# Dynamical nuclear spin polarization in a quantum dot
with an EDSR-driven electron spin

Peter Stano1,2 Takashi Nakajima1 Akito Noiri1 Seigo Tarucha1,3 Daniel
Loss1,3,4 1Center for Emergent Matter Science, RIKEN, 2-1 Hirosawa, Wako-shi,
Saitama 351-0198, Japan  2Institute of Physics, Slovak Academy of Sciences,
845 11 Bratislava, Slovakia  3RIKEN Center for Quantum Computing, 2-1
Hirosawa, Wako, Saitama, 351-0198 Japan  4Department of Physics, University of
Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

###### Abstract

We analyze the polarization of nuclear spins in a quantum dot induced by a
single-electron spin that is electrically driven to perform coherent Rabi
oscillations. We derive the associated nuclear-spin polarization rate and
analyze its dependence on the accessible control parameters, especially the
detuning of the driving frequency from the electron Larmor frequency. The
arising nuclear-spin polarization is related to the Hartmann-Hahn effect known
from the NMR literature with two important differences. First, in quantum dots
one typically uses a micro magnet, leading to a small deflection of the
quantization axes of the electron and nuclear spins. Second, the electric
driving wiggles the electron with respect to the atomic lattice. The two
effects, absent in the traditional Hartmann-Hahn scenario, give rise to two
mechanisms of nuclear-spin polarization in gated quantum dots. The arising
nuclear-spin polarization is a resonance phenomenon, achieving maximal
efficiency at the resonance of the electron Rabi and nuclear Larmor frequency
(typically a few or a few tens of MHz). As a function of the driving
frequency, the polarization rate can develop sharp peaks and reach large
values at them. Since the nuclear polarization is experimentally detected as
changes of the electron Larmor frequency, we often convert the former to the
latter in our formulas and figures. In these units, the polarization can reach
hundreds of MHz/s in GaAs quantum dots and at least tens of kHz/s in Si
quantum dots. We analyze possibilities to exploit the resonant polarization
effects for achieving large nuclear polarization and for stabilizing the
Overhauser field through feedback.

##  I Introduction

Spin qubits in semiconducting quantum dots [1] are pursued as promising qubit
hosts [2, 3]. The advantage of semiconducting spin qubits is that they can be
controlled electrically. For example, single-qubit gates exploit electric
dipole spin resonance (EDSR) where an oscillating electric field drives spin
rotations through either the material spin-orbit interaction [4, 5], or a
designed micro magnet [6, 7]. The manipulation thus proceeds by applying a
resonant radio-frequency (rf) field through local gates instead of a global
field typical for electron spin resonance (ESR) experiments.

Since the first experiments with gated spin qubits, it has been routinely
observed that some form of nuclear spin polarization often accompanies
electrical manipulation [8, 9] including a resonant driving [10, 5]. Since in
semiconductors nuclear spins have a strong impact on spin qubits, limiting
their lifetime and coherence [11, 12, 13, 14, 15, 16], a lot of research went
into understanding the electron-nuclear spin interactions (see, for example,
the reviews in Refs. [17, 18, 19, 20] and the references therein). A possible
control of nuclear spins through the arising nuclear polarization received
particular attention [13, 21, 22, 23, 24, 25]. While a certain degree of
control was demonstrated [26, 27, 28], overall it remained limited [29, 30,
31, 32] and the large nuclear noise persists as the major issue of III-V
materials to be dealt with [33, 34, 35, 36].111Nuclear spins are one of the
main reasons to switch from element–III-V to element-IV quantum dots. However,
even here nuclear spins might still remain as a performance limit of spin
qubits made with electrons [37, 38] as well as holes [39].222Since our focus
is on gated semiconducting dots manipulated electrically, we will make only
sporadic comments to works on self-assembled dots accessed optically, where
the nuclear-spin-control program continues unabated [40, 41, 42, 43, 44].

