# Interplay of charge noise and coupling to phonons in adiabatic electron
transfer between quantum dots

Jan A. Krzywda<EMAIL_ADDRESS>Łukasz Cywiński  Institute of Physics,
Polish Academy of Sciences, al. Lotników 32/46, PL 02-668 Warsaw, Poland

###### Abstract

Long-distance transfer of quantum information in architectures based on
quantum dot spin qubits will be necessary for their scalability. One way of
achieving it is to simply move the electron between two quantum registers.
Precise control over the electron shuttling through a chain of tunnel-coupled
quantum dots is possible when interdot energy detunings are changed
adiabatically. Deterministic character of shuttling is however endangered by
coupling of the transferred electron to thermal reservoirs: sources of
fluctuations of electric fields, and lattice vibrations. We theoretically
analyse how the electron transfer between two quantum dots is affected by
electron-phonon scattering, and interaction with sources of $1/f$ and Johnson
charge noise in both detuning and tunnel coupling. The electron-phonon
scattering turns out to be irrelevant in Si quantum dots, while a competition
between the effects of charge noise and Landau-Zener effect leads to an
existence of optimal detuning sweep rate, at which probability of leaving the
electron behind is minimal. In GaAs quantum dots, on the other hand, coupling
to phonons is strong enough to make the phonon-assisted processes of interdot
transfer dominate over influence of charge noise. The probability of leaving
the electron behind depends then monotonically on detuning sweep rate, and
values much smaller than in silicon can be obtained for slow sweeps. However,
after taking into account limitations on transfer time imposed by need for
preservation of electron’s spin coherence, minimal probabilities of leaving
the electron behind in both GaAs- and Si-based double quantum dots turn out to
be of the same order of magnitude. Bringing them down below $10^{-3}$ requires
temperatures $\leq\\!100$ mK and tunnel couplings above $20$ $\mu$eV.

##  I Introduction

In quantum computing architectures based on voltage-controlled quantum dots
(QDs), developed in GaAs/AlGaAs [1, 2, 3], Si/SiGe [4, 5], and silicon MOS [6,
7, 8, 9] structures, scalability will be possible only if quantum information
is transferred between few-qubit registers, separated by distances much larger
than the typical QD size. This is caused by short-distance character of
exchange interaction needed for two-qubit gates, and spatial extent of wiring
needed for controlled application of voltages to the gates defining the dots,
which together put limits on density of a qubit array [10]. Coupling of
electron spins to microwave photons is a possible mean of coherent coupling of
spin qubits in GaAs [11] and silicon [12, 13, 14, 15]. A conceptually simpler
alternative, which has been recently pursued in experiments [16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28], is to simply transfer an electron spin
qubit over a large (at least a few micrometer) distance.

We focus here on electron transfer along a chain of tunnel-coupled QDs [22,
23, 24, 25, 26, 27, 28]. The shuttling is then caused by controlled tilting of
energy levels of neighboring QDs that makes an electron move from one dot to
the other. The basic step in such a process is single electron transfer
between two tunnel-coupled QDs. In a simplified situation, in which we neglect
spin and valley (in case of Si) degrees of freedom of the electron, the basic
physics is captured by the Hamiltonian acting in a two-dimensional Hilbert
space spanned by $\ket{L(R)}$ states, corresponding to electron localized in a
local ground states of energy $E_{L(R)}$ in left (right) dot:

$\hat{H}=\frac{\epsilon}{2}\hat{\sigma}_{z}+\frac{t_{c}}{2}\hat{\sigma}_{x}\,\,,\
$ (1)

where $\epsilon=E_{L}-E_{R}$ is the so-called interdot detuning of energy,
$t_{c}$ is the tunnel coupling between the QDs,
$\hat{\sigma}_{z}=\outerproduct{L}{L}-\outerproduct{R}{R}$, and
$\hat{\sigma}_{x}=\outerproduct{L}{R}+\outerproduct{R}{L}$. For
$\epsilon\\!\ll\\!-t_{c}$ the lowest energy state is localized in the $L$ dot,
and this is the state that we take as an initial one in all the considerations
below. For $\epsilon\\!\gg\\!t_{c}$ the lowest-energy state is localized in
the $R$ dot, and one of course expects that for very slow change of $\epsilon$
from negative to positive values, the evolution will be adiabatic and the
system will end up in this state. For a linear sweep,
$\epsilon\\!\propto\\!v\tau$, where $v$ is the rate of change of detuning, and
constant $t_{c}$, we are dealing with classical Landau-Zener model [29], for
which the probability of having the electron in an exited state for
$\epsilon(\tau)\rightarrow\infty$ (i.e. leaving the electron behind in the L
dot) is given by

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

so that a near-perfect adiabatic transfer occurs when $t_{c}^{2}/v\\!\gg\\!1$,
i.e. when the sweep rate is low.

Changing the interdot detunings slowly is thus an obvious way to perform an
on-demand deterministic transfer of an electron spin qubit. Of course, the
total shuttling time should be much shorter than the spin coherence time of a
moving electron, and according to Eq. (2) this requirement will put a lower
bound on values of $t_{c}$ characterizing the chain of QDs. However, another
issue needs to be addressed before we can claim to have a realistic estimate
of sweep rate $v$ giving the smallest possible probability of error $Q$ in
transfer between a pair of dots. Electrons are affected by charge noise
unavoidable in semiconductor nanostructures, and coupled to lattice
vibrations. As we show in this paper, interactions with sources of electric
field noise and phonons in realistic Si- and GaAs-based structures are
dominating the physics of charge transfer in a wide range of sweep rates, with
nonadiabatic effects described by Landau-Zener theory being relevant only for
very fast sweeps.

Since our focus here is on open system character of an electron tunneling
between two quantum dots, we use the above-described simplest possible two-
level model of the closed system. Taking into account the spin degree of
freedom and spin-orbit cupling that affects its dynamics during the electron
motion in GaAs [30] (and to a smaller extent in silicon [30, 31]), and then a
valley degree of freedom in Si [32, 33, 34], leads to 4- or 8-level models
with multiple anticrossings of states [30, 35, 36, 37, 38, 39, 31, 40, 41].
The two-level model used here exhibits a simpler behavior in closed system
case, and using it will typically lead to an underestimation of unwanted
effects due to not-slow-enough sweeps (for a closed system), and coupling to
environment (for an open system). The results given in this paper consequently
correspond to the best-case scenario for given $t_{c}$ and assumed magnitudes
of charge noise and temperature.

The physics of Landau-Zener effect in presence of coupling to environment has
obviously been a subject of multiple works. Dissipative adiabatic evolution
affected by coupling to bosonic baths having Ohmic spectrum was most often
considered [42, 43, 44, 45, 46, 47]. It is known [48, 49] that coupling to
zero-temperature bath suppresses the final occupation of the higher-energy
state (“the electron being left behind in the initial dot” in the physical
scenario of interest here), while at finite temperature this occupation can be
enhanced [50, 51, 52, 43, 53]. Such effects of coupling to low-temperature
reservoirs were discussed in many physical contexts [54, 42, 55]. Stochastic
modifications of LZ parameters were also considered [56, 57, 58], including
fast classical fluctuations [59] and noise characterized by non-trivial
spectral density [60, 61, 62, 63, 39], including 1/f type noise, the tail of
which also resulted in incoherent transitions between the states [64, 39, 65].
In this paper we focus on quantum dots based on silicon and GaAs, and employ
realistic models of charge noise (having both Johnson/Ohmic and $1/f$ type
spectra, and coupling to both $\epsilon$ and $t_{c}$), and phonon interaction
with an electron confined in a double quantum dot. We use the Adiabatic Master
Equation [66, 53, 42], in which the influence of the environment (actually a
few distinct reservoirs in the case discussed here) is modeled with energy-
dependent rates of transitions between instantaneous eigenstates of the slowly
changing Hamiltonian of the system. For negligible probability of coherent
Landau-Zener excitation, this approached reduces to a simple differential rate
equation [67, 68]), which we solve in a way analogous to the one described in
[53].

During the detuning sweep, the energy gap between eigenstates of instantaneous
Hamiltonian varies between $t_{c}\sim$ $10$ $\mu$eV and largest value of
$\epsilon\\!\sim\\!1$ meV. With temperatures in experiments typically around
$100$ mK, corresponding to thermal energy of $\approx\\!10$ $\mu$eV, we should
expect a nontrivial role of temperature dependence of rates of energy
absorption and emission by the reservoirs. Note that in our previous work [39]
we have focused on influence of classical (i.e. high-temperature) $1/f$ charge
noise on electron transfer. Here we address the situation of lower
temperatures/larger tunnel couplings, taking into account the quantum limit
[69] of both $1/f$ noise from two-level fluctuators present in the
nanostructure, and Johnson noise from reservoirs of free electrons, while
furthermore considering the coupling of the moving electron to phonons.
Coupling to all these thermal reservoirs gives transition rates,
$\Gamma_{+/-}(\Omega)$ for transfer of energy $\Omega$ from/to the
environment, that nontrivially depend on $\Omega$. The detailed balance
between them, which reads $\Gamma_{+}/\Gamma_{-}\\!=\\!e^{-\beta\Omega}$, has
the following general consequence for the dynamics of the system. With the
system initially in ground state, transitions into an excited are
exponentially suppressed for large negative detunings, and they start to
become increasingly efficient as we approach the anticrossing of levels, at
which the gap is minimal and equal to $t_{c}$. This effect of enhancement of
excitation rate at the anticrossing is additionally strengthened in the
considered system by the fact that an electron delocalized between the two
dots is more susceptible to both charge noise and interaction with phonons (as
the transitions between states localized in each dot that govern the dynamics
in far-detuned regimes are suppressed by small overlap of wavefunctions). The
finite occupation of the “wrong” dot generated during passing through
$|\epsilon|\\!\lesssim\\!t_{c}$ region can then be diminished (“healed” in the
terminology used below) by processes of energy emission into the reservoirs
that dominate over processes of energy absorption by them when
$\epsilon\\!\gg\\!k_{B}T$. Arriving at the final result of interplay between
environment-induced excitation near the anticrossing, and the subsequent
energy relaxation (the environment-assisted dissipative tunneling into the
“correct” final state), requires consideration of realistic coupling to all
the reservoirs at temperatures and sweep rates relevant for experiments in
quantum dots. Such a careful consideration is the goal of this paper.

Our key qualitative result concerning application to realistic quantum dots,
is that in Si-based structures (both Si/SiGe and SiMOS) the dominant process
disturbing the adiabatic evolution close to anticrossing of levels is due to
charge noise (with coupling to phonons giving transition rates $3$ order of
magnitude smaller than those estimated for charge noise), and the finite
probability of leaving the electron behind is subsequently diminished by
relaxation processes due to charge noise and phonons that occur at large
detunings only when the transfer is very slow. On the other hand, in
GaAs/AlGaAs structures the piezoelectric coupling to phonons dominates over
coupling to charge noise over a wide range of detunings, and consequently the
processes involving energy exchange between the transferred electron and
lattice vibrations dominate the physics of the problem. The longer the charge
transfer takes, the more time the system spends in far-detuned regime in which
the energy gap exceeds thermal energy, and the closer it gets to a thermalized
state characterized by small occupation of higher-energy level, i.e. of the
electron being in the wrong dot. Phonons thus help in maintaining a
deterministic character of the charge transfer. These conclusions are quite
robust against modifications of parameters of high-frequency properties of
Johnson and $1/f$ type charge noises considered here.

The article is organized in the following way, in Sec. II we set up the
problem for the closed system and discuss the adiabatic condition for its
dynamics, introduce the Adiabatic Master Equation as an approach to open
system dynamics, and discuss a few physically transparent (and, as we show
later, relevant for the case of electron transfer in silicon- and GaAs-based
quantum dots) approximate solutions of this equation. In Section III we
calculate the detuning-dependent transition rates between instantaneous
eigenstates of the two-level Hamiltonian. We perform calculations for coupling
to phonons, and finite-temperature environments that cause charge noise of
both Johnson and $1/f$ type in detuning and tunnel coupling. We give there a
discussion of expected amplitude of $1/f$ noise at GHz frequencies relevant
for transitions during electron transfer in realistic GaAs- and silicon-based
quantum dots. Finally, in Section IV we use these rates to calculate the
dynamics of the electron driven adiabatically through an anticrossing of
levels associated with the two dots, and show a qualitative difference between
resulting probability of “leaving the electron behind” between GaAs- and
silicon-based quantum dots. In the last Section we discuss some of the
implications of these results for experimental efforts aimed at using chains
of quantum dots for coherent shuttling of electron spin qubits.

##  II Model of System’s dynamics

###  II.1 Adiabatic condition for closed system

We consider two energy levels that in the double quantum dot case correspond
to the lowest-energy orbital states localized in each of the two dots,
$\ket{L}$ and $\ket{R}$. In case of silicon QDs we assume that the valley
splitting is large enough for us to consider a single anticrossing of two
lowest-energy valley-orbital levels. We also neglect the spin degree of
freedom - interplay between the nonadiabatic effects in charge transfer and
dynamics of the spin of the transferred electron will be discussed elsewhere
[70]. We therefore work with the model defined by Hamiltonian from Eq. (1), in
which we now assume that $\epsilon$ and $t_{c}$ depend on time $\tau$.

