\mathbf{k}}\eta_{\mathbf{k},\lambda}v_{\mathbf{k},\lambda}^{(j)}(b_{\mathbf{k},\lambda}+b_{-\mathbf{k},\lambda}^{\dagger})e^{i\mathbf{k}\mathbf{r}},$
(75)

where we sum over piezoelectric (p) and deformation (d) phonons and their
polarisation: longitudinal (L) and transverse (T), as well as the wavevector
$\mathbf{k}$. The couplings are given by
$\eta_{\mathbf{k},\lambda}=\sqrt{k/2\varrho c_{\lambda}V}$ and :

$v_{\mathbf{p},\lambda}^{(p)}=\frac{\chi_{p}}{k}\,,\,v_{\mathbf{d},L}^{(d)}=\Xi_{d}+\Xi_{u}\frac{k_{z}^{2}}{k^{2}}\,,\,v_{\mathbf{d},T}^{(d)}=-\frac{k_{x,y}k_{z}}{k^{2}}$
(76)

Using Fermi Golden rule the Spectral density of phonons is given by:

$\displaystyle S_{i}(\Omega)$
$\displaystyle=2\pi\sum_{\mathbf{k},\lambda,j}\big{|}F_{i,\mathbf{k}}(\mathbf{r})v_{\mathbf{k},\lambda}^{(j)}\eta_{\mathbf{k},\lambda}\big{|}^{2}\delta\big{(}\Omega-\omega_{\mathbf{k},\lambda}\big{)}$
(77)

where $F_{x,\mathbf{k}}(\mathbf{r})=2\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{R}$,
$F_{z,\mathbf{k}}(\mathbf{r})=\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{L}-\bra{R}e^{i\mathbf{k}\mathbf{r}}\ket{R}$.
We use linear dispersion relation
$\omega_{\mathbf{k},\lambda}=c_{\lambda}|k|$. The spectral densities can now
computed numerically by substituting (C), substituting coupling constants
(77), and changing sum into an integral, i.e.
$\sum_{\textbf{k}}=\frac{V}{(2\pi)^{3}}\int\text{d}\mathbf{k}$.

Figure 8: Spin relaxation rate in GaAs (a) and Si (b). In both panels we
compare two tunnel couplings ($2t_{c}=40\mu$eV (solid line), $2t_{c}=100\mu$eV
(dashed), as well as two spin splittings $E_{Z}=30\mu$eV (line without dots),
$E_{Z}=10\mu$eV (lines with the dots). Finally for GaAs (a) we compare
contributions from intrinsic SOC, i.e $|a|=1\mu$eV, $b=0$ (red) against
synthetic $|a|=0,b=1\mu$eV (black). Similarly for Si (b) we compare intrinsic
SOC $|a|=0.1\mu$eV, $b=0$ (green) against sythetic $|a|=0,b=1\mu$eV (yellow).

To compute the form factors $F_{i,\mathbf{k}}(\mathbf{r})$, we employ Hund-
Mulliken approximation, and write hybridised states in terms of isolated
Gaussian wavefunctions

$\psi_{L_{0}/R_{0}}(\mathbf{r})=\bra{x}\ket{L_{0}/R_{0}}\psi_{yz}(y,z)=\frac{\psi_{yz}(y,z)}{(\pi
L^{2})^{1/4}}e^{-\frac{(x\pm d/2)^{2}}{2L^{2}}},$ (78)

parameterized by dots separation $d$ and isotropic dot size $L$. Due to finite
overlap, the eigenstates are the linear combination of bare wavefunctions.

$\displaystyle\ket{L}$ $\displaystyle=\ket{L_{0}}-\frac{\gamma}{2}\ket{R_{0}}$
$\displaystyle\ket{R}$
$\displaystyle=\ket{R_{0}}-\frac{\gamma}{2}\ket{L_{0}},$ (79)

where $\gamma\approx\bra{L_{0}}\ket{R_{0}}=\exp(-d^{2}/4L^{2})\ll 1$ is the
overlap between the wavefunctions.