We revisit here the nuclear spin polarization induced by an EDSR-driven and
coherently precessing electron spin in an isolated quantum dot. We consider
the coherent regime with the electron spin Rabi frequency large compared to
the relevant decay times, either the electron lifetime in the dot or its spin
Rabi decay time. This regime of well-defined Rabi rotations (or strong
driving) is the essential difference to previous works on this topic [45, 22,
46] which implicitly or explicitly considered the limit of weak driving.333See
especially Footnote [18] in Ref. [45]

The physics’ essence is closely related to the Hartmann-Hahn resonance [47],
well known from nuclear magnetic resonance (NMR): dynamical nuclear spin
polarization (DNSP)444In line with the literature on spin qubits, we will use
the name ‘dynamical nuclear spin polarization’ (DNSP) rather than the
‘dynamical nuclear polarization’ (DNP) used in the NMR community. arises when
the Rabi frequency of the driven electron is equal to the Larmor frequency of
nuclear spins.555Since the effect exists in several flavors, it might be
useful to mention further names that are used: The original work, Ref. [47],
considered two different nuclear species, both of which are driven. The spin
‘cross-polarization’ then arises when their Rabi frequencies are equal, a
condition called also ‘double resonance’ [48]. Ref. [49] coined the acronym
‘NOVEL’ for the variant where one of the spins is electronic, being driven,
and the other is nuclear, not driven. This is the situation we consider in a
quantum dot. However, important differences preclude using the existing NMR
results: 1) since the electron-spin driving is electrical, the electron shifts
in space with respect to the atomic lattice; 2) there is often a micro-magnet
gradient giving dispersion to nuclear Larmor frequencies and spin quantization
axes; 3) unlike the dipole-dipole interaction relevant in NMR, the electron-
hyperfine spin-spin interaction is isotropic, preserving the total spin. As an
illustration of the difference, if the electron is driven purely magnetically
(ESR) and the spin quantization axes of all nuclei and as well as the electron
are collinear, the DNSP effects that we describe would not be
present.666However, we reason that such a highly idealized situation does not
describe realistic experiments even if they do not employ micro magnets. The
DNSP arises, and our formulas apply also in this scenario; see Sec. V for
details.

The article focuses on a detailed derivation of the DNSP rate, but it also
contains measured data on it. The derivation is presented in Secs. II–IV with
auxiliaries delegated to Apps. A–J. The main result is the polarization rate
given in Eq. (47a). It is derived for a generic material (we present theory
plots for GaAs and Si), the electron spin 1/2,777The formula covers also the
case of a hole spin, if the hole-spin–nuclear-spin interaction tensor is
known. We discuss the hole-spin scenario in App. I. and nuclear spins of
arbitrary magnitude and isotopic composition. The effects that we describe
here are resonance phenomena and very sharp resonance peaks result in the
theory if applied naively. When fitting experimental results, one needs to
account for additional ‘smearing’ effects as discussed in Section V. In Sec.
VI, we analyze the dependence of the polarization rate on the detuning from
the resonance to implement feedback to control the nuclei, similarly to
previous works along this line [45, 22, 46, 27, 28, 50].

We uncover two mechanisms of the DNSP: the first is due to the electron
spatial displacement due to the electric field, the second due to the
misalignment of the quantization axes for the electron and the nuclei induced
by the micro-magnet magnetic-field gradient. The two mechanisms coexist and
interfere, making the polarization-rate dependence on parameters involved.
Nevertheless, in GaAs with the Hartmann-Hahn resonance condition fulfilled,
the polarization rate can reach hundreds of MHz per second (we convert the
nuclear polarization to the change of the electron precession frequency due to
the induced Overhauser field).

An important question is whether one expects sizable DNSP in natural Si. While
the rates are orders of magnitude smaller than in GaAs, the effect might be
observable because of longer spin coherence times in Si. We estimate that the
rates can reach tens of kHz/s, and even more in smaller dots. On the other
hand, our estimates given in Sec. VI.3 suggest that, unlike in GaAs, the
arising DNSP does not appreciably affect gate fidelities in Si.