For any value of $\epsilon(\tau)$ and tunnel coupling $t_{c}(\tau)$, the
Hamiltonian $\hat{H}(\tau)$ has eigenstates

$\displaystyle\ket{+,\theta(\tau)}$
$\displaystyle=\cos[\theta(\tau)/2]\ket{R}+\sin[\theta(\tau)/2]\ket{L}$
$\displaystyle\ket{-,\theta(\tau)}$
$\displaystyle=\cos[\theta(\tau)/2]\ket{L}-\sin[\theta(\tau)/2]\ket{R},$ (3)

where $\theta(\tau)=\arccot(-\epsilon(\tau)/|t_{c}(\tau)|)$. The discussion of
nonadiabatic effects due to time-dependence of $\epsilon$ and $t_{c}$, or
effects of interaction with the environment, is most transparent if we
transform the state of the system into an “adiabatic frame” [71]: instead of
working with $\ket{\psi(\tau)}$ which fulfills
$i\partial_{\tau}\ket{\psi(\tau)}=\hat{H}(\tau)\ket{\psi(\tau)}$ we work with
$|\tilde{\psi}(\tau)\rangle\\!\equiv\\!\hat{U}[\theta(\tau)]\ket{\psi(\tau)}$,
where a time-dependent unitary operator

$\hat{U}[\theta(\tau)]=\exp[i\hat{\sigma}_{y}\theta(\tau)/2]\,\,.$ (4)

transforms the $R/L$ states into the instantaneous eigenstates of
$\hat{H}(\tau)$: $\ket{+(-),\theta(\tau)}=\hat{U}[\theta(\tau)]\ket{R(L)}$.
One can see that for a perfectly adiabatic evolution of the system, for which
an initial superposition of eigenstates of $\hat{H}(\tau_{i})$ at given
$\tau_{i}$ evolves into the same superposition of eigenstates of
$\hat{H}(\tau_{f})$ at the final time $\tau_{f}$, the transformed
$|\tilde{\psi}(\tau)\rangle$ state is time-independent. Indeed, the evolution
in the adiabatic frame is controlled by

$\mathcal{H}(\tau)=\hat{U}[\theta(\tau)]\hat{H}(\tau)\hat{U}^{\dagger}[\theta(\tau)]-i\hat{U}[\theta(\tau)]\left(\frac{\partial\hat{U}^{\dagger}[\theta(\tau)]}{\partial\tau}\right)\,\,,$
(5)

which for the system discussed here reads

$\hat{\mathcal{H}}(\tau)=\frac{\Omega(\tau)}{2}\hat{\varsigma}_{z}-\frac{\dot{\theta}(\tau)}{2}\hat{\varsigma}_{y},$
(6)

where $\hat{\varsigma}_{z}$, $\hat{\varsigma}_{y}$ are Pauli operators in
$\ket{+,\theta(\tau)},\ket{-,\theta(\tau)}$ basis of instantaneous eigenstates
of the time-dependent Hamiltonian $\hat{H}(\tau)$,
$\dot{\theta}\\!=\\!\mathrm{d}\theta(\tau)/\mathrm{d}\tau$, and the
instantaneous energy splitting is

$\Omega(\tau)=\sqrt{\epsilon^{2}(\tau)+t_{c}^{2}(\tau)}\,\,.$ (7)

We assume the electron is initialized in the ground state at large negative
detuning $\epsilon(-\tau_{\infty})\ll t_{c}$, such that the initial state
$\ket{\psi(-\tau_{\infty})}=\ket{-,\theta(-\tau_{\infty})}\\!\approx\\!\ket{L}$.
Due to non-negligible coupling between the adiabatic states during the
system’s evolution (i.e. a nonzero $\dot{\theta}(\tau)$ term in Eq. (6)), a
non-zero occupation of excited state $\ket{+,\theta(\tau)}$ can be generated.
When the detuning sweep terminates at large $\epsilon(\tau_{\infty})\gg
t_{c}$, the occupation of excited state defines the transfer error, i.e. the
probability of the electron being left behind in the $L$ dot:

$Q=|\bra{\psi(\tau_{\infty})}\ket{+,\theta(\tau_{\infty})}|^{2}\approx|\innerproduct{\psi(\tau_{\infty})}{L}|^{2}.$
(8)

The calculation of $Q$ for an electron coupled to environments relevant for
semiconductor-based gated quantum dots is the main goal of this paper.

For constant tunnel coupling $t_{c}$, and for $\epsilon(\tau)=v\tau$ we are
dealing with the well-known Landau-Zener model [72], in which $Q$ is given by
$Q_{LZ}$ from Eq. (2). We concentrate here on the adiabatic regime, defined by
$Q_{\text{LZ}}\\!\ll\\!1$, which implies $t_{c}^{2}/v\\!\gg\\!1$, and means
that the ratio of “transverse” and “longitudinal” terms in the effective
Hamiltonian, Eq. (6), fulfills

$\frac{\dot{\theta}}{\Omega}=\frac{vt_{c}}{\Omega^{3}}<\frac{v}{t_{c}^{2}}\ll
1\,\,.$ (9)

This is the adiabatic condition for the dynamics of closed and noise-free
system. When it is fulfilled during detuning sweep, the electron remains at
all times in the ground state $\ket{-,\theta(\tau)}$, which means it
physically moves from the state initially localized in the left dot
$\ket{-,\theta(-\tau_{\infty})}\\!=\\!\ket{L}$, to a final state
$\ket{-,\theta(\tau_{\infty}))}\\!=\\!\ket{R}$, located in the right dot.

###  II.2 Dynamics of an open system

We use here Adiabatic Master Equation (AME) approach [51, 66, 53, 55], in
which transitions caused by the environment occur between the instantaneous
eigenstates of $\hat{H}(\tau)$, which are given by Eqs. (II.1). Our focus on
the adiabatic regime ($t_{c}^{2}\gg v$), combined with relatively weak
coupling to charge noise (with noise RMS $\sigma\ll t_{c}$) and short
intrinsic correlation time of phonon bath allow us to use here a Lindbladian
form of AME [43, 42], which reads:

$\partial_{\tau}{\hat{\varrho}}_{A}=i[\hat{\mathcal{H}}(\tau),\hat{\varrho}_{A}]+\Gamma_{+}(\tau)\mathcal{D}[\hat{\varsigma}_{+}]\hat{\varrho}_{A}+\Gamma_{-}(\tau)\mathcal{D}[\hat{\varsigma}_{-}]\hat{\varrho}_{A},$
(10)

where
$\hat{\varrho}_{A}=\hat{U}_{y}(\theta)\hat{\varrho}_{\text{LR}}\hat{U}_{y}^{\dagger}(\theta)$
and $\varrho_{\text{LR}}$ is the density matrix of the system before switching
the description to the “adiabatic” frame,
$\hat{\varsigma}_{+}=\outerproduct{+,\theta}{-,\theta}$,
$\hat{\varsigma}_{-}=\outerproduct{-,\theta}{+,\theta}$, and

$\mathcal{D}[\hat{o}]\hat{\varrho}=\hat{o}\hat{\varrho}o^{\dagger}-\frac{1}{2}\anticommutator{\hat{o}^{\dagger}\hat{o}}{\hat{\varrho}}$
(11)

is the Linbladian associated with operator $\hat{o}$ and time-dependent
relaxation/excitation rate $\Gamma_{\pm}(\tau)$. In this approach these rates
depend on time though their dependence on the value of instantaneous energy
splitting $\Omega(\tau)$ from Eq. (7), i.e.
$\Gamma_{\pm}(\tau)\\!=\\!\Gamma_{\pm}[\Omega(\tau)]$. Below we will use both
notations, $\Gamma_{\pm}(\tau)$ and $\Gamma_{\pm}[\Omega(\tau)]$, depending on
context. In particular, if noise-induced excitations dominate over the Landau-
Zener effect due to deterministic time-dependence of $\hat{H}(\tau)$, i.e.
$Q_{\text{noise}}\gg Q_{\text{LZ}}$, the unitary evolution can be safely
neglected and Eq. (10) reduces to a simple rate equation

$\dot{Q}(\tau)\approx\Gamma_{+}(\tau)-Q(\tau)\big{(}\Gamma_{-}(\tau)+\Gamma_{+}(\tau)\big{)},$
(12)

where $Q(\tau)=\bra{+,\theta}\hat{\varrho}_{A}(\tau)\ket{+,\theta}$ denotes
occupation of the higher energy state $\ket{+,\theta}$ at time $\tau$.

Given the initial condition $Q(-\tau_{\infty})=0$, the solution to Eq. (12)
reads

$\displaystyle Q$
$\displaystyle=\int_{-\tau_{\infty}}^{\tau_{\infty}}\text{d}\tau\,\Gamma_{+}(\tau)e^{-\int_{\tau}^{\tau_{\infty}}\Gamma_{+}(\tau^{\prime})+\Gamma_{-}(\tau^{\prime})\text{d}\tau^{\prime}}=$
$\displaystyle=\int_{-\tau_{\infty}}^{\tau_{\infty}}\text{d}\tau\Gamma_{+}(\tau)e^{-\chi(\tau,\tau_{\infty})}.$
(13)

where

$\chi(\tau,\tau_{\infty})=\int_{\tau}^{\tau_{\infty}}\left(\Gamma_{+}+\Gamma_{-}\right)\text{d}\tau^{\prime}\,\,.$
(14)

###  II.3 Approximate solutions

Let us now discuss a few physically motivated approximate solutions for the
probability of ending up in the excited state at the end of the sweep $Q$,
i.e. the probability that the electron remains in the initial dot. We start
with a simplest perturbative approach to rate equation (12), assuming
$\Gamma_{\pm}\tau_{\infty}\\!\ll\\!1$. In the lowest order one can write:

$Q^{(1)}=\int_{-\tau_{\infty}}^{\tau_{\infty}}\Gamma_{+}(\tau)\text{d}\tau,$
(15)

As the energy needed for transition from ground to excited state comes from
thermal fluctuations of environment, the excitation rate $\Gamma_{+}$ is
strongly suppressed at low temperatures, when
$k_{B}T\\!\ll\\!t_{c}\\!\leq\Omega(\tau)$. At these temperatures the rate of
energy relaxation into the environment, $\Gamma_{-}(\Omega)$, is temperature-
independent, as the thermal occupation factor for environmental states of
energy $\Omega\\!\gg\\!k_{B}T$ is zero, and $\Gamma_{-}(\Omega)$ depends then
only on density of environmental states and coupling matrix elements. For all
the environments considered in this paper, these dependencies lead to a power-
law behavior of the rates, $\Gamma_{-}(\Omega)\\!\propto\\!\Omega^{a}$ with
$a\\!\in\\![-3,3]$ depending on the transition mechanism and range of
$\Omega$, see derivations in the next Section. As we assume the environment to
be in thermal equlibrium, the detailed balance condition, which reads
$\Gamma_{+}(\Omega)=\Gamma_{-}(\Omega)e^{-\beta\Omega}$, leads to
$\Gamma_{+}(\Omega)\propto\Omega^{a}e^{-\beta\Omega}$ with
$e^{-\beta\Omega}\\!\ll\\!1$ at low temperatures.

The excitation process takes then place in a narrow range of detunings around
the avoided crossing, as $\Gamma_{+}(\Omega)$ very quickly decreases when
$|\epsilon|$ increases. As $\Omega(\tau)\approx\beta t_{c}+\beta
v^{2}\tau^{2}/2t_{c}$ for $\epsilon\\!\ll\\!t_{c}$, we neglect in this regime
the $\epsilon$ dependence of $\Gamma_{+}(\Omega)$ and replace it with value
for $\Omega\\!=\\!t_{c}$ (equivalently: for $\tau\\!=\\!0$), while we keep it
in the thermal factor. The integrand in (15) can then be approximated as
$\Gamma_{+}(\tau)\approx\Gamma_{-}(0)e^{-\beta\Omega(\tau)}$, and the
integration can be done over a range of $|\epsilon|\\!\ll\\!t_{c}$. In this
way we obtain the Single Excitation Approximation Limit (SEAL):

$\displaystyle Q_{\text{SEAL}}$ $\displaystyle=\Gamma_{-}(0)e^{-\beta
t_{c}}\int_{-\infty}^{\infty}e^{-\frac{\beta
v^{2}t^{2}}{2t_{c}}}\text{d}\tau\,\,,$ $\displaystyle=\frac{\sqrt{2\pi
k_{\text{B}}Tt_{c}}}{v}\Gamma_{-}(0)e^{-\beta t_{c}}\,\,,$ (16)

which assumes that at most a single quantum jump from ground to excited state
takes place in the avoided crossing region.

The SEAL approximation does not take into account possibility of electron
transition in the opposite direction, i.e. from excited to ground state, which
would lead to partial recovery of ground state occupation - an effect that we
will refer to as a “healing” of excitation that occurred close to the
anticrossing. This effect is captured by the $\exp[-\chi(\tau)]$ factor in Eq.
(II.2) with $\chi(\tau)$, given in Eq. (14)), evaluated in the low-temperature
limit of $\Gamma_{-}\\!\gg\\!\Gamma_{+}$. The effect of transitions occuring
during the part of the sweep when $\epsilon(\tau)\\!>\\!t_{c}$ is captured by
a Healed Excitation Approximation Limit (HEAL):

$Q_{\text{HEAL}}\approx
Q_{\text{SEAL}}\exp{-\int_{0}^{\tau_{\infty}}\Gamma_{-}(\tau)\,\text{d}\tau}\,.$
(17)

The physical picture expected to hold at low $T$ is thus the following. A
finite $Q$ is generated due to coupling to a thermal reservoir near the
anticrossing, and then processes of emission of energy into this reservoir
lead to a diminishing of its final value at the end of the sweep, making the
final state of the system closer to the one following from an ideal adiabtic
evolution. Such a healing process results in environment-assisted inelastic
tunneling into ground state at the end of the driving, see Fig. 1. In Sec. IV
we will demonstrate in which regimes of parameters the SEAL/HEAL solutions are
applicable for realistic DQD devices.

Note that up to this moment we have not specified any particular form of
relaxation/excitation rates $\Gamma_{\pm}(\tau)$, which makes above
approximations suitable also for other systems described in terms of the L-Z
Hamiltonian (1), in the adiabatic limit ($t_{c}^{2}\gg v$) and coupled to
environment at relatively low-temperature ($t_{c}\gtrsim k_{\text{B}}T$).

Figure 1: A schematic picture of adiabatic transition between two quantum dots
in presence of charge noise and phonon bath. In panel (a) we show energy of
instantaneous states $\ket{\pm,\theta}$(red/blue lines) as a function of
detuning $\epsilon$ (lower axis) and orbital angle
$\theta=\acot(-\epsilon/t_{c})$ (upper axis). Despite detuning sweep being
adiabatic ($t_{c}^{2}\gg v$), the electron initialized in the left dot (blue
circle) can still end up with non-zero occupation of excited state localized
in right dot $Q$ (red circle), as a result of coupling to environment, which
leads to incoherent transitions between eigenstates of the instantaneous
Hamiltonian characterized by the rates $\Gamma_{\pm}(\Omega)$. At low
temperatures, the excitation rate is non-negligible only in vicinity of
avoided crossing, where the gap is smallest, $\Omega(0)=t_{c}$, while
relaxation accounts for recovery of ground state occupation (the “healing” of
the excitation at larger detuning. As the detuning is changed, the character
of eigenstates of the instantaneous Hamiltonia, $\ket{\pm,\theta}$, is
modified from dot-like character at $|\epsilon|\gg t_{c}$ to orbital-
molecular-like at $|\epsilon|\ll t_{c}$, which is accompanied by dominant role
of transverse (interdot) and longitudinal transitions, respectively, see Sec.
III.1. To illustrate the difference between them, in panel (b) we plot
$\cos^{2}\theta=|\bra{+,\theta}\sigma_{z}\ket{+,\theta}|^{2}$ (dashed blue)
and $\sin^{2}\theta=|\bra{+,\theta}\sigma_{x}\ket{+,\theta}|^{2}$ (dashed
green) factor that determine the relative importance of $\Gamma_{t}$ and
longitudinal $\Gamma_{\epsilon}$ relaxation channels, respectively. Insets
with green and blue frames schematically representing transition mechanisms
dominant in regimes of $\epsilon\\!\ll\\!-t_{c}$, $|\epsilon|\\!\ll\\!t_{c}$,
and $\epsilon\\!\gg\\!t_{c}$.