##  Appendix D Dephasing due to low-frequency noise

We provide here more detailed derivation of Eqs (29) and (30) , which shows
that the most relevant contributions to spin-dephasing for low-frequency noise
is caused by the fluctuation of detuning noise and spin-splitting noise. We
start by assuming the electron stays in the two lowest-energu instantaneous
states, for which the energies are given by:

$E_{-s}=\frac{1}{2}E_{Z}-\frac{1}{2}\Omega_{s}(t)$ (80)

where $\Omega_{s}(t)=\sqrt{(\epsilon+\sigma_{s}\Delta
E_{Z}/2)^{2}+(2t_{c})^{2}}$. We define the spin splitting noise as

$\omega_{\text{spin}}=E_{-\uparrow}-E_{-\uparrow}$ (81)

and introduce the quasistatic noise, i.e. replace
$\epsilon\to\epsilon+\delta\epsilon$, $t_{c}\to\delta t_{c}$, $E_{Z}\to
E_{Z}+\frac{1}{2}(\delta E_{Z,l}+\delta E_{Z,r})$ and $\Delta E_{Z}\to\Delta
E_{Z}+\delta E_{Z,l}-\delta E_{Z,r}$. Together the phase acquired between two
spin components during a transition between $0$ and $t_{f}$ is given by:

$\varphi=\int_{0}^{t_{f}}\omega_{\text{spin}}(t)\text{d}t=(E_{Z}+\delta
E_{Z})t_{f}+\int_{0}^{t_{f}}\tfrac{\Omega_{\downarrow}(t)-\Omega_{\uparrow}(t)}{2}\text{d}t,$
(82)

where in the first term $\delta E_{Z}=\tfrac{1}{2}(\delta E_{Z,L}+\delta
E_{Z,R})$, which corresponds to low-frequency fluctuations in spin-splitting.

The integrand in the second term is related to the difference between spin-
dependent orbital energies
$\Delta\Omega(t)=\Omega_{\downarrow}(t)-\Omega_{\uparrow}(t)$ and can be
written as:

$\displaystyle\Delta\Omega(t)$
$\displaystyle=\sqrt{[\epsilon+\delta\epsilon-\tfrac{1}{2}(\Delta
E_{Z}+\delta\Delta E_{Z})]^{2}+(2t_{c}+\delta 2t_{c})^{2}}$
$\displaystyle-\sqrt{[\epsilon+\delta\epsilon+\tfrac{1}{2}(\Delta
E_{Z}+\delta\Delta E_{Z})]^{2}+(2t_{c}+\delta 2t_{c})^{2}},$ (83)

where we denoted $\delta\Delta E_{Z}=\delta E_{Z,L}-\Delta E_{Z,R}$. We now
exploit the fact that the noise terms are small in comparison to noiseless
orbital splitting,

$\Omega_{0,s}(t)=\sqrt{(\epsilon(t)+\tfrac{1}{2}\sigma_{s}\Delta
E_{Z})^{2}+(2t_{c})^{2}},$ (84)

which in linear order in the noise terms $\delta\epsilon/\Omega_{0,s}(t)$,
$\delta t_{c}/\Omega_{0,s}(t)$ and $\delta\Delta E_{Z}/\Omega_{0,s}(t)$ allows
to write:

$\displaystyle\Delta\Omega(t)\approx\Delta\Omega_{0}(t)+\delta\epsilon\,$
$\displaystyle[\cos\theta_{\downarrow}(t)-\cos\theta_{\uparrow}(t)]$
$\displaystyle-\delta\Delta E_{Z}\,$
$\displaystyle[\cos\theta_{\downarrow}(t)+\cos\theta_{\downarrow}(t)]$
$\displaystyle+\delta t_{c}\,$
$\displaystyle[\sin\theta_{\downarrow}(t)-\sin\theta_{\uparrow}(t)],$ (85)

where
$\Delta\Omega_{0}(t)=\Delta\Omega_{0,\downarrow}(t)-\Delta\Omega_{0,\uparrow}(t)$,
$\cos\theta_{s}(t)=(\epsilon+\sigma_{s}\Delta E_{Z}/2)/\Omega_{0,s}(t)$,
$\sin\theta_{s}(t)=2t_{c}/\Omega_{0,s}(t)$. Finally we use the fact that
$\Delta E_{Z}\ll 2t_{c}$, such that combination of trigonometric functions in
leading order in $\Delta E_{Z}/2t_{c}$ are given by:

$\displaystyle\cos\theta_{\downarrow}(t)-\cos\theta_{\uparrow}(t)$
$\displaystyle\approx-\frac{\Delta E_{Z}}{\Omega_{0}}\sin^{2}[\theta_{0}(t)],$
$\displaystyle\cos\theta_{\downarrow}(t)+\cos\theta_{\uparrow}(t)$
$\displaystyle\approx 2\cos[\theta_{0}(t)],$
$\displaystyle\sin\theta_{\downarrow}(t)-\sin\theta_{\uparrow}(t)$
$\displaystyle\approx\frac{\Delta E_{Z}}{2\Omega_{0}}\sin[2\theta_{0}(t)],$
(86)

where the right-hand side was expressed in terms of spin-less quantities, i.e.
$\Omega_{0}=\sqrt{\epsilon^{2}+(2t_{c})^{2}}$ while
$\cos\theta_{0}=\epsilon/\Omega_{0}$ and $\sin\theta_{0}=2t_{c}/\Omega_{0}$.