Concerning the experiment, the measurements were performed by driving a single
electron spin in a double dot GaAs sample with a micro magnet using the Pauli
spin blockade as the spin detection. While we find the qualitative
correspondence to the theory satisfactory, the measured data are noisy and do
not show clear resonance peaks. We believe that this is because of strong
feedback: the polarized nuclear spins change the DNSP rate by changing the
EDSR resonance frequency. It is only through compensating for this effect in
the experiment (that is, readjusting the driving frequency to the actual value
of the hyperfine field) that polarization rates could have been measured. The
compensation precision is limited and, therefore, the correspondence of the
theory and measurements is only qualitative concerning the shape of the curve
for the DNSP rate. On the other hand, the magnitude of the observed rate
aligns with the theory almost without any fitting, using the material
constants and parameters of the dot obtained independently.

Figure 1: DNSP in a GaAs quantum dot. The measured data (points) show the
polarization rate observed in an EDSR-driven single-electron quantum dot. The
three color curves plot Eq. (47) for the three isotopes of GaAs as given in
the plot legend. The following parameters were used in the evaluation of the
theory expressions: external field $B=1$ T, Rabi frequency at resonance 6.5
MHz, pulse time $T_{\mathrm{pulse}}=1$ $\mu$s, cycle time
$T_{\mathrm{cycle}}=20$ $\mu$s, dot displacement $d=0.5$ nm, dot in-plane size
$l=34$ nm, dot out-of-plane size $l_{z}=10$ nm, longitudinal magnetic field
gradient $\nabla_{||}B=1$ T/$\mu$m, transverse magnetic field gradient
$\nabla_{\perp}B=0.3$ T/$\mu$m. Finally, we used energy density $G_{\Sigma}$
with a Lorenzian profile and included an additional smearing of $2\pi\times
250$ kHz according to the discussion in Sec. V. For better comparison to the
data, the theoretically calculated rates were multiplied by 1/2.

##  II Electron spin coupled to ensemble of nuclear spins

We consider an electron confined in a quantum dot interacting with nuclear
spins of the atoms of the semiconducting host. We now list the elements of the
problem.

###  II.1 Quantum dot

On top of a homogeneous field of a solenoid coil, a micro magnet fabricated
nearby the quantum dot adds an inhomogeneous component, together resulting in
a spatially dependent magnetic field $\mathbf{B}(x,y,z)$. For the DNSP rates
studied here, one can neglect the spin-orbit effects (both from the intrinsic
spin-orbit interactions and from the magnetic field inhomogeneity) on the
electron wave function and take it as separable to the spin part and the
orbital part. We take the latter as

$\Psi(\mathbf{r},z)=\frac{1}{\sqrt{\pi}l}\exp[-(\mathbf{r}-\mathbf{r}_{0})^{2}/2l^{2}]\psi(z).$
(1)

The Gaussian form in the in-plane (the 2DEG plane) coordinates
$(x,y)\equiv\mathbf{r}$ corresponds to harmonic confinement with the scale $l$
and the minimum at $\mathbf{r}_{0}$. Together with the effective mass $m$, the
length scale $l$ defines the in-plane orbital energy $\hbar^{2}/ml^{2}$. The
wave-function profile along the coordinate $z$ (out-of-plane), $\psi(z)$, will
not be important and is left unspecified except of assigning it a
corresponding length scale $l_{z}$. With that, we define the quantum dot
effective volume $V_{D}=2\pi l^{2}l_{z}$ and the effective number of nuclei
within the quantum dot $N_{\mathrm{tot}}=V_{D}/v_{0}$ (see App. A for the
definition of $V_{D}$ and the definition motivation). Here $v_{0}=a_{0}^{3}/8$
is the volume per atom in a zinc-blende or diamond lattice. $N_{\mathrm{tot}}$
counts all atomic nuclei, irrespective of their spin. The spin depends on the
isotope. Introducing the isotope fractions $\phi_{i}$, the number of atoms of
isotope $i$ in the dot is $N_{i}=\phi_{i}N_{\mathrm{tot}}$. The total number
of spin-carrying nuclei is large, up to a million in a typical GaAs gated dot
and ten thousand in a Si dot with natural isotopic concentrations.