##  III Transition rates for an adiabatically transferred electron

###  III.1 General properties

We consider now a transfer of an electron between two quantum dots that is
driven by a detuning sweep slow enough to be adiabatic in the closed system
limit. After turning on a weak coupling to an environment, the transition
rates $\Gamma_{\pm}(\tau)$ in the Adiabatic Master Equation (AME) from Eq.
(10) are evaluated at given $\tau$ as if the system described by the
instantaneous Hamiltonian $\hat{H}(\tau)$ from Eq. (1), was subjected to an
off-diagonal coupling with an environment for a long enough time for Fermi
Golden Rule (FGR) calculation to be applicable. Thus the general form of
electron-environment coupling
$\hat{V}=\tfrac{1}{2}(\hat{V}_{t}\hat{\sigma}_{x}+\hat{V}_{\epsilon}\hat{\sigma}_{z})$
in the $\ket{L/R}$ basis, at given $\theta(\tau)$ should be expressed in the
basis of eigenstates of instantaneous Hamiltonian, $\ket{\pm,\theta}$, using
$\hat{\mathcal{V}}(\theta)=\hat{U}_{y}^{\dagger}(\theta)\hat{V}\hat{U}_{y}(\theta)$,
which leads to

$\hat{\mathcal{V}}(\theta)=\tfrac{1}{2}(\hat{\mathcal{V}}_{x}\,\hat{\varsigma}_{x}+\hat{\mathcal{V}}_{z}\,\hat{\varsigma}_{z}),$
(18)

where
$\hat{\mathcal{V}}_{x}=(\hat{V}_{t}\cos\theta+\hat{V}_{\epsilon}\sin\theta)$
and
$\hat{\mathcal{V}}_{z}=(\hat{V}_{\epsilon}\cos\theta-\hat{V}_{t}\sin\theta)$.
This means that at every $\tau$ we do the FGR calculation for
$\hat{\mathcal{V}}_{x}\hat{\varsigma}_{x}/2$ coupling, where
$\hat{\varsigma}_{x}$ acts in basis of eigenstates of the instantaneous
$\hat{H}(\tau)$. With the environmental Hamiltonian given by $\hat{H}_{E}$, we
calculate then the quantum spectral density for the operator
$\hat{\mathcal{V}}_{x}(t)=e^{i\hat{H}_{E}t}\hat{\mathcal{V}}_{x}e^{-i\hat{H}_{E}t}$,
given by [73, 69]

$S^{Q}_{\mathcal{V}}(\omega)=\int_{-\infty}^{\infty}\langle\hat{\mathcal{V}}_{x}(t)\hat{\mathcal{V}}_{x}(0)\rangle
e^{i\omega t}\mathrm{d}t$ (19)

where $\langle\ldots\rangle\\!=\\!\mathrm{Tr}_{E}(\hat{\rho}_{E}\ldots)$ is
the averaging over the environmental density matrix $\hat{\rho}_{E}$. The rate
of excitation of the system, i.e. a transition that involves taking energy
$\Omega(\tau)$ from the environment, is then given by [73, 69]

$\Gamma_{+}(\tau)=\frac{1}{4}S^{Q}_{\mathcal{V}}[-\Omega(\tau)]\,\,,$ (20)

while the relaxation rate is

$\Gamma_{-}(\tau)=\frac{1}{4}S^{Q}_{\mathcal{V}}[\Omega(\tau)]\,\,.$ (21)

For an environment in thermal equilibrium which we consider here, we have
$\hat{\rho}_{E}\\!\propto\\!e^{-\beta\hat{H}_{E}}$ and the detailed balance
condition, $S^{Q}_{V}(\Omega)\\!=\\!S^{Q}_{V}(-\Omega)e^{\beta\Omega}$, and
thus
$\Gamma_{-}[\Omega(\tau)]\\!=\\!\Gamma_{+}[\Omega(\tau)]e^{\beta\Omega(\tau)}$,
is fulfilled.

As the longitudinal $\hat{V}_{\epsilon}$ and transverse $\hat{V}_{t}$
couplings in dots basis are often of different physical origin, we assume
$\langle\hat{V}_{\epsilon}\hat{V}_{t}\rangle=0$, so that the transition rate
can be written as
$\Gamma_{\pm}(\tau)=\Gamma_{\epsilon,\pm}(\tau)+\Gamma_{t,\pm}(\tau)$, where
we introduced

$\displaystyle\text{(Longitudinal) :}\quad\Gamma_{\epsilon,\pm}(\tau)$
$\displaystyle=\frac{1}{4}\sin^{2}\theta\,\,S^{Q}_{\epsilon}[\mp\Omega(\tau)],\,$
$\displaystyle\text{(Transverse) :}\quad\Gamma_{t,\pm}(\tau)$
$\displaystyle=\frac{1}{4}\cos^{2}\theta\,\,S^{Q}_{t}[\mp\Omega(\tau)]$ (22)

contributions, defined using spectral densities of $\hat{V}_{\epsilon}$ and
$\hat{V}_{t}$ operators,
$S_{\epsilon}^{Q}(\omega)=\int\langle\hat{V}_{\epsilon}(t)\hat{V}_{\epsilon}(0)\rangle
e^{i\omega t}\text{d}t$ and $S_{t}^{Q}(\omega)=\int\langle
V_{t}(t)\hat{V}_{t}(0)\rangle e^{i\omega t}\text{d}t$. The
$\hat{V}_{\epsilon}\hat{\sigma}_{z}$ coupling that is longitudinal in the
$\ket{L/R}$ basis (the dot basis) appears due to fluctuations of detuning or
phonons coupling to the operator $\hat{\sigma}_{z}$. It is most efficient at
causing transitions between $\ket{\pm,\theta}$ states when the latter have a
molecular-orbital character, i.e. $\theta\\!\approx\\!\pi/2$,
$\sin^{2}\theta\\!\approx\\!1$,
$\ket{\pm,\pi/2}=\tfrac{1}{2}(\ket{L}\pm\ket{R})$, and
$|\epsilon|\\!\ll\\!t_{c}$. On the other hand, the transverse coupling
$\hat{V}_{t}\hat{\sigma}_{x}$ is due to fluctuations of tunnel coupling or
phonons coupling to the operator $\hat{\sigma}_{x}$. It leads to transitions
of interdot character between the states $\ket{L}\leftrightarrow\ket{R}$ that
correspond to $\ket{\pm,\theta}$ states at $\theta\\!\ll\\!1$ and
$\theta\\!\approx\\!\pi$ (i.e. $\epsilon\\!\ll\\!-t_{c}$ and
$\epsilon\\!\gg\\!t_{c}$, respectively), see Fig. 7b. Below we will see that
for all the considered mechanisms, the transverse processes are weaker than
the longitudinal ones, i.e. $S_{t}^{Q}(\Omega)\ll S_{\epsilon}^{Q}(\Omega)$,
so the latter could become relevant only in a very far-detuned regime.

For an electron in a double quantum dot, the relevant mechanisms of
transitions between the eigenstates are due to coupling of electron charge to
two reservoirs: lattice vibrations (phonons) and sources of fluctuations of
electric fields – free electrons in metallic electrodes and ungated regions of
semiconductor quantum well being the sources of Johnson noise, and bound
charges switching between a discrete number of states being the sources of
$1/f$ type noise [74]. Due to their distinct physical origin, we neglect
correlations between different transition mechanisms and write the relaxation
rate as:

$\Gamma_{-}=\Gamma_{-}^{(\text{ph})}+\Gamma_{-}^{(\text{1/f})}+\Gamma_{-}^{(\text{Joh})}.$
(23)

In the above, we separated charge noise contribution into
$\Gamma^{(\text{1/f})}$ due to tail of 1/f-like noise from two-level
fluctuators [65] in the quantum well interface and $\Gamma^{(\text{Joh})}$ due
to Johnson’s noise caused by wiring in vicinity of quantum dots [75]. As a
direct consequence of Eq. (23) the exact formula for leaving electron in the
initial dot reads:

$\displaystyle
Q=\int_{-\tau_{\infty}}^{\tau_{\infty}}\text{d}\tau\sum_{m}\Gamma_{+}^{(m)}(\tau)\exp(-\sum_{m^{\prime}}\chi^{(m^{\prime})}(\tau,\tau_{\infty})),$
(24)

where indices $m,m^{\prime}$ stands for phonon, 1/f or Johnson’s mechanisms,
while
$\chi^{(m)}(a,b)=\int_{a}^{b}\Gamma_{+}^{(m)}(\tau^{\prime})+\Gamma_{-}^{(m)}(\tau^{\prime})\text{d}\tau^{\prime}$.

Let us discuss now the quantum noise spectra relevant for the two types of
reservoirs being the sources of charge noise, and the lattice vibrations.

###  III.2 Charge noise

The way in which sources of charge noise couple to the electron in a DQD is
most easily visible if we consider the high-temperature (or low energy
transfer) limit of $\beta\Omega\\!\ll\\!1$. The quantum spectral density
becomes then symmetric in frequency,
$S^{Q}_{V}(\Omega)\\!=\\!S^{Q}_{V}(-\Omega)$ (so that
$\Gamma_{+}\\!=\\!\Gamma_{-}$), and it can be identified with a classical
power spectral density of a classical stochastic process describing the
fluctuations of the electric fields caused by the dynamics of the reservoir.
These processes manifest themselves as time-dependent corrections to
parameters of $\hat{\mathcal{H}}(\tau)$: $\delta\epsilon(\tau)$ and $\delta
t(\tau)$ for detuning and tunnel coupling noise, respectively. As long as the
amplitude of the noise is small ($\delta t,\delta\epsilon\ll t_{c}$), the
modification of the instantaneous splitting
$\Omega(\tau)=\sqrt{(v\tau+\delta\epsilon)^{2}+(t_{c}+\delta t)^{2}}$ is
negligible. However, time variation of $\delta\epsilon$ and $\delta t$
activates coupling between the eigenstates of the instantaneous Hamiltonian
from Eq. (6)), as in the lowest order in $\delta\epsilon$ and $\delta t$ we
have

$\displaystyle\dot{\theta}$
$\displaystyle=\frac{\partial}{\partial\tau}\acot(-\frac{v\tau+\delta\epsilon}{t_{c}+\delta
t})\approx\frac{\sin\theta{\delta\dot{\epsilon}}+\cos\theta{\delta\dot{t}}}{\Omega_{0}}\,\,,$
(25)

where $\Omega_{0}\\!=\\!\sqrt{v^{2}\tau^{2}+t_{c}^{2}}$, and the last
approximation relies on $t_{c}^{2}\gg v$ assumption to neglect contributions
not larger than the noiseless coupling
$\dot{\theta}_{0}=vt_{c}/\Omega_{0}^{2}\ll\Omega$, see Eq. (9). As we neglect
correlations between $\delta\epsilon$ and $\delta t$, we treat the transitions
induced by these two noises independently. Taking then into account that the
classical spectrum
$S^{cl}_{\dot{x}}(\omega)\\!=\\!\int\langle\dot{x}(t)\dot{x}(0)\rangle
e^{i\omega t}\mathrm{d}t$ (where $\langle\ldots\rangle$ denotes now averaging
over realizations of noise) is related to the classical spectrum of $x(t)$
noise by $S^{cl}_{\dot{x}}(\omega)\\!=\\!\omega^{2}S^{cl}_{x}(\omega)$, and
that $\sin^{2}\theta\\!=\\!t_{c}^{2}/\Omega_{0}^{2}$ and
$\cos^{2}\theta\\!=\\!(v\tau)^{2}/\omega_{0}^{2}$, we have

$\displaystyle\Gamma_{\pm,\epsilon}(\tau)$
$\displaystyle=\frac{1}{4}\frac{t_{c}^{2}}{\Omega^{2}_{0}(\tau)}S^{cl}_{\epsilon}[\mp\Omega_{0}(\tau)]\,\,,$
(26) $\displaystyle\Gamma_{\pm,t}(\tau)$
$\displaystyle=\frac{1}{4}\frac{(v\tau)^{2}}{\Omega^{2}_{0}(\tau)}S^{cl}_{t}[\mp\Omega_{0}(\tau)]\,\,.$
(27)

In these equations the $\mp\Omega_{0}$ arguments can be, of course, replaced
by $|\Omega_{0}|$, as the classical spectra are symmetric in frequency. In
Appendix A we give an alternative derivation of these results (in the spirit
of methods used previously in [39, 61]). We also show there that the AME
calculation using these rates agrees very well with direct averaging of
evolution due to $\hat{H}(\tau)$ averaged over realizations of classical noise
with experimentally relevant parameters (discussed below in this Section). In
this way we check the applicability of AME to the system of interest in this
paper in the classical noise/high temperature regime.

Eqs. (26) and (27) connect the rates as given $\tau$ with (classical) spectra
of appropriate noise at $\pm\Omega_{0}$ frequencies. Extension of AME to
regime of lower temperatures/higher $\Omega_{0}$ amounts then to replacing the
classical spectra, $S^{cl}(\pm\Omega_{0})$, by their quantum counterparts,
$S^{Q}(\pm\Omega_{0})$. Let us now discuss the classical and quantum regimes
for the two charge noise spectra relevant for semiconductor quantum dots in
GHz range ($t_{c}\sim\\!10$ $\mu$eV) of energies.

First we consider electric fluctuations from electron gas in metallic gates,
the Johnson-Nyquist noise of general form [76, 69, 77]:

$S^{Q,(J)}_{\epsilon}(\omega)=\frac{\Re{Z}}{R_{q}}\frac{\omega}{1-e^{-\beta\omega}}$
(28)

where $R_{q}$ is the inverse of conductance quantum $R_{q}=\pi/e^{2}=13$
$\text{k}\Omega$ and $Z$ is the impedance of a noise source, which we model
here as an ideal resistor (R) of the impedance given by typical for microwaves
$Z_{R}=R=50\Omega$. The temperature-dependent part of Eq. (28) reduces to
Bose-Einstein distribution $n(\omega)=1/(e^{\beta\omega}-1)$ for negative
frequencies $\omega<0$ (absorption) and $n(\omega)+1$ for $\omega>0$
(stimulated and spontaneous emission). In the $\gtrsim$ GHz frequency range
relevant here, Johnson noise from a lossy transmission line discussed in [78]
for Si/SiGe quantum dot, gives at most an order of magnitude larger noise
power.

Next, we consider $1/f$-type fluctuations of electric field due to two-level
fluctuators (TLFs) localized in the insulating regions of the nanostructure
[74]. We focus first on noise in detuning, as there are numerous measurements
of spectrum of this noise in DQDs. Due to very high spectral weight at low
frequencies such a $1/f$ noise dominates the dephasing of qubits the energy
splitting of which depends on electric fields [74, 79]. Here, however, we
focus on high (GHz range) positive and negative frequency behavior of
$S^{Q,(1/f)}(\omega)$ that is of $1/|\omega|^{\alpha}$ character at very low
frequencies. The behavior of quantum noise caused by an ensemble of TLFs at
such frequencies depends on microscopic details of these fluctuators and the
distribution of their parameters, see [65] and references therein.