Figure 9: Dephasing during adiabatic transition, caused by the slow
fluctuations of dots detuning and dot-dependent spin splittings for
$2t_{c}=40\mu$eV and $v=5\mu$eV/ns. We simulate phase error by integrating the
adiabatic spin splitting $E_{\uparrow-}(t)-E_{\downarrow-}(t)$ from Eq. (2)
and averaging the result over 1/f noise in detuning with
$S_{1}=1\mu$eV${}^{2}/$Hz (solid lines, filled dots) and
$S_{1}=0.1^{2}\mu$eV${}^{2}/$Hz (dashed lines, hollow dots), and over spin
splitting fluctuations in left and right dot modeled by two independent
Ornstein-Uhlenbeck noise processes with $\sigma=\sqrt{2}/T_{2}^{*2}$ and
correlation time $\tau_{c}=1$s for two values of $T_{2}^{*}=2\mu$s (red) and
$T_{2}^{*}=20\mu$s (green). We compare numerical results (dots) against
analytical quasistatic predictions (lines) given by Eq. (29) and Eq. (30).

We highlight that for the symmetric sweep, the quasistatic contribution from
$\delta\Delta E_{Z}$ and $\delta t_{c}$ vanishes, and we are left with :

$\varphi=\varphi_{0}+\delta\varphi_{E_{Z}}+\delta\varphi_{\epsilon},$ (87)

where $\varphi_{0}=\int_{0}^{t_{f}}E_{Z}+\Delta\Omega_{0}(t)\text{d}t$ is the
noiseless phase difference, $\delta\varphi_{E_{Z}}=(\delta E_{Z,L}+\delta
E_{Z,R})t_{f}/2$ and

$\delta\varphi_{\epsilon}=2t_{c}\Delta
E_{Z}\delta\epsilon\,\int_{0}^{t_{f}}\frac{\text{d}t}{\sqrt{\epsilon(t)^{2}+(2t_{c})^{2}}}=\frac{\Delta
E_{Z}t_{f}\,\delta\epsilon}{\sqrt{v^{2}t_{f}^{2}+(2t_{c})^{2}}},$ (88)

which in the typical limit of $vt_{f}\gg 2t_{c}$ reduces to:

$\delta\varphi_{\epsilon}\approx\frac{\Delta E_{Z}}{v}\,\delta\epsilon,$ (89)

such that for sufficiently long and symmetric sweep of detuning the small
contribution to dephasing due to low-frequency noise can be written as:

$\langle\delta\varphi^{2}\rangle=\langle(\delta\varphi_{E_{Z}}+\delta\varphi_{\epsilon})^{2}\rangle=\left(\frac{t_{f}}{T_{2}^{*}}\right)^{2}+\left(\frac{\Delta
E_{Z}}{v}\right)^{2}\sigma_{\epsilon}^{2},$ (90)

where we assumed that $\langle\delta E_{Z}\delta\epsilon\rangle=0$, and the
spin splitting fluctuations in the dots have the same amplitude but are
statistically independent
$\sqrt{2}/T_{2}^{*}=\sigma_{E_{Z,L}}=\sigma_{E_{Z,R}}$.

###  D.1 Numerical test

In this section we perform numerical test of an adiabatic dephasing. To do so
we average $\varphi$, given by Eq. (87), over $\delta\epsilon$, $\delta
t_{c}$, $\delta E_{Z,L}$, $\delta E_{Z,R}$. The results are depicted in Fig.
9, where numerically simulated $\langle\delta\varphi^{2}\rangle$ (dots) is
shown as a function of noiseless difference in spin splittings $\Delta E_{Z}$
between the two dots.

We plot the results for two different values of $T_{2}^{*}=2,20\,\mu$s and two
different amplitude of detuning noise
$\langle\delta\epsilon^{2}\rangle=5,0.5\,\mu$eV, which would correspond to a
1/f noise in detuning with a typically measured spectral density at 1Hz given
by $\sqrt{S(1\text{Hz})}\approx 0.5,0.05\,\mu$eV/Hz. As it can be seen from
the Figure, the numerical averages (dots) follows quasistatic predictions
(black solid and dashed lines) for sufficiently large $\Delta E_{Z}$, for
which the charge noise contribution dominates over fluctuations of Zeeman
splittings. By simultaneous averaging over spin splitting fluctuations and
detuning noise (previous section), we also showed that in the relevant error
range (small $Q$), the charge noise and nuclear noise contribution to the
transfer error $Q$ can be treated additively.

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