###  II.2 Nuclei

Concerning atoms, we need to distinguish different isotopes as they differ in
their nuclear-spin characteristics. We use the following notation. The atoms
within the quantum dot are indexed by subscript $n$. When the individual
position of the nucleus is not relevant, we trade the individual index $n$ for
the isotope index $i$. (The latter is a function of the former, $i=i(n)$, but
we omit the argument for notational clarity.) In GaAs
$i\in\\{^{69}\mathrm{Ga},^{71}\mathrm{Ga},^{75}\mathrm{As}\\}$, while $n$ is
an integer going from one to about a million. A quantity $X$ specified for a
given nucleus then reads $X_{n}$ or $X_{i}$. For notational clarity, we
sometimes omit the nuclear index on the spin operator entirely,
$\mathbf{I}_{n}$ or $\mathbf{I}_{i}$ $\to$ $\mathbf{I}$. There are also
quantities that are defined only with the isotope index $i$, for example, the
material isotopic fractions $\phi_{i}$.

The nuclear spin is coupled to the magnetic field through the Zeeman term,

$H_{n}^{Z}=-g_{n}\mu_{N}\mathbf{B}_{n}\cdot\mathbf{I}_{n}.$ (2)

Here, $g_{n}$ is the nuclear $g$ factor, $\mu_{N}$ is the nuclear magneton,
$I_{n}$ is the nuclear spin magnitude (not necessarily 1/2), and
$\mathbf{I}_{n}$ is the vector of nuclear spin operators. Among these, the $g$
factor and spin magnitude depend only on the isotope, so that the atom index
$n$ could be traded for the isotope index $i$. Importantly, the magnetic field
$\mathbf{B}_{n}=\mathbf{B}(x_{n},y_{n},z_{n})$ depends on the location of the
atom because of the micro-magnet induced gradients. They are parameterized by
$\nabla\mathbf{B}$, a second-rank tensor defined by
$(\nabla\mathbf{B})_{ij}=\nabla_{i}B_{j}$. While the gradients are small,
$l|\nabla\mathbf{B}|\ll B$, taking them into account is crucial for one of the
DNSP mechanisms. Finally, we define the unit vector $\mathbf{z}_{n}$ pointing
along $\mathrm{sgn}(g_{n})\mathbf{B}_{n}$, being the direction of the nuclear
spin in the ground state of $H_{n}^{Z}$. With that, we rewrite Eq. (2) as

$H_{n}^{Z}=-\hbar\omega_{n}\mathbf{I}_{n}\cdot\mathbf{z}_{n},$ (3)

where the angular Larmor frequency $\omega_{n}$ is positive independently on
the sign of the $g$ factor, a form which will be useful in the derivations
below.

###  II.3 Electron and its hyperfine interaction with nuclei

The DNSP arises due to a coupling of the electron and nuclear spins. It takes
the form of the Fermi-contact, or hyperfine, interaction,

$H_{\mathrm{hf}}=\sum_{n}A_{n}v_{0}|\Psi(\mathbf{r}_{n},z_{n})|^{2}\mathbf{I}_{n}\cdot\mathbf{s}.$
(4)

Here, $A_{n}$ is an isotope-dependent constant, and $\mathbf{s}$ is the vector
of electron spin operators. We consider a spin one-half, $s=1/2$, and use the
spin operator $\mathbf{s}=\boldsymbol{\sigma}/2$ with $\boldsymbol{\sigma}$
the Pauli sigma matrices. Once the electron orbital degrees of freedom have
been separated and specified by Eq. (1), the spin is the remaining degree of
freedom. It is described by the Hamiltonian