Here, as in [80], where Si/SiGe charge qubit in a DQD was considered, we take
$\alpha\\!=\\!1$ with noise amplitude directly extrapolated from the low-
frequency regime, i.e. for positive-frequency quantum spectrum we have

$S^{Q,(1/f)}_{\epsilon}(\omega>0)=s_{1}(T)\frac{\omega_{1}}{\omega},$ (29)

where $\omega_{1}=2\pi$/s and $s_{1}(T)=S^{cl}(\omega_{1})$ is a commonly
reported classical spectral density at $f=1$Hz, which at electron temperature
of $T=100$mK in typical Si/SiGe device is given by
$s_{1}(100\text{mK})\approx(0.3-2)^{2}\,\mu$eV2/Hz [81, 82, 83, 84, 85]. As
$s_{1}(T)\propto T$ scaling was observed in experiments on quantum dots [83,
85, 84], we assume here
$s_{1}(T)\\!=\\!s_{1}(100\text{mK})\tfrac{T}{100\text{mK}}$. The negative-
frequency quantum spectrum follows from Eq. (29) using the detailed balance
condition. It is commonly believed that charge disorder in Si-MOS should have
larger amplitude, for example $s_{1}(100$mK$)\approx 10\mu$eV2/Hz was measured
in [86] at $T=300$mK. However, following [87] and references therein, we
assume that the $1/f$ noise amplitude in Si-MOS can be made comparable or even
smaller then Si/SiGe [83].

Let us stress that the character of noise generated by an ensemble of TLFs
above $\sim$ MHz frequency is not universal, as its amplitude and exponent
varies between the DQDs materials and devices. In particular, recent
measurements of charge noise in Si/SiGe [88] and Si-MOS [89] showed $1/f$ and
$1/f^{0.7}$ scaling up to $100$MHz and $1$MHz respectively, which contrasted
with few orders of magnitude weaker amplitude of charge noise at MHz
frequencies in some of GaAs singlet-triplet qubits [90, 91]. Additionally, in
neither experiment a linear scaling of spectral density with temperature was
seen at highest frequencies, and in particular the Si/SiGe case showed only
weak dependence on the temperature, confirmed also elsewhere for SiMOS [92,
87], which stood in contrast to GaAs device, where $S(\omega)\propto T^{2}$
and the spectrum became flat, i.e $\alpha\to 0$ as $T$ was increased [90]. A
recent theoretical study [65] of qubit relaxation caused by interaction with
an ensemble of TLFs coupled to thermal bath (which create $1/f$ noise at low
frequencies) showed that at high positive frequencies (between MHz and GHz,
depending on temperature), a crossover first to $S^{Q}(\omega)\propto
1/\omega^{2}$, and then to a flat or Ohmic spectrum (depending on details of
distribution of energy splitting of the TLFs) occurs. One can thus expect that
in measurement of high-frequency quantum noise, it is difficult to distinguish
the noise caused by TLFs from other sources of electric field fluctuations, as
a flat spectrum has been observed already at MHz frequency in SiMOS QD
spectroscopy [93]. Let us note that one of the models of distribution of
energies of TLFs considered in [65] led to $S^{Q}(\omega>0)\\!\propto\\!T$ at
high frequencies. In light of the above discussion, we use the above model to
estimate the relevance of the tail of 1/f noise in silicon-based devices and
set $s_{1}(0.1\text{K})\\!=\\!1^{2}\mu$eV2/Hz. We will use the same spectrum
for GaAs, probably overestimating the noise in this case, but below we will
show that for GaAs quantum dots the influence of electron-phonon coupling
dominates over that of charge noise having even such a large amplitude.

For the charge noise in tunnel coupling, we assume that it is uncorrelated
with the noise in detuning, as the two are caused by TLFs from distinct
spatial regions. We parametrize the ratio of rms of fluctuations of the noise
in $t_{c}$ and $\epsilon$ by
$\eta=\sqrt{S_{t}(\omega)/S_{\epsilon}(\omega)}\\!\approx\\!0.1$, with its
value motivated by semiconductor quantum dots experiments [94, 95, 90, 88],
and typical values of level arm used to control the electronic gates during
shuttling [25]. We conclude this section by giving the explicit forms of
longitudinal and transverse contributions to relaxation rates due to charge
noise

$\displaystyle\Gamma_{-,\epsilon}(\tau)$
$\displaystyle=\frac{1}{4}\left(\frac{t_{c}}{\Omega(\tau)}\right)^{2}S^{Q}_{\epsilon}[\Omega_{0}(\tau)]\,\,,$
(30) $\displaystyle\Gamma_{-,t}(\tau)$
$\displaystyle=\frac{\eta^{2}}{4}\left(\frac{v\tau}{\Omega(\tau)}\right)^{2}S^{Q}_{\epsilon}[\Omega_{0}(\tau)]\,\,,$
(31)

which are applicable for both $1/f$ and Johnson noise. The corresponding
excitation rates are obtained via detailed balance condition
$\Gamma_{+}(\Omega)=\Gamma_{-}(\Omega)e^{-\beta\Omega}$.

###  III.3 Electron-phonon interaction

In semiconductors, another mechanism responsible for transitions between the
$\ket{\pm,\theta}$ states is associated with energy exchange between the
electron and lattice vibrations. Phonons are assumed to be in thermal
equilibrium, with their free Hamiltonian given by
$H_{\text{ph}}=\sum_{\mathbf{k},\lambda}\omega_{\mathbf{k},\lambda}b^{\dagger}_{\mathbf{k},\lambda}b_{\mathbf{k},\lambda}$,
where $\lambda$ and $\mathbf{k}$ represents phonon polarizations and
wavevector respectively. The electron-phonon interaction is given by [96]:

$\displaystyle\mathcal{H}_{\text{el-ph}}$
$\displaystyle=\sum_{j,\mathbf{k},\lambda=L,T}\sqrt{\frac{|\mathbf{k}|}{2\varrho
c_{\lambda}V}}\,v^{(j)}_{\mathbf{k},\lambda}\big{(}\hat{b}_{\mathbf{k},\lambda}+\hat{b}_{-\mathbf{k},\lambda}^{\dagger}\big{)}e^{i\mathbf{k}\mathbf{r}}\,\,,$
(32)

in which $\varrho$ denotes crystal density, $V$ the crystal volume, and
$c_{\lambda}$ is the speed of $\lambda$-polarized phonons. The coupling
$v_{\mathbf{k},\lambda}^{(j)}$ stands for piezoelectric ($j=p$) and
deformation potential ($j=d$), evaluated for transverse ($\lambda=T$) and
longitudinal ($\lambda=L$) polarizations of phonons:

$v^{(\text{p})}_{\mathbf{k},\lambda}=\frac{\chi_{p}}{k},\quad
v^{(\text{d})}_{\mathbf{k},L}=\Xi_{d}+\Xi_{u}\left(\frac{k_{z}}{k}\right)^{2},\quad
v^{(\text{d})}_{\mathbf{k},T}=-\Xi_{u}\frac{k_{xy}k_{z}}{k^{2}},$ (33)

where $\chi_{p}$ is piezoelectric constant, while $\Xi_{d}$, $\Xi_{u}$ are
dilatation and shear deformation potentials respectively. In GaAs and Si the
coupling to phonons takes a very different form, namely Si lacks the dominant
in the GaAs piezoelectric coupling $\chi_{p}^{\text{SiGe}}=0$ [97], while the
opposite is true for shear deformation potential since
$\Xi_{u}^{\text{GaAs}}=0$.

We evaluate the matrix elements of interaction from Eq. (32) in the two-
dimensional space spanned by $\ket{L/R}$ states (see Appendix B for details),
obtain the $\hat{V}_{t}\hat{\sigma}_{x}+\hat{V}_{\epsilon}\hat{\sigma}_{z}$
form of coupling discussed in Sec. III.1, and arrive at quantum spectra
associated with longitudinal ($\hat{V}_{\epsilon}$) and transverse
($\hat{V}_{t}$) couplings to phonons:

$S_{\epsilon/t}^{\text{(ph)}}[\Omega(\tau)]=4\pi\sum_{j,\mathbf{k},\lambda}\frac{k|v_{\mathbf{k},\lambda}^{(j)}|^{2}}{\varrho
c_{\lambda}V}|\mathcal{M}_{\epsilon/t}(\mathbf{k})|^{2}\frac{\delta(\Omega(\tau)-\omega_{\mathbf{k},\lambda})}{1-e^{-\beta\omega_{\mathbf{k},\lambda}}},$
(34)

where the matrix elements read
$\mathcal{M}_{\epsilon}(\mathbf{k})=\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{L}-\bra{R}e^{i\mathbf{k}\mathbf{r}}\ket{R}$,
and
$\mathcal{M}_{t}(\mathbf{k})=\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{R}+\bra{R}e^{i\mathbf{k}\mathbf{r}}\ket{L}$,
while the temperature-dependent term reduces to Bose-Einstein distribution
$n(\Omega)$ for $\Omega<0$ (absorption) and to $n(\omega)+1$ for $\Omega>0$
(emission). The transition rates are given by

$\displaystyle\Gamma_{\pm,\epsilon}^{(\text{ph})}(\tau)$
$\displaystyle=\frac{1}{4}\sin^{2}\theta\,S_{\epsilon}^{(\text{ph})}[\mp\Omega(\tau)]$
(35) $\displaystyle\,\Gamma_{\pm,t}^{(\text{ph})}(\tau)$
$\displaystyle=\frac{1}{4}\cos^{2}\theta\,S_{t}^{(\text{ph})}[\mp\Omega(\tau)]\,\,.$
(36)

For further calculation we need to specify a model of
$\bra{\mathbf{r}}L_{0}/R_{0}\rangle$ wavefunctions localized in the uncoupled
dots. We assume that they are separable and Gaussian:

$\bra{\mathbf{r}}\ket{L_{0}/R_{0}}=\frac{1}{(\pi^{3}r_{xy}^{4}r_{z}^{2})^{\frac{1}{4}}}\exp{-\frac{(x\mp\tfrac{\Delta
x}{2})^{2}+y^{2}}{2r_{xy}^{2}}-\frac{z^{2}}{2r_{z}^{2}}},$ (37)

where the full width at half maximum (FWHM) of electron wavefunction, which
defines dots diameter, is given by $2r_{xy}$ in planar, and $2r_{z}$ in the
growth direction of structure, while the $\Delta x$ gives the distance between
the dots. Next we use Hund-Mulliken approximation [98, 99], to generate a set
of orthogonal states in DQDs system that fulfill $\bra{L}\ket{R}=0$, which can
be done by setting:

$\ket{L/R}=\ket{L_{0}/R_{0}}-g\ket{R_{0}/L_{0}},$ (38)

where $g\\!\approx\\!\frac{1}{2}\bra{L_{0}}\ket{R_{0}}=\tfrac{1}{2}e^{-\Delta
x^{2}/4r_{xy}^{2}}\ll 1$.

As the energy quanta exchanged between the electron and the lattice are
$<\\!1$ meV, we take into acocunt only the acoustic phonons with
$\omega_{\mathbf{k},\lambda}=c_{\lambda}|\mathbf{k}|$. The relaxation rates
due to electronphonon interaction are then given by:

$\displaystyle\Gamma_{-,\epsilon}^{\text{(ph)}}(\Omega)$
$\displaystyle=\sum_{\lambda,j}\frac{\Omega\,t_{c}^{2}}{8\pi^{2}\varrho
c_{\lambda}^{5}}\int\text{d}\Omega_{\mathbf{k}}\big{|}v_{\mathbf{k}_{\lambda},\lambda}^{(j)}\big{|}^{2}|F_{\epsilon}(\mathbf{k}_{\lambda})|^{2}[n(\Omega)+1]$
(39) $\displaystyle\Gamma_{-,t}^{\text{(ph)}}(\Omega)$
$\displaystyle=\sum_{\lambda,j}\frac{\Omega\,\epsilon^{2}}{8\pi^{2}\varrho
c_{\lambda}^{5}}\absolutevalue{\bra{L_{0}}\ket{R_{0}}}^{2}\int\text{d}\Omega_{\mathbf{k}}\big{|}v_{\mathbf{k}_{\lambda},\lambda}^{(j)}\big{|}^{2}|F_{t}(\mathbf{k}_{\lambda})|^{2}$
$\displaystyle[n(\Omega)+1]\,\,,$ (40)

where the integration over solid angle of resonant wavevector
$\mathbf{k}_{\lambda}$, with length $k_{\lambda}=\Omega/c_{\lambda}$ was
denoted by
$d\Omega_{\mathbf{k}}=d\vartheta_{\mathbf{k}}d\varphi_{\mathbf{k}}\sin\vartheta_{\mathbf{k}}$,
while the form factors read:

$\displaystyle|F_{t}(\mathbf{k})|^{2}$
$\displaystyle=\exp(-\frac{k_{xy}^{2}r_{xy}^{2}+k_{z}^{2}r_{z}^{2}}{2})\,\big{(}1-\cos(\tfrac{k_{x}\Delta
x}{2})\big{)}^{2}$ $\displaystyle|F_{\epsilon}(\mathbf{k})|^{2}$
$\displaystyle=\exp(-\frac{k_{xy}^{2}r_{xy}^{2}+k_{z}^{2}r_{z}^{2}}{2})\,\sin^{2}(\tfrac{k_{x}\Delta
x}{2}).$ (41)

The common term $\exp(-(k_{xy}^{2}r_{xy}^{2}+k_{z}^{2}r_{z}^{2})/2)$ is the
Fourier transform of the electron wavefunction, while the main difference
between the longitudinal and transverse relaxation is the overlap of bare dots
wavefunctions, $\absolutevalue{\bra{L_{0}}\ket{R_{0}}}^{2}=e^{-\Delta
x^{2}/2r_{xy}^{2}}\ll 1$ which makes the transverse relaxation mechanism
orders of magnitude weaker, i.e.
$\Gamma_{-,t}^{(\text{ph})}\ll\Gamma_{-,\epsilon}^{(\text{ph})}$, unless
detuning is so large that $\theta$ is close enough to $\pi$ for
$\sin^{2}\theta$ term in Eq. (35) suppresses
$\Gamma_{-,\epsilon}^{(\text{ph})}$ to the degree that is becomes smaller than
$\Gamma_{-,t}^{(\text{ph})}$.

Quantity | Symbol | Values
---|---|---
Tunnel coupling | $t_{c}$ |  $5-60\mu$eV
Effective electron temperature | $T$ |  $50-500$mK
Detuning sweep rate | $v$ |  $1-3000\mu$eV/ns
Initial detuning | $\epsilon_{i}$ |  $-500\mu$eV
Final detuning | $\epsilon_{f}$ |  $500\mu$eV
Time of detuning sweep | $\epsilon_{f}-\epsilon_{i}=\Delta\epsilon/v$ |  $0.3-1000$ns
Transverse/longitudinal noise ratio | $\sqrt{S_{t}(\omega)/S_{\epsilon}(\omega)}$ | $0.1$
Resistance of noisy resistor (Johnson noise) | $Z_{R}$ | $50\Omega$
1/f noise amplitude at $T=0.1$K |  $s_{1}(0.1$K$)$ | $1\mu\text{eV}^{2}/\text{Hz}$
Dots separation | $\Delta x$ |  $50$nm (GaAs), $100$nm (SiGe), $150$nm (Si-MOS)
Spread of electron wavefunction in XY plane | $r_{xy}$ |  $40$nm (GaAs), $20$nm (Si/SiGe, Si-MOS)
Width of quantum well | $2r_{z}$ |  $20$nm (GaAs),$5$nm (Si/SiGe, Si-MOS)
Table 1: Parameters used in the paper. Figure 2: Phonon relaxation rates in
(a) GaAs, (b) Si/SiGe, (c) Si-MOS double quantum dot devices as a function of
dots detuning at fixed tunnel couplings: $t_{c}=10\mu$eV (solid line),
$20\mu$eV (dashed-dotted) and $40\mu$eV (dashed). Contributions from different
phonon mechanisms are shown with a distinct color: piezoelectric longitudinal
coupling $\Gamma_{\epsilon}^{(\text{piez})}$ (green), piezoelectric transverse
coupling $\Gamma_{t}^{(\text{piez})}$ (yellow), deformation longitudinal
coupling $\Gamma_{\epsilon}^{(\text{def})}$ (red) and deformation transverse
coupling $\Gamma_{t}^{(\text{def})}$ (blue). Longitudinal phonons couples
orbital-like states in vincinity of avoided crossing, while tranverse phonons
couples dot-like states in detuned regime. Each panel represents different
device with parameters given in Tab. 1.

As the size of quantum dot in planar direction $r_{xy}$ is larger then size in
the $z$ direction, $2r_{z}\ll r_{xy}$, its value can be extracted from
splitting between the ground and first excited dot state $\Delta E$, i.e.
$r_{xy}=\sqrt{1/m^{*}\Delta E}$. In Si at $\Delta E=1$meV an estimate of
$r_{xy}\sim 20$nm is consistent with reported values of $r_{xy}\approx 15$nm
[100], $13$nm [82] in Si/SiGe and $r_{xy}\approx 21$nm [101], $18$nm [86] in
Si-MOS. The GaAs dots are typically larger ($r_{xy}\approx 55$nm [102], $21$
nm [103]), mostly due to smaller effective mass, i.e.
$m^{*}_{\text{Si}}/m^{*}_{\text{GaAs}}\approx 3$. The typically reported
values of $2r_{z}\approx 20$nm [102] in GaAs are also larger than those in
Si/SiGe, $2r_{z}\approx 4$nm [104], $6$nm [82]. We assume here the extent of
electron’s wavefunction in the $z$ direction in Si-MOS is similar to that in
Si/SiGe, and for both we take it as $2r_{z}=5$nm. Finally, smaller dots allow
for decreasing the distance between the sites from typical for GaAs $\Delta
x\sim 150$nm [102], $\sim 110$ nm [103] to Si/SiGe values of $\Delta x\sim
100$nm [5], to Si-MOS $\Delta x\sim 50$nm [5]. The distances between the dots
are correlated with reported values of $t_{c}$, the largest of which are
achieved in SiMOS structures, with examples of $t_{c}\approx 450\mu$eV and
$50\mu$eV for dots separated by $\Delta x\sim 40$nm [28] and $\sim 100$nm
[101] respectively. However, recently $t_{c}\\!\approx\\!40\mu$eV was achieved
in Si/SiGe across an array of quantum dots with $r_{xy}\\!\sim\\!10$nm and
$\Delta x\\!\approx\\!70$nm [25]. In GaAs, tunnel coupling of
$t_{c}\\!\approx\\!20-40\mu$eV was measured in an array of eight quantum dots
with $\Delta x\\!\approx\\!150$nm [2] for array of 8 QDs. Representative
parameters for each nanostructure that we will use in subsequent calculations
are given in Tab.1.

We now evaluate numerical values of relaxation rates from Eqs. (39) and(40)
for above-discussed parameters of “typical” GaAs, Si/SiGe and Si-MOS double
quantum dots given in table 1, In Fig. 2 we plot zero-temperature electron
relaxation rate due to scattering with phonons,
$\Gamma_{-}^{(\text{ph})}(\Omega)$, as a function of detuning $\epsilon$ (let
us recall that $\Omega=\sqrt{t_{c}^{2}+\epsilon^{2}}$) for three values of
tunnel coupling, $t_{c}=10,20,40\mu$eV. It is clear that the scattering of a
single electron in a DQD in each of considered nanostructures is dominated by
a different mechanisms. In polar GaAs, the piezoelectric coupling dominates
over the deformation potential one, with the fastest relaxation at low
detuning, where the transitions occur between molecular-orbital type states.
The relaxation rate, for the energies below $c/\Delta x\\!\approx\\!50\mu eV$
shows oscillatory behaviour due to
$|F_{\epsilon}|^{2}\propto\sin^{2}(k_{x}\Delta x/2)$ term, see Eq. (III.3).
For larger detunings, when the energy transfer $\Omega\\!\approx\\!\epsilon$,
the relaxation rate decreases as its mostly longitudinal character that makes
it $\propto(t_{c}/\epsilon)^{2}$, is combined with phonon spectral density
$\propto\epsilon^{3}$ and piezoelectric coupling
$|v^{(\text{piez}})|^{2}\propto\epsilon^{-2}$, to produce an overall
$\Gamma_{-}^{(\text{piez})}\propto(t_{c})^{2}/\epsilon$ scaling in the far
detuned regime $\epsilon\gg t_{c}$, until $\epsilon\\!\approx\\!500\mu$eV when
phonon bottleneck effects start to become strongly viisble. On the other hand,
in Si/SiGe a weaker deformation potential scattering gives
$\Gamma_{\epsilon}^{(\text{def})}$ that first increases with $\epsilon$, and
then becomes suppressed by phonon bottleneck effect at large detunings. The
relaxation time $1/\Gamma_{-}$ falls below 100 ns for $\epsilon\\!\sim\\!100$
$\mu$eV only for the largest considered $t_{c}=40\mu$eV. Finally, in Si-MOS
the smaller interdot distance makes the transverse relaxation more efficient.
Due to its $\Gamma_{t,-}^{(\text{def})}\propto\epsilon^{3}$ scaling up to
phonon bottleneck energy of about $1$ meV, it becomes the dominant process at
larger detunings. Such a transverse relaxation rate weakly depends on tunnel
coupling (note the presence of single blue lines in Fig. 3, and requires
overlap between wavefunctions of L/R dots, which is not large enough in the
other nanostructures: $\Gamma^{(\text{piez})}_{t}$ might be relevant in GaAs
only at highest detunings, see Fig. 3a, and the transverse process never
becomes of similar order of magnitude as the longitudinal one in the
considered Si/SiGe structures.

###  III.4 Comparison of the transition rates

Figure 3: Relaxation (solid lines) and excitation (dashed lines) rates as a
function of detuning at typical tunnel coupling $t_{c}=20\mu$eV and
temperature of $T=100$mK. Transition rates due to phonons are drawn using
green (GaAs), black (Si/SiGe) and blue (Si-MOS) colors, while transition rates
due to common for all nanostructures charge noise is depicted using red
(Johnson) and yellow (1/f) colors. Both excitations and relaxations in GaAs
are dominated by electron-phonon coupling. In Si the excitations are commonly
caused by charge noise (either 1/f or Johnson of similar amplitude), while the
relaxation at finite detuning relies on Johnson noise in Si/SiGe and
relatively stronger interdot phonons in Si-MOS, where the dots are closer.
Figure 4: Transition rates relevant for electron charge transfer in GaAs
(green), Si/SiGe (red) and Si-MOS (blue): the excitation rate at avoided
crossing $\Gamma_{+}(0)$ ($\epsilon=0$) (a), relaxation rates in detuned
regime $\Gamma_{-}(\epsilon=100\mu$)eV (b) and $\Gamma_{-}(\epsilon=100\mu$)eV
(c) as a function of tunnel coupling $t_{c}$ for selection of temperatures
$T=50,100,500$mK (solid, dashed-dotted, dashed) lines respectively. The
excitation rate due to piezoelectric phonons in GaAs in $T>100$mK is the only
non-monotonic function of tunnel coupling. Otherwise excitation rates
decreases for larger $t_{c}$ due to exponential factor $\Gamma_{+}(0)\propto
e^{-\beta t_{c}}$, while relaxation rates increases due to dominant role of
longitudinal mechanisms $\Gamma_{-}(\Omega)\propto t_{c}^{2}/\Omega^{2}$.
Since $\Omega=\sqrt{t_{c}^{2}+\epsilon^{2}}$ increase is stronger at lower
$\epsilon$. The only discrepancy between Si/SiGe and Si-MOS is visible in the
relaxation rate at far detuned regime ($\epsilon=400\mu$eV) due to presence of
interdot phonons in the latter.

Let us know use the results of the previous Sections and compare the relative
importance of various types of environments on the discussed DQD structures.
In Fig. 3 we plot the relaxation rate $\Gamma_{-}[\Omega(\epsilon)]$ (the
excitation rate $\Gamma_{+}(\Omega)\\!=\\!\Gamma_{-}(\Omega)e^{-\beta\Omega}$)
with solid (dashed) lines as functions of detuning for all the considered
mechanisms using the above-discussed representative parameters for GaAs,
Si/SiGe, and SiMOS structures, and temperature $T=100$mK and tunnel coupling
$t_{c}=20\mu$eV. As expected from discussion in Sec. II.3, the excitation
rates are the largest at the anticrossing, and they become suppressed
exponentially with $\epsilon$ increasing above $t_{c}$. In that regime the
relaxation overwhelmingly dominates over excitation, but the dynamics of the
electron will depend on the value of total $\Gamma_{-}$: the electron tranfer
error will depend on the ratio of timescale of environment-assisted inelastic
tunneling between the dots in the far-detuned regime, $1/\Gamma_{-}$, and the
duration of the detuning sweep. Note that for $t_{c}\\!=\\!20$ $\mu$eV the
requirement of $Q_{LZ}\\!<\\!10^{-4}$ means $v\\!<\\!200\mu$eV/ns, so the
total time of detuning sweep over a meV range is $5$ns. This will give a
ballpark estimate what timescales we should compare $1/\Gamma_{-}$ to.

In GaAs the coupling to phonons (the green line in Fig. 3 dominates the
relaxation, with influence of Johnson noise possibly becoming dominant at
highest considered detunings. As discussed above,
$\Gamma_{-}^{(\text{piez})}\propto(t_{c})^{2}/\epsilon$ for most of the
considered range of $\epsilon$, so for the healing of excitation to be
significant the time spent at moderate detunings, up to about 200 $\mu$eV (see
Fig. 3), has to be larger than average relaxation time in this range,
$1/\Gamma_{-}\\!\sim\\!1-10$ ns.

The situation is more complex in Si nanostructures. For parameters of Si/SiGe
DQDs it is the Johnson noise - red line in Fig. 3 \- that dominates (more
visibly at lower $\epsilon)$) over the relaxation due to deformation potential
coupling to phonons (the black line in Fig. 3). The detuning dependence of
this process is rather weak. When $t_{c}/k_{\text{B}}T\gg 1$ (in Fig. 3 we
have $t_{c}/k_{\text{B}}T\approx 2.3$), the Johnson noise from $50\Omega$
resistor gives $\Gamma^{(\text{J})}_{-,\epsilon}\propto t_{c}^{2}/\epsilon$
for stronger longitudinal process, and and
$\Gamma^{(\text{J})}_{-,t}\propto\epsilon$ in case of weaker transverse one.
For their assumed ratio, the relaxation the rates become equal at
$\epsilon_{J}=10t_{c}$, which means $\Gamma^{(J)}_{-}$ slowly decreases as
$\epsilon^{-1}$ up to $\epsilon=400\mu$eV, and then it starts to slowly
increase with $\propto\epsilon$. The relaxation time for the assumed amplitude
of Johnson noise is $\sim\\!100$ ns in the relevant detuning range.

Finally, for SiMOS the smaller interdot distance assumed for this architecture
makes $\Gamma_{t,-}^{(\text{piez})}\propto\epsilon^{3}$ the dominant
relaxation process at large detunings: as show in Fig. 3 this relaxation
channel dominates over the one due to Johnson noise for
$\epsilon\\!\gtrsim\\!200$ $\mu$eV. The relaxation times at large detunings
approach $\sim\\!10$ ns, so phonon-assisted interdot tunneling might be an
efficient mechanism of healing of charge noise-induced excitation that
occurred close to the anticrossing in SiMOS.

The other mechanisms only weakly contribute to relaxation, as longitudinal 1/f
noise relaxation rate is strongly attenuated with increasing detuning, as
$\Gamma_{-,\epsilon}^{(1/f)}\propto t_{c}^{2}/\epsilon^{3}$ at
$\epsilon>t_{c}$, while small overall strength and weak detuning dependence of
longitudinal phonon processes in Si/SiGe,
$\Gamma_{-.\epsilon}^{(\text{def})}\propto t_{c}^{2}\epsilon$, produces
relaxation times above $100$ ns only approaching the order of magnitude of
contribution of Johnson noise around $\epsilon=200\mu$eV.

Let us now discuss the tunnel coupling and temperature dependence of the total
$\Gamma_{+}$ rate at $\epsilon\\!=\\!0$, and of the total $\Gamma_{-}$ rate at
moderate and high detunigs, $\epsilon\\!=\\!100$ and $400$ $\mu$eV,
respectively. The relaxation rates at moderate detuning have a common
dependence on $t_{c}$ inherited from the tunneling dependence of the dominant
there longitudinal process, i.e. $\Gamma_{-}\propto t_{c}^{2}$. This is not
the case at larger detuning, where transverse processes that are weakly
dependent on $t_{c}$ can dominate. Similarly for considered here $\beta
t_{c}\gg 1$, temperature dependence of relaxation is very weak. We illustrate
both statements in Fig. 4a and 4b where we plot relaxation rates at
$\epsilon=100,400\mu$eV as a function of tunnel coupling. As it can be seen
difference between Si/SiGe and SiMOS is visible at large detuning where for
small tunnel couplings interdot phonons provide order of magnitude faster
relaxation rate in the latter.

In Fig. 4c we illustrate the relevant excitation rate $\Gamma_{+}(0)$,
computed at the avoided-crossing at $T=50,100,500$mK. In GaAs the only
relevant mechanism is the coupling between orbital states provided by the
phonons, which has a strong scaling with tunnel coupling
$\Gamma_{+}^{(\text{piez)}}(0)\propto t_{c}^{3}e^{-\beta t_{c}}$ as long as
$t_{c}\ll c/\Delta x\sim 50\mu$eV, wher $t_{c}^{3}$ dependence is provided by
the piezoelectric coupling ($t_{c}$) and the resonance term
$\sin^{2}(k_{x}\Delta x/2)$ ($t_{c}^{2}$). As a result, at smaller
temperatures, excitation rate in GaAs shows a non-monotonic behaviour as a
function of tunnel coupling. In Si, the excitation at $\epsilon\ll t_{c}$ is
caused only by charge noise, and hence for amplitude of this noise used here
for both Si/SiGe and SiMOS, it results in the same rate, in which
contributions from $\Gamma_{+}^{(1/f)}(0)\propto e^{-\beta t_{c}}/t_{c}$ and
$\Gamma_{+}^{(\text{J})}(0)\propto t_{c}e^{-\beta t_{c}}$ are combined. The
latter becomes more relevant at larger $t_{c}$, for which however the overall
charge noise is attenuated due to exponential factor, as it can be seen in
Fig. 4c by a decrease of excitation rate in Si.