$H_{e}^{Z}=g_{e}\mu_{B}\mathbf{B}_{e}\cdot\mathbf{s},$ (5)

where $g_{e}$ is the $g$ factor and $\mu_{B}$ is the Bohr magneton. Equation
(5) is the analog of Eq. (2) (the overall sign is opposite due to the opposite
electric charge), but there are differences concerning the field
$\mathbf{B}_{e}$. Namely, in the lowest approximation that we adopted by Eq
(1), it is a sum of two contributions. The first is the spatial average of the
magnetic field within the quantum dot,

$\langle\mathbf{B}\rangle=\int|\Psi(\mathbf{r},z)|^{2}\mathbf{B}(\mathbf{r},z)\,\mathrm{d}\mathbf{r}\,\mathrm{d}z.$
(6)

The second is the statistical average of Eq. (4), the Overhauser field, which
we specify introducing polarizations $p_{n}$,

$\langle\mathbf{I}_{n}\rangle=p_{n}I_{n}\mathbf{z}_{n}.$ (7)

To make progress, we adopt further approximations. The goal of this article is
to calculate the nuclear spin polarization $p_{n}$, or its rate of change, the
DNSP rate. However, we are not interested in polarizations of individual
atoms, which are not observable anyway, but rather in their collective effect
on the electron spin. Therefore, we assign all atoms of a given isotope the
same polarization

$p_{n}\to p_{i},$ (8)

drastically reducing the set of unknowns. Compared to this approximation, in
reality the nuclei in the center of the dot will be polarized more and on the
outskirts less. In the derivations below, we repeatedly average over the
nuclei (or over the dot coordinates) in this spirit. The second approximation
is to neglect the deflection of the Overhauser field from the average external
field concerning the electron Zeeman energy. This deflection is a higher-order
effect (in the magnetic field gradients) and neglecting it is in line with
using Eq. (1). We thus write the electron Zeeman term

$H_{e}^{Z}=-\hbar\omega_{e}\mathbf{s}\cdot\mathbf{z}_{e},$ (9)

with $\mathbf{z}_{e}$ a unit vector along
$-\mathrm{sgn}(g_{e})\langle\mathbf{B}\rangle$ and the positive Zeeman energy

$\hbar\omega_{e}=|g_{e}\mu_{B}\langle\mathbf{B}\rangle|+\sum_{i}\mathrm{sgn}(g_{e})p_{i}\phi_{i}I_{i}|A_{i}|.$
(10)

To arrive at this form, we assumed that the polarizations are small, so that
the magnitude of the first term is bigger than the second (see App. B for the
derivation).

###  II.4 EDSR

The last basic element is the EDSR driving. Applying an oscillating electric
field
$\mathbf{E}(t)=\mathbf{E}_{0}\cos(\omega_{\mathrm{rf}}t-\phi_{\mathrm{rf}})$
drives the electron in space. The micro-magnet-field gradients result in an
effective oscillating magnetic field. Since the driving frequency is small
compared to the electron orbital confinement energy,
$\hbar\omega_{\mathrm{rf}}\ll\hbar^{2}/ml^{2}$, the drive is adiabatic with
respect to the electron orbital degrees of freedom and results in a time-
dependent displacement of the dot center $\mathbf{r}_{0}$ by

$\mathbf{d}(t)=\frac{e\mathbf{E}(t)l^{2}}{\hbar^{2}/ml^{2}}.$ (11)

The EDSR drive can thus be taken into account by using Eq. (1) with a time-
dependent center, $\mathbf{r}_{0}\to\mathbf{r}_{0}+\mathbf{d}(t)$, in Eqs. (4)
and (6). The replacement in Eq. (4) will lead to one of the DNSP mechanisms
(as we will see below), while in Eq. (6), it gives an effective oscillating
magnetic field