##  IV Probability of leaving the electron behind

Let us use now the above-derived transition rates to calculate the central
quantity of this paper – occupation of higher energy state after detuning
sweep $Q$, i.e. the probability of leaving the electron in the initial dot.

Figure 5: Probability of occupying excited state $Q$, i.e. leaving the
electron in the initial dot after detuning sweep, as a function of sweep rate
$v$ for fixed tunnel coupling and temperature in two semiconductor DQD
devices: GaAs A (Green) and SiGe A for Johnson noise originating from ideal
resistor (red) and from transmission line (pink). In the four panels we show
combinations of tunnel couplings $t_{c}=10,20\mu$eV (columns) and temperatures
$T=50,100$mK (rows). Squares correspond to numerical solution of Adiabatic
Master Equation, where we have used filled (hollow) squares to denote
adiabatic (non-adiabatic) regime. Dashed line corresponds to Single Excitation
Approximation Limit $Q_{\text{SEAL}}$, see Eq. (16), while the solid line is
the Healed Excitation Approximation Limit, $Q_{\text{HEAL}}$, see Eq. (17).
Dotted black line shows the Landau-Zener result $Q_{\text{LZ}}=\exp(-\pi
t_{c}^{2}/2v)$. Remaining parameters are given in Tab. 1 in the Appendix.
III.3.

.

We assume the relevant part of detuning sweep starts and terminates at
$\epsilon_{f}=\pm 500\mu$eV, since at $\epsilon\geq 10t_{c}$ the dots become
uncoupled, i.e. the approximation of constant $t_{c}$ breaks down [94, 105],
and the detuning sweep rate used in an experiment can be increased [25]. In
Fig. 5 we compare the results of a numerical solution of Adiabatic Master
Equation (AME) from Eq. (10), depicted as squares, against the approximation
of single excitation at avoided crossing without relaxation process,
$Q_{\text{SEAL}}$ from Eq. (15), shown as dashed lines, and the approximation
of an excitation followed by relaxation processes only, $Q_{\text{HEAL}}$ from
Eq. (17), shown as solid lines. The dotted line is the Landau-Zener formula
$Q_{\text{LZ}}$ from Eq. (2). In the four panels we show results for
combinations of tunnel coupling and temperatures: $t_{c}=10,20\mu$eV
(columns), $T=50,100m$K (rows). With hollow squares we mark the AME results in
the region where $Q_{\text{SEAL}}$ and $Q_{\text{HEAL}}$ are no longer an
upper and a lower bound on $Q$, as probability of Landau-Zener transition
dominates. We stress that in this region the applicability of AME in secular
approximation used here is limited [42, 43], however a correction to the L-Z
formula computed using different methods correction is expected to be small
for prediminantly longitudinal relaxation $\Gamma_{\epsilon}\gg\Gamma_{t}$
[51, 53, 45].

Let us now concentrate on the region in which $Q$ is dominated by effects
related to interaction with the thermal environment, where the AME results are
plotted as filled squares. For both Si-based devices in a region of moderate
$v$ we observe that $Q\propto 1/v$. This suggests that the value of $Q$
follows from a finite excitation probability in a limited range of detunings
(near the anticrossing), and the occupation of the excited state grows with
increased time spent in this region. In agreement with this picture, $Q\approx
Q_{\text{SEAL}}$ (dashed line), and the electron undergoes a single transition
from ground to excited state in vicinity of the anticrossing. We note such a
transition from ground to excited state at $\epsilon\ll t_{c}$ in SI-based
devices is caused solely by charge noise. As the sweep rate is decreased, an
increasing time spent during the electron trasnfer in the far-detuned regime,
$\epsilon\\!\gg\\!t_{c}$, allows for a significant recovery of ground state
occupation by the relaxation mechanism, which is reflected by a deviation from
a SEAL approximation and $Q\sim Q_{\text{HEAL}}$ (solid lines) for smallest
sweep rates. The healing effect is stronger for the SiMOS device, due to
effective phonon relaxation between the dot-like eigenstates at large
detunings. The agreement between the result of evaluation of AME, and the
approximation is more visible at lower $T$ (higher $t_{c}$), since this
agreement is expected to improve as $t_{c}/k_{\text{B}}T\gg 1$. In contrast,
in GaAs device, $Q$ (plotted in green color) decreases monotonically as the
sweep rate $v$ gets smaller. This is a consequence of a much stronger coupling
to environment (specifically piezoelectric coupling to phonons), which on one
hand increases probability of transition from ground to excited state in
vicinity of the avoided crossing (dashed line), but on the other hand allows
for subsequent relaxation even for not-too-high $v$. As a result there is no
region in which $Q\sim Q_{\text{SEAL}}\sim 1/v$, however as long as
$Q_{\text{SEAL}}\leq 0.1$, i.e. the probability of excitation-relaxation-
excitation sequence is relatively low ($<Q_{\text{SEAL}}^{2}$), the result can
be well approximated by taking into account only relaxation processes
modifying the excitation generated at the anticrossing, i.e.
$Q\\!\approx\\!Q_{\text{HEAL}}$. Finally, when the electron transfer time is
long enough to allow for a second transition from ground to excited state,
i.e. when $Q_{\text{SEAL}}\geq 0.1$, the HEAL formula gives only a lower bound
for results of the AME, as visible at low $v$ when comparing the squares and
solid lines.

Figure 6: Probability of leaving the electron behind in the case of high
tunnel coupling $t_{c}=40\mu$eV. We compare results for $T=500$mK (a) and
$T=100$mK (b), since large tunnel coupling in general should allow for
relatively efficient transfer in higher temperature.  Figure 7: Optimal
transfer in Si. In panel (a) we plot optimal sweep rate for Si/SiGe and Si-MOS
devices $v_{\text{opt}}$ obtained as a solution to Eq. (42) for
$T=50,100,500$mK (solid, dashed and dotted-dashed lines). In panel (b) we plot
probability of leaving the electron behind after the detuning sweep with a
rate $v_{\text{opt}}$, as a function of tunnel coupling $t_{c}$ and for the
same selection of temperatures. We compare results for Si against phonon
dominated transfer with the same sweep rate in GaAs (Green) as a reference.

An obvious way to increase the efficiency of charge transfer, or equivalently
decrease $Q$, would be to bring the $Q_{\text{SEAL}}$ result down, as for
$\Gamma_{+}<\Gamma_{-}$ it gives an upper bound of excited state occupation
induced by environmental fluctuations, i.e. $Q<Q_{\text{SEAL}}$ in the
adiabatic regime where $Q>Q_{\text{LZ}}$. This can be achieved by lowering the
temperature or increasing the tunnel coupling. In Fig. 6 we show a rather
optimistic result of probability of leaving the electron in the left dot
evaluated for the largest $t_{c}$ reported in the array of Si/SiGe quantum
dots [25], $t_{c}=40\mu$eV. As a reference we compare it to the other
materials considered, and plot results for $T=100,500$mK, as larger tunnel
couplings should in principle allow for working at higher temperatures [106,
107, 108]. We stress that a calculation for $T=50$mK (not shown) gives $Q\leq
10^{-6}$ for $v<400\mu$eV/ns. For Si nanostructures the behaviour at higher
temperatures is qualitatively similar to than shown in Fig. 5, with a local
minimum of $Q=10^{-4}$,$10^{-2}$ at $v=400\mu$eV/ns, $800\mu$eV/ns for
$T=100$mK, $500$mK respectively. In GaAs the large value of $t_{c}$ results in
strong coupling between transferred electron and the environment, which at
higher temperatures causes flattening of $Q$ as a function of $v$. This effect
can be attributed to reaching thermal equilibrium of
$Q_{\text{eq}}(\epsilon=0)=\Gamma_{+}(0)/(\Gamma_{+}(0)+\Gamma_{-}(0))\sim
0.3$ around the avoided crossing, followed by slower relaxation at larger
$\epsilon$.

A local minimum of $Q(v)$ is thus expected in both Si-based nanostructures
considered here. The value of $Q$ at this minimum can be estimated as the
intersection of $Q_{\text{SEAL}}$ and $Q_{\text{LZ}}$, i.e.

$Q_{\text{SEAL}}(v_{\text{opt}})=Q_{\text{LZ}}(v_{\text{opt}}),$ (42)

the solution to which is expressed in terms of Lambert $W$ function [109] as
$v_{\text{opt}}=\pi t_{c}^{2}/2W(a)$ where $W(a)$ satisfies equation
$W(a)e^{W(a)}=a$ for $a=\tfrac{\pi}{4}\sqrt{\beta t_{c}^{3}}/\Gamma_{+}(0)$.
Since typically $a\gg 1$, the asymptotic form of $W(a)=\ln(a)-\ln(\ln(a))$ in
the low temperature limit $\beta t_{c}>1$ allows to write
$v_{\text{opt}}\sim\pi k_{\text{B}}Tt_{c}/2$. As the value of
$Q_{\text{SEAL}}$ is independent of coupling to environment at larger
detunings, the sweep rate $v_{\text{opt}}$ which minimizes $Q$ depends on the
charge noise amplitude at $t_{c}$, which is assumed here to be the same in
Si/SiGe and Si-MOS.

In Fig. 7a we show how $v_{\text{opt}}$ in Si varies with $t_{c}$ for
$T=50,100,500$ mK. We see that $v_{\text{opt}}$ increases as the
$Q_{\text{LZ}}$ curve shifts to higher $v$ (due to an increase of $t_{c}$), or
noise-induced excitations $Q_{\text{SEAL}}$ become stronger (here due to an
increase of $T$). Next, in Fig. 7b we use $v_{\text{opt}}$ to compare the
corresponding transfer error in Si $Q(v_{\text{opt}})$ (red) against analogous
quantity in GaAs (green), as a function of $t_{c}\in 5-60\mu$eV. In the Figure
we have put together results of the AME (solid, dashed, dotted dashed lines)
for three different temperatures $T=50,100,500$ mK. The probability of losing
the electron $Q$ in SiGe appears to be below the value for GaAs for the tunnel
couplings apart from the smallest ones of $t_{c}\sim 5-10\mu$eV, where the
corresponding optimal sweep rate ($v_{\text{opt}}\sim 10-50\mu$eV$/$ns
depending on the temperature) is large enough to make the Landau-Zener
$Q_{LZ}$ probability stay above the phonon-induced $Q$. Of course, $Q$ in GaAs
can be made lower by using $v\\!<\\!v_{\text{opt}}$, but there are other
factors that are limiting $v$ from below in GaAs devices (see discussion in
the next Section). Similarly, in Si quantum dots charge transfer can be in
principle improved by going to much lower sweep rates $v<1\mu$ev/ns, however
it would make the few-nanosecond transfer impossible as it has been
demonstrated by showing the sweep time interval on the right y-axis of Fig.
7a.

The value of optimal sweep rate and corresponding minimum of transfer error
$Q(v_{\text{opt}})$ in Si depends on the amplitude of charge noise at
frequency corresponding to tunnel coupling (which is in the GHz range), where
its influence dominates over that of phonons. We concentrate here on the
amplitude of $1/f$ noise, the amplitude of which can vary by at least an order
of magnitude between different Si DQD devices. In Fig. 8 we plot a minimal
transfer error at typical electron temperature $T=100$mK as a function of
square root of $1/f$ spectral density evaluated at $1$Hz and at $T=100$mK,
which we have previously taken as constant
$\sqrt{s_{1}(100\text{mK})}=1\mu$eV/$\sqrt{\text{Hz}}$ (see Sec. III.2). We
plot the results for range of $t_{c}$ considered here, and emphasize that
noise amplitude can be directly related to excitation rate at $\Omega=t_{c}$
with the following formula:

$\Gamma_{+}^{(1/f)}(t_{c})\big{[}1/\text{ns}\big{]}\approx\frac{2s_{1}[\mu\text{eV}^{2}/\text{Hz}]}{t_{c}[\mu\text{eV}]}\exp(-\frac{t_{c}}{k_{\text{B}}T})\,\,,$
(43)

where $s_{1}=s_{1}(100\text{mK})$ for brevity and square brackets denoted
units in which the quantities should be substituted. The excitation rate
obtained using this Equation can be directly used in the SEAL formula, given
by Eq. (16), the result of which was illustrated in Fig. 8 using dashed lines.
As expected, $Q_{\text{SEAL}}$ agrees well with the results of adiabatic
Master equation (dots) for relatively small error
$Q(v_{\text{opt}})\\!\ll\\!1$. Next we analyze transition between $1/f$ and
Johnson noise dominated excitations. The latter can be seen in Fig. 8 as a
flattening of the solid lines, which represents results of adiabatic master
equation with both $1/f$ and Johnson noise, from $R=50\Omega$ resistor,
contributions. By comparing solid and dashed lines, we conclude that amplitude
of $1/f$ noise at which it starts to dominate over Johnson noise becomes
larger when the tunnel coupling is increased, which can be deduced from
scaling of respective excitation rates, i.e. $\Gamma_{+}^{(1/f)}(t_{c})\propto
1/t_{c}$ and $\Gamma_{+}^{(1/f)}(t_{c})^{(\text{J})}\propto t_{c}$ for
$t_{c}\gg\text{k}_{B}T$. As the optimal sweep rate $v_{\text{opt}}$ is too
high to allow for any phonon-mediated suppression of $Q$ in Si DQDs, the
difference between AME and SEAL visible for large noise amplitude is
attributed to subsequent relaxation (and further transitions) caused by $1/f$
noise of either large amplitude ($t_{c}\geq 10\mu$eV) or at relatively high
temperature ($t_{c}=5\mu$eV, for which $\beta t_{c}\sim 0.5$).

Figure 8: Probability of leaving the electron behind using optimal
$v_{\text{opt}}$ sweep rate, as a function of 1/f noise spectral density
measured at $1$ Hz and selection of tunnel couplings $t_{c}=5,10,20,40\mu$eV
at $T=100$mK. For assumed in Sec. III.2 model of high-frequency 1/f noise we
compare results of adiabatic master equation with (solid line) and without
(dots) additional contribution from Johnson noise against $Q_{\text{SEAL}}$
approximation with 1/f noise only (dashed lines). For the noise in tunnel
couplings we assumed $s_{1}^{t}=(0.1)^{2}s_{1}^{\epsilon}$.

##  V Discussion and summary

We have presented a theory of the dynamics of a system undergoing a Landau-
Zener transition in presence of weak transverse and longitudinal couplings to
thermal environments: sources of noise of $1/f$ and Johnson type, and a bath
of noninteracting bosons, specifically acoustic vibrations of a three-
dimensional crystal. Our focus was on the regime in which the deterministic
change of parameters of the Hamiltonian is slow enough to neglect the Landau-
Zener coherent excitation, and the effectively nonadiabatic character of the
evolution can be caused only by interaction with the environment. A general
theory based on adiabatic Master equation (AME) was then applied to a case of
electron transfer between a pair of voltage-controlled semiconductor quantum
dots, for which we took into account realistic parameters for electron-phonon
interaction and both Johnson and $1/f$ charge noise. We have calculated
transition rates between system’s eigenstates as function of interdot detuning
$\epsilon$, and used them in AME calculation to obtain the probability of
failure of charge transfer between the two dots, $Q$, as a function of
detuning sweep rate $v$.