$\mathbf{B}_{\mathrm{rf}}(t)=(\mathbf{d}(t)\cdot\nabla_{\mathbf{r}_{0}})\langle\mathbf{B}\rangle.$
(12)

The component of $\mathbf{B}_{\mathrm{rf}}(t)$ perpendicular to the average
field $\mathbf{B}_{e}$ is denoted as

$-g_{e}\mu_{B}\left[\mathbf{B}_{\mathrm{rf}}(t)\right]_{\perp}\equiv-2\hbar\omega_{RR}\mathbf{b}\cos(\omega_{\mathrm{rf}}t-\phi_{\mathrm{rf}}).$
(13)

The equation defines the unit vector $\mathbf{b}$ and the Rabi angular
frequency at resonance $\omega_{RR}$. The Rabi oscillations of the electron
due to this term, induced by the electric field, are called EDSR.

All quantities that were defined in this section and will be used in the
following are collected for reference in Tab. J in App. J.

##  III Electron-nuclear spin pair

We consider DNSP arising in the following repeated experiment. The electron
spin is initialized to the ground state of $H_{Z}^{e}$ (using the electron
reservoir, not nuclei), and then EDSR driven for a fixed time, of order
$\mu$s, at a fixed detuning $\omega_{\Delta}=\omega_{\mathrm{rf}}-\omega_{e}$
of order tens of $2\pi$ $\times$ MHz. Ref. [51] gives a detailed description
of these steps and their implementation.888The regularity of re-initalization
of the electron spin is crucial for auto-focusing in experiments such as Ref.
[52]. On the other hand, assuming random re-initialization times was important
for the description in Ref. [53]. In our model, the (ir)regularity of the
moments at which the electron spin is initialized is irrelevant (although it
matters into what state the electron spin is initialized): The DNSP is
happening continuously during the electron Rabi precession.

We derive the polarizations $p_{i}$ and the corresponding rates

$\Gamma_{i}=\partial_{t}p_{i},$ (14)

proceeding in two steps: First, we consider an isolated nucleus $n$, of the
isotope $i$, in contact with a driven electron. We solve for its dynamics.
Second, we average the arising polarization rate over the dot, in line with
Eq. (8). Considering the nuclei polarization rates as independent is a good
approximation as long as only a small fraction of the electron spin is
transferred to the nuclear ensemble over one experimental cycle (after which
the electron is reinitialized).999Ref. [54] went beyond the approximation of
independent rates and considered the electron spin being dissipated into the
nuclear ensemble as a whole. This condition is well fulfilled in all our
numerical examples and plots.

The restriction to a single nucleus allows us to simplify the notation. We
introduce a shorthand notation for the hyperfine coupling (the Knight field)
as

$J_{n}(t)=A_{n}v_{0}|\Psi_{n}(t)|^{2},$ (15)

where we denoted the time dependence explicitly. The Hamiltonian for the
electron-nuclear pair is

$\begin{split}H=&-\hbar\omega_{n}\mathbf{I}\cdot\mathbf{z}_{n}-\hbar\omega_{e}\mathbf{s}\cdot\mathbf{z}_{e}\\\
&-2\hbar\omega_{RR}\mathbf{s}\cdot\mathbf{b}\cos(\omega_{\mathrm{rf}}t-\phi_{\mathrm{rf}})+J_{n}(t)\delta\mathbf{I}\cdot\mathbf{s}.\end{split}$
(16)

The first two terms are the Zeeman energies, Eqs. (2) and (5), the third is
the EDSR-driving term, Eq. (13), and the last is the hyperfine coupling,
originating from Eq. (4). In this term, we subtracted the statistical average,
defining $\delta\mathbf{I}=\mathbf{I}-\langle\mathbf{I}\rangle$, since the
average has been included in $\omega_{e}$. For further convenience, all
frequencies in the above equation are defined as positive. Inverting a sign,
for example of a $g$-factor, would be reflected by inverting the corresponding
unit vector $\mathbf{z}$.101010We find that while the $g$-factor signs are not
entirely irrelevant as they show up in the formulas below, they do not lead to
qualitative differences. Rather, inverting a $g$-factor maps the problem to an
equally relevant scenario for all questions that we consider. See especially
Sec. VI. While the hyperfine coupling $J_{n}$ is signed, neither the DNSP rate
nor the feedback through Eq. (10) will depend on the sign.