When $v$ is below the value at which the Landa-Zener transition is activated,
only a finite temperature of environment allows for energy absorption
necessary for modification of $Q$, since otherwise electron would stay in the
ground state. This absorption most likely takes place in the vicinity of the
anticrossing, where the thermal energy needed for transition is the smallest.
A specific feature of the system under consideration is that the dominant
coupling to the environment is most effective at the anticrossing, making this
effect even stronger. Consequently during the process of electron transfer
caused by sweeping the detuning, a finite Q is generated at the anticrossing,
when $|\epsilon|\\!\leq\\!t_{c}$. Then, for larger positive detunings the
electron relaxation processes dominate over the excitation processes, and
suppression of $Q$ is expected.

In the considered DQDs there are two possible scenarios. In Si-based dots,
coupling to charge noise dominates, and the transition timescale are longer
than the typical transfer times, so that the final $Q$ is very close to the
value generated near the anticrossing, which is $\propto\\!1/v$ (proportional
to the time spent neart the anticrossing), so it exhibits a dependence on $v$
qualitatively opposite to the one for Landau-Zener effect dominating at large
$v$. Only at lowest $v$ the energy relaxation starts to be efficient at
lowering $Q$, with this effect being stronger in SiMOS compared to Si/SiGe
structures. The competition between the environment-induced excitation and
Landau-Zener effect leads then to appearance of optimal $v$, at which $Q$ is
minimal. In GaAs, on the other hand a strong piezoelectric coupling to phonons
dominates, transition timescales are shorter than the charge transfer time and
consequently many transitions take place, and the final $Q$ monotonically
decreases with decreasing $v$, approaching a value exponentially small in
final $\epsilon$, reflecting approaching a thermal occupation of the ground
state.

The main qualitative theoretical result of the paper, which could also apply
to systems other than double quantum dots, is thus that a system described by
a Landau-Zener model, when coupled to a thermal environment can realize two
possible scenarios: one qualitatively similar to the Landau-Zener effect, but
with dependence of $Q$ on $v$ renormalized by environment, and another in
which dependence of $Q$ on $v$ is nomonotonic, and there is an optimal sweep
rate that minimizes $Q$. The main conclusion specific to the considered case
of GaAs and Si-based quantum dots is that for $T\\!\approx\\!50$ mK, in GaAs
case $Q$ can be made smaller than $10^{-4}$ by choosing $v$ smaller than
$\approx\\!10$ $(100)$ $\mu$eV/ns for $t_{c}\\!=\\!10$ $(20)$ $\mu$eV, while
in case of Si having $Q\\!=\\!10^{-4}$ requires $t_{c}\\!>\\!20$ $\mu$eV, and
optimal $v$ of a few tens of $\mu$eV/ns. Large tunnel couplings and low
temperatures are crucial for having small $Q$. In Si-based DQDs there is a
possibility of further suppression of $Q$ by decreasing the level of charge
noise at GHz frequencies, corresponding to $t_{c}\\!\approx\\!10$ $\mu$eV
energy splitting at the anticrossing.

A process of a controlled electron transfer between two quantum dots is
relevant for ongoing attempts at construction of quantum buses based on chains
of many tunnel-coupled dots [23, 25, 28, 40, 70]. Let us now discuss the
implications of the results of this paper for prospects of coherent shuttling
of electron-based spin qubits across $N\\!\approx\\!100$ quantum dots. This
number of dots in a 1D chain is motivated by requirement of having
$\approx\\!10$ $\mu$m distance between few-qubit registers in a realistic
architecture of a quantum computer based on gate-controlled QDs [10] and
typical interdot separation $\lesssim 100$ nm.

When the goal of charge shuttling is an on-demand transfer of qubits, which
should be highly coherent, and which are to take part in further coherent
manipulations after being moved from one register to another, the
deterministic character of the shuttling is necessary. Any randomness in qubit
arrival times will complicate the application of subsequent coherent
operations involving that qubit. Furthermore, any stochastic component in the
duration of the qubit transfer will introduce a random contribution to the
phase of the qubit. More in-depth discussion of relationship between the
indeterministic character of electron shuttling and spin qubit dephasing will
be given in [70]; here it is enough to realize that large probability of
electron arriving at the end of $N$-dot chain at a time other than the desired
one, will cause major problems in the context of quantum information
processing, and we will treat it here as an error probability. Assuming that
$Q\\!\ll\\!1/N$, the probability of the electron arriving at the end of the
chain not at the desired time, i.e the probability of qubit transfer-
associated error, is $Q_{N}\approx NQ$.

Our results for Si-based quantum dots show that for tunnel couplings in
$5-40\mu$eV range, as recently reported in first experimental realizations of
electron shuttling over a few dots, achieving $Q_{N}\\!\approx\\!10^{-3}$ will
be possible for $t_{c}\\!\geq\\!30$ $\mu$eV and at $T\\!\leq\\!50$ mK. Note
that a high-fidelity charge transfer between two dots in Si MOS structure was
demonstrated experimentally using $t_{c}=450\mu$eV [28], but maintaining such
a strong tunnel coupling in a 1D array of $N\\!\approx\\!100$ quantum dots
will be challenging.

In GaAs, on the other hand, $Q_{N}$ can be made much smaller by decreasing the
detuning sweep rate $v$, so that the time of interdot transfer becomes longer
than $\sim\\!10$ ns. This in fact also holds for Si-based dots, only $v$ has
to be made at least a further order of magnitude smaller. However, for such
slow transfers one has to start worrying about well-known mechanisms of spin
dephasing that affect the coherence of a static electron localized in a QD. In
both the considered materials interaction with nuclei leads to dephasing
$T_{2}^{*}$ time of the order or $10$ ns for GaAs [110] and a few hundreds of
ns for natural Si [111, 112, 113] (and up to tens of microsecond for
isotopically purified silicon with about $10^{3}$ ppm of spinful 29Si [111,
114, 82]). For isotopically purified Si QDs in vicinity of micromagnets, their
spatially inhomogeneous magnetic fields together with charge noise lead to
$T_{2}^{*}\\!\approx\\!20$ $\mu$s. Let us use now $T_{2}^{*}\\!=\\!10$ ns
($10$ $\mu$s) for GaAs and Si. In order to avoid significant spin dephasing
during the interdot charge transfer, the time of the latter has to be much
shorter than $T_{2}^{*}$. Assuming that the range of detuning sweep
corresponding to the transfer is $\sim\\!1$ meV, the sweep rates have to
fulfill $v\\!\gg\\!0.1$ $\mu$eV/ns for Si, and $v\\!\\!\gg 100$ $\mu$eV/ns in
GaAs. In Fig. 5 we see that it means that in GaAs this lower bound on $v$
severely restricts the possibility of lowering $Q$ by making the transfer
slower, and in fact a tradeoff between amount of spin dephasing and a finite
value of $Q$ due to Landau-Zener effect that dominates the behavior of $Q(v)$
for $v\\!\geq\\!100$ $\mu$eV/ns has to be made. In silicon, the lower bound on
$v$ is much smaller than $v_{opt}$, so a local minimum of $Q$ visible in the
Figure is attainable - but the viability of strategy of lowering $Q$ by using
$v\\!<\\!1$ $\mu$eV/ns depends on the efficiency of electron relaxation due to
charge noise and electron-phonon relaxation (compare red and blue lines,
corresponding to Si/SiGe and SiMOS in the Figure) and an exact value of
$T_{2}^{*}$. All these observations suggest that from the point of view of
coherent transfer of a spin qubit, Si-based quantum dot architectures could
have an advantage over GaAs-based ones.

Let us finish by stressing the main message following from our calculations
for realistic GaAs and Si-based quantum dots: the dynamics of inter-dot
electron transfer is very strongly affected by electron’s interaction with
charge noise in Si-based systems and phonons in case of GaAs-based ones.
Effects of energy exchange with these environments have to be taken into
account to correctly describe the basic physics of electron transfer in the
currently available devices. More subtle effects appearing in closed-system
description, associated with spin-orbit and valley-orbit (in case of Si)
interactions, could become relevant if levels of charge noise are
significantly suppressed compared to the currently encountered ones.

###### Acknowledgements.

We would like to thank Lars Schreiber and Lieven Vandersypen for discussions
that motivated us to focus on the problem addressed in the paper, and Piotr
Szańkowski for multiple comments on earlier versions of this manuscript. This
work has been funded by the National Science Centre (NCN), Poland under
QuantERA programme, Grant No. 2017/25/Z/ST3/03044. This project has received
funding from the QuantERA Programme under the acronym Si QuBus.

##  Appendix A Correction due to dynamics of classical noise

Here we provide detailed calculations of occupation of excited state in the
classical limit, i.e. where the fluctuations of detuning and tunnel coupling
can be modeled by stochastic contribution to Hamiltonian (1), i.e.

$\hat{H}=\frac{\epsilon+\delta\epsilon}{2}\hat{\sigma}_{z}+\frac{t+\delta
t}{2}\hat{\sigma}_{x}.$ (44)

As pointed out in the main text, in the limit of weak noise corrections comes
from noise dynamics, which in the adiabatic frame modifies off-diagonal
element of Hamiltonian (6), written explicitly as

$\dot{\theta}\approx\frac{\sin\theta\delta\dot{\epsilon}+\cos\theta\delta\dot{t}}{\Omega_{0}},$
(45)

where $\Omega_{0}=\sqrt{\epsilon^{2}+t_{c}^{2}}$,
$\cos\theta=-\epsilon/\Omega_{0}$, $\sin\theta=t_{c}/\Omega_{0}$ and
$\delta\dot{\epsilon}=\partial_{\tau}\delta\epsilon$.

###  A.1 Leading order perturbation theory

We evaluate the leading order excitation probability $Q^{(1)}$ due to the
noisy term. We use first order time-dependent perturbation theory in adiabatic
basis
$|\tilde{\psi}(\tau)\rangle=a_{-}(\tau)\ket{-,\theta}+a_{+}(\tau)\ket{+,\theta}$,
and assuming $a_{+}=\lambda a_{+}^{(1)}+\lambda^{2}a_{+}^{(2)}+\ldots$, we
compute leading order the correction to occupation of excited state as
$Q^{(1)}=\langle|a_{+}^{(1)}(\tau)|^{2}\rangle$, which equals

$Q^{(1)}=\frac{1}{4}\int_{-\infty}^{\infty}\Big{\langle}\dot{\theta}(\tau_{1})\dot{\theta}(\tau_{2})\Big{\rangle}\,e^{i\int_{\tau_{2}}^{\tau_{1}}\Omega_{0}(\tau^{\prime})\text{d}\tau^{\prime}}\text{d}\tau_{1}\text{d}\tau_{2},$
(46)

where $\langle\ldots\rangle$ denotes classical averaging over noise
realizations. The substitution of Eq. (25) into Eq. (46) results in four
distinct contributions:

$Q^{(1)}=Q^{(1)}_{\epsilon\epsilon}+Q^{(1)}_{tt}+Q^{(1)}_{\epsilon
t}+Q^{(1)}_{t\epsilon},$ (47)

which correspond to auto- or cross-correlation function of respective noise
derivative
$Q_{xy}^{(1)}\sim\langle{\delta\dot{x}}(\tau_{1}){\delta\dot{y}}(\tau_{2})\rangle$,
where $x,y=\epsilon$ or $t$. For assumed here stationary noises, it is
convenient to use the Fourier transform of correlation function, which for the
noise derrivative can be expressed in terms of power spectral density of noise
(PSD) $S_{xy}(\omega)=\int_{-\infty}^{\infty}\langle\delta x(\tau)\delta
y(0)\rangle e^{-i\omega\tau}\text{d}\tau$, i.e.
$S_{\dot{x}\dot{y}}(\omega)\equiv\omega^{2}S_{xy}(\omega)$ [76]. This allows
us to write the correction as

$\displaystyle
Q_{xy}^{(1)}=\int_{-\infty}^{\infty}\frac{\text{d}\omega}{8\pi}S_{xy}(\omega)\,F_{x}(\omega)F_{y}^{*}(\omega),$
(48)

in which we introduced filtering function $F_{x}(\omega)$ using Eq. (46):

$F_{x}(\omega)=\int_{-\infty}^{\infty}\text{d}\tau\,f_{x}(\tau)\frac{\omega}{\Omega(\tau)}\,\exp{i\omega\tau+i\int_{0}^{\tau}\Omega(\tau^{\prime})\text{d}\tau^{\prime}},$
(49)

with $f_{\epsilon}(\tau)=\sin\theta=t_{c}/\Omega$ and
$f_{t}(\tau)=\cos\theta=-v\tau/\Omega$.

###  A.2 Stationary Phase approximation

We evaluate the integral (49) for $x=\epsilon,t$, in leading order of
stationary phase approximation, where we seek for time at which the argument
of exponent
$\varphi(\tau)=\omega\tau+\int_{0}^{\tau}\Omega(\tau^{\prime})\text{d}\tau$ is
stationary, i.e. $\partial_{\tau}\varphi(\tau)=0$, from that
$\omega=-\Omega(\tilde{\tau})$, which takes place at
$\pm\tilde{\tau}=\sqrt{(\omega^{2}-t_{c}^{2})/v^{2}}$. Additionally since
$\Omega\geq t_{c}$, the $\omega$ is strictly negative and smaller then
$-t_{c}$. The second derivative, of the phase evaluated at $\tilde{\tau}$
reads $\partial_{\tau}^{2}\varphi(\tau)|_{\tau=\tilde{\tau}}=\pm
v\sqrt{1-t_{c}^{2}/\omega^{2}}$. In the leading order, the integral (49)
reads:

$\displaystyle F_{\epsilon}\approx$
$\displaystyle\frac{t_{c}}{\omega}\int_{-\infty}^{\infty}\exp{i\varphi(\tilde{\tau})+i\frac{v}{2}\sqrt{1-t_{c}^{2}/\omega^{2}}(\tau-\tilde{\tau})^{2}}+$
$\displaystyle+\exp{i\varphi(-\tilde{\tau})-i\frac{v}{2}\sqrt{1-t_{c}^{2}/\omega^{2}}(\tau+\tilde{\tau})^{2}}\text{d}\tau$
(50) $\displaystyle F_{t}\approx$
$\displaystyle\frac{-v\tilde{\tau}}{\omega}\int_{-\infty}^{\infty}\exp{i\varphi(\tilde{\tau})+i\frac{v}{2}\sqrt{1-t_{c}^{2}/\omega^{2}}(\tau-\tilde{\tau})^{2}}+$
$\displaystyle-\exp{i\varphi(-\tilde{\tau})-i\frac{v}{2}\sqrt{1-t_{c}^{2}/\omega^{2}}(\tau+\tilde{\tau})^{2}}\text{d}\tau$
(51)

Now we perform Gaussian integration,
$\int\text{d}xe^{iax^{2}}=\sqrt{\frac{\pi}{ia}}$, using which integrand terms
differ by a phase $\sqrt{1/\pm i}=e^{\mp\pi/4}$. Since
$\varphi(\tilde{\tau})=-\varphi(-\tilde{\tau})$ the result can be written as

$\displaystyle F_{\epsilon}(\omega)$
$\displaystyle=\frac{2t_{0}\cos(\varphi(\tilde{\tau})-\pi/4)}{\omega}\sqrt{\frac{2\pi}{v}}\left(1-\frac{t_{c}^{2}}{\omega^{2}}\right)^{-1/4},$
$\displaystyle F_{t}(\omega)$
$\displaystyle=\frac{-2iv\tilde{\tau}\sin(\varphi(\tilde{\tau})-\pi/4)}{\omega}\sqrt{\frac{2\pi}{v}}\left(1-\frac{t_{c}^{2}}{\omega^{2}}\right)^{-1/4}$
(52)

First we consider diagonal part ($x=y$) of Eq. (48), in which
$|F_{\epsilon}|^{2}\propto\cos^{2}(\varphi-\pi/4)$ and
$|F_{t}|^{2}\propto\sin^{2}(\varphi-\pi/4)$. Due to rapidly oscillating nature
of both functions, we replace them by average values of $\cos^{2}\varphi$ and
$\sin^{2}\varphi$ equal 1/2, which leads to:

$\displaystyle Q_{\epsilon\epsilon}^{(1)}$
$\displaystyle=\frac{1}{2v}\int_{-\infty}^{-t_{c}}\text{d}\omega
S(\omega)\frac{t_{c}^{2}}{\omega^{2}}\left(1-\frac{t_{c}^{2}}{\omega^{2}}\right)^{-1/2}$
$\displaystyle Q_{tt}^{(1)}$
$\displaystyle=\frac{1}{2v}\int_{-\infty}^{-t_{c}}\text{d}\omega
S(\omega)\frac{\omega^{2}-t_{c}^{2}}{\omega^{2}}\left(1-\frac{t_{c}^{2}}{\omega^{2}}\right)^{-1/2},$
(53)

where the strictly negative value of $\omega$, reflects absorption of energy
quanta. Finally we conclude by showing that cross-correlation is negligibly
small. We use the argument that $F_{t}^{*}=-F_{t}$ is strictly imaginary, and
as a result we have

$Q_{\epsilon t}^{(1)}+Q_{t\epsilon}^{(1)}=\int_{-\infty}^{\infty}(S_{\epsilon
t}(\omega)-S_{t\epsilon}(\omega))F_{\epsilon}(\omega)F_{t}^{*}(\omega),$ (54)

where the integrand is equivalent to imaginary part of cross-spectrum, and
hence vanishes for $\langle\delta\epsilon(\tau)\delta t\rangle=\langle\delta
t\delta\epsilon\rangle$. Non-trivial imaginary part of cross-spectrum results
only from causal relation between $\delta t$, $\delta\epsilon$ [115], however
even in such special case we argue that
$F_{\epsilon}(\omega)F_{t}^{*}(\omega)\propto\cos(\varphi+\pi/4)\sin(\varphi+\pi/4)$
which due to zero average is expected to be much smaller than auto-correlation
contributions. As a result corrections to occupation of excited state due to
weak classical noise can be written as:

$\displaystyle Q_{\epsilon\epsilon}^{(1)}$
$\displaystyle=\frac{1}{2v}\int_{t_{c}}^{\infty}\text{d}\Omega\frac{S_{\epsilon}(-\Omega)}{\sqrt{1-t^{2}/\Omega^{2}}}\left(\frac{t_{c}^{2}}{\Omega^{2}}\right)$
$\displaystyle Q_{tt}^{(1)}$
$\displaystyle=\frac{1}{2v}\int_{t_{c}}^{\infty}\text{d}\Omega\,S_{t}(-\Omega)\sqrt{1-\frac{t_{c}^{2}}{\Omega^{2}}},$
(55)

using which we recovered high frequency limit of [61] where lower bound of the
integrals reflects the minimal energy needed for the excitation to occur. Due
to the dominant role of longitudinal component $\delta\epsilon$ we omit here
contributions from frequencies below $t_{c}$, which are relevant only for
transverse $\delta t$ noise [61, 60]. In particular corrections from quasi-
static noise in tunnel coupling vanishes in assumed here weak noise ($\delta
t\\!\ll\\!t_{c}$) and adiabatic ($t_{c}^{2}>v$) limits [57].

Figure 9: Probability of occupying higher energy state after detuning sweep
$Q$ in presence of white (filled) and 1/f (hollow dots) classical noise in
detuning/tunneling as a function of sweep rate for tunnel couplings
$t_{c}=10\mu$eV (green) and $20\mu$eV (red). Points correspond to numerical
simulation of Schrodinger equation averaged over realization of classical
noise process. Lines correspond to analytical expression $Q^{(\infty)}$ with
the rates calculated according to Eq. (A.3) for the noise in detunning
$\delta\epsilon$ (solid) and tunnel coupling $\delta t$ (dashed). For
illustration we used arbitrary parameters for detuning noise
$S^{\epsilon}_{0.1}T/T_{0.1}=(1.5)^{2}\mu$eV2/Hz (1/f) and
$2Jk_{\text{B}}T=(0.3)^{2}\mu$eV2/Hz (White part of Johnsons noise). The
tunnel coupling fluctuations are reduced by a factor of 10, i.e.
$S^{\epsilon}(\omega)=(10^{2})S^{t}(\omega)$. To emulate high temperature
limit we set terminal sweep rate to $\epsilon_{f}=100\mu$eV, which corresponds
of thermal energy at $T\approx 1.2$K .

###  A.3 Transition rates

The first order calculation can be interpreted as a probability of single
transition from ground to excited state during adiabatic transfer, and as such
can be written as an integral of transition rate
$Q^{(1)}=\int\text{d}\tau^{\prime}\Gamma_{\infty}(\tau^{\prime})$, see Eq.
(15). An explicit form of $\Gamma_{\infty}$ can be deduced from Eq. (A.2) as

$\displaystyle\Gamma_{\infty,\epsilon}(\tau)$
$\displaystyle=\frac{1}{4}\left(\frac{t_{c}}{\Omega(\tau)}\right)^{2}\,S_{\epsilon}^{\text{cl}}\big{(}\Omega(\tau)\big{)}$
$\displaystyle\Gamma_{\infty,t}(\tau)$
$\displaystyle=\frac{1}{4}\bigg{(}1-\left(\frac{t_{c}}{\Omega(\tau)}\right)^{2}\bigg{)}\,S_{t}^{\text{cl}}\big{(}\Omega(\tau)\big{)}.$
(56)

Finally, we prove that a result obtained by substituting
$\Gamma_{\pm}=\Gamma_{\infty}=\Gamma_{\infty,\epsilon}+\Gamma_{\infty,t}$ into
rate equation Eq. (12), which results in high temperature solution:

$Q^{(\infty)}=\frac{1}{2}\left(1-\exp{-2\int_{\tau_{i}}^{\tau_{f}}\Gamma_{\infty}(\tau^{\prime})\text{d}\tau^{\prime}}\right),$
(57)

is equivalent to an evolution driven by Hamiltonian Eq. (1), averaged over
realizations of classical fluctuations of parameters. In Fig. 9 we have
separately plotted contributions from detuning noise $\delta\epsilon$ (solid
line), and tunnel coupling noise $\delta t$ (dashed line), as a result of
$1/f$ noise (hollow dots) and white noise (filled dots). Independently of the
considered noise type, in the fast sweep rate limit ($v\gg t_{c}^{2}$) we
recover the Landau-Zener solution, for which $Q_{\text{LZ}}$ depends on the
relation between $v$ and $t_{c}$ only, and thus for sufficiently large $v$ the
results group according to tunnel couplings $t_{c}=10\mu$eV (green) and
$t_{c}=20\mu$eV (red). In the low sweep rate limit, for $1/f$ noise in
detuning (solid line, hollow dots) we obtain results from [39], for which
$Q_{\epsilon}^{(\infty)}$ ($Q^{(\infty)}$ with
$\Gamma_{\infty}=\Gamma_{\infty,\epsilon}$) is independent of $t_{c}$. The
same applies to $1/f$ noise in tunnel coupling (dashed lines, hollow dots),
for which a 10-fold decreased noise amplitude (compared to the case of
detuning noise) translates into almost 2 orders of magnitude lower
$Q_{t}^{(\infty)}$. For the white part of Johnson’s noise, distinction between
different $t_{c}$ is much more visible for noise in detuning (solid lines,
filled dots), since larger $t_{c}$ significantly increases the time spent in
vicinity of avoided crossing $\approx\\!2t_{c}/v$, during which the
longitudinal transitions $\Gamma_{\infty,\epsilon}\propto(t_{c}/\Omega)^{2}$
are most effective. In the case of noise in tunnel coupling the opposite is
true, since larger $t_{c}$ only slightly decreases time spent outside of the
avoided crossing region, while $\Gamma_{\infty,t}\propto(v\tau/\Omega)^{2}$.

##  Appendix B Details of phonon relaxation rate

We now turn to evaluation of zero-temperature phonon relaxation rate in more
details. First we show how orbital and interdot phonon-related processes
emerge when using the $\ket{\pm,\theta}$ basis of eigenstates of instantaneous
Hamiltonian. Next we investigate the elements for Gaussian choice of electron
wavefunctions, and discuss the role of harmonic and Hund-Muliken
approximation.

###  B.1 Interdot and orbital processes

We start with the Phonon spectral density, given by Eq. (34), which predicts
$S(\omega)\propto|\bra{-}e^{i\mathbf{k}\mathbf{r}}\ket{+}|^{2}$. We now
evaluate the matrix element, by pluging in adiabatic basis given by Eq.
(II.1), which results in:

$\displaystyle\bra{-,\theta}e^{i\mathbf{k}\mathbf{r}}$
$\displaystyle\ket{+,\theta}=$
$\displaystyle\cos\theta\Re{\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{R}}+\Im{\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{R}}$
$\displaystyle+\frac{1}{2}\sin\theta(\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{L}-\bra{R}e^{i\mathbf{k}\mathbf{r}}\ket{R}),$
(58)

where $\tan\theta=-\epsilon/t_{c}$, and consecutive terms corresponds to
interdot ($\hat{\sigma}_{x}$, $\hat{\sigma}_{y}$) and orbital
$\hat{\sigma}_{z}$ coupling in dots basis respectively. In absence of large
magnetic field in $z$ direction, wavefunctions can be assumed real, hence
$\Im{\bra{L}e^{i\mathbf{k}\mathbf{r}}\ket{R}}=0$. The exact form of matrix
element depends on the assumed form of wavefunctions, i.e.
$\psi_{L/R}(\mathbf{r})=\bra{\mathbf{r}}\ket{L/R}$, which will be investigated
below.

###  B.2 Hund-Mulliken approximation

Before invoking concrete form of wavefunction of an electron localized in a
QD, let us comment on the so called Hund-Mulliken approximation, in which one
uses orthogonalized orbitals
$\ket{L/R}=\mathcal{N}(\ket{L_{0}/R_{0}}-g\ket{R_{0}/L_{0}})$ built from bare
wavefunction of electrons in isolated quantum dots: $\ket{L_{0}/R_{0}}$, with
$\mathcal{N}$ being normalization constant. The parameter $g$ is a function of
the overlap $l=\bra{L_{0}}\ket{R_{0}}\ll 1$, with its value given by the
orthogonality condition:

$\bra{L}\ket{R}=\mathcal{N}^{2}(l-2g+lg^{2})=0$ (59)

from which $g=(1-\sqrt{1-l^{2}})/l\sim l/2$ for $l\ll 1$. Consistently we
concentrate on leading order in $g$ or $l$, according to which and
$1=\bra{L}\ket{L}=\mathcal{N}^{2}(1-2gl+g^{2})$ we take $\mathcal{N}\sim 1$.
Assuming real wavefunction $\ket{L_{0}/R_{0}}$, we substitute the
orthogonalized states into Eq. (B.1) from which we obtain:

$\displaystyle\bra{-}e^{i\mathbf{k}\mathbf{r}}\ket{+}\sim$
$\displaystyle\sim\cos\theta\bigg{(}\bra{L_{0}}e^{i\mathbf{k}\mathbf{r}}\ket{R_{0}}-g\big{(}\bra{L_{0}}e^{i\mathbf{k}\mathbf{r}}\ket{L_{0}}+\bra{R_{0}}e^{i\mathbf{k}\mathbf{r}}\ket{R_{0}}\big{)}\bigg{)}$
$\displaystyle+\frac{1}{2}\sin\theta\bigg{(}\bra{L_{0}}e^{i\mathbf{k}\mathbf{r}}\ket{L_{0}}-\bra{R_{0}}e^{i\mathbf{k}\mathbf{r}}\ket{R_{0}}\bigg{)},$
(60)

where in the latter term correction linear in the overlap $g$ cancels.

###  B.3 Harmonic approximation

Finally we substitute concrete form of isolated wavefunctions, and evaluate
matrix element $\bra{-}e^{i\mathbf{k}\mathbf{r}}\ket{+}$. We assume the
wavefunction is indepedent in all three directions
($\Psi_{L_{0}/R_{0}}(\mathbf{r})=\psi_{L_{0}/R_{0}}(x,y)\psi_{z}(z)$) and in
has a Gaussian shape:

$\displaystyle\psi_{L_{0}/R_{0}}(x,y)=\frac{1}{(\pi^{2}r_{xy}^{4})^{1/4}}\exp{-\frac{(x\pm\Delta
x/2)^{2}+y^{2}}{2r_{xy}^{2}}}$ $\displaystyle\psi_{z}(z)=\frac{1}{(\pi
r_{z}^{2})^{1/4}}\exp{-\frac{z^{2}}{2r_{z}^{2}}},$ (61)

such that for electron wavefunction, FWHM${}_{x}\approx 2r_{xy}$ and
FWHM${}_{z}\approx 2r_{z}$. In such case Eq. (B.2) reads:

$\displaystyle\bra{-}e^{i\mathbf{k}\mathbf{r}}\ket{+}$
$\displaystyle\sim\exp{-\frac{k_{xy}^{2}r_{xy}^{2}+k_{z}^{2}r_{z}^{2}}{4}}\times$
$\displaystyle\bigg{(}$ $\displaystyle\cos\theta\exp{-\frac{\Delta
x^{2}}{4r_{xy}^{2}}}\big{(}1-\cos(k_{x}\Delta x/2)\big{)}+$
$\displaystyle\qquad\qquad-i\sin\theta\sin(k_{x}\Delta x/2)\bigg{)},$ (62)

where we used that in harmonic approximation $g=l/2=\tfrac{1}{2}e^{-\Delta
x^{2}/4r_{xy}^{2}}$. Interdot and orbital relaxation are given by real and
imaginary part of above matrix element and hence cause relaxation
independently.

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