##  IV Polarization rate

We now proceed with the derivation of the polarization rate using Eq. (16). As
already noted, the calculation is related to some results of the NMR and
molecular-chemistry literature [55, 56, 54, 57]. Nevertheless, there are
important differences, which we point out on the way.

###  IV.1 The origin of the DNSP

The first step is to gauge away the time-dependent driving, transforming to a
rotating reference frame, $\Psi^{\prime}=U\Psi$. It is useful to transform
both the electron and nuclear spins, with the following unitary,

$U(t)=\exp(-i\mathbf{s}\cdot\mathbf{z}_{e}\omega_{\mathrm{rf}}t)\exp(-i\mathbf{I}\cdot\mathbf{z}_{n}\omega_{\mathrm{rf}}t).$
(17)

Adopting the rotating-wave approximation in the third term of Eq. (16) gives
the transformed Hamiltonian

$\begin{split}H^{\prime}=&-(\hbar\omega_{n}-\hbar\omega_{\mathrm{rf}})\mathbf{I}\cdot\mathbf{z}_{n}-(\hbar\omega_{e}-\hbar\omega_{\mathrm{rf}})\mathbf{s}\cdot\mathbf{z}_{e}\\\
&-\hbar\omega_{RR}\,\mathbf{s}\cdot{\mathbf{y}_{e}}+\delta\mathbf{I}\cdot
J_{n}^{\prime}(t)\cdot\mathbf{s}.\end{split}$ (18)

We have defined ${\mathbf{y}_{e}}$ as the vector $\mathbf{b}$ rotated by angle
$\phi_{\mathrm{rf}}$ around axis $\mathbf{z}_{e}$, and the transformed
hyperfine tensor by the relation

$U(t)J_{n}(t)\,\delta\mathbf{I}\cdot\mathbf{s}\,U(t)^{\dagger}=\sum_{ij}[J_{n}^{\prime}(t)]_{ij}\delta
I_{i}s_{j}.$ (19)

Here, the differences to the existing derivations can be appreciated. First,
were the quantization axes of the electron and the nucleus parallel, which
would be the case for a magnetic field constant in space, the transformation
$U$ would commute with the spin-spin interaction

$[J^{\prime}_{n}(t)]_{ij}=J_{n}(t)\delta_{ij},$ (20)

This result arises because the hyperfine interaction Eq. (4) conserves the
total (electron plus nuclear) spin. In this case, the transformation into the
rotating frame does not generate time-dependent terms. In our problem, the
Knight field $J_{n}$ is still time-dependent due to the spatial displacement
of the electron, making the wave function modulus $|\Psi_{n}|^{2}$ in Eq. (15)
time-dependent. This additional time dependence is the second difference to
the existing results. For a typical NMR scenario with two nuclei in the
lattice of a crystal or in a molecule, their mutual interaction in the
laboratory frame is constant. That would here correspond to an ESR[58] (and
not EDSR) driving of the electron, by which the transformed hyperfine tensor
would become not only diagonal in spin indexes but also time-independent

$[J_{n}^{\prime}(t)]_{ij}=J_{n}(0)\delta_{ij},$ (21)

Under such conditions, the transformed Hamiltonian in Eq. (18) would be time-
independent and no DNSP effects would arise.111111On the other hand, both ESR
[10] and EDSR [5] experiments in a gated quantum dot showed signatures of
DNSP. In the former, an oscillating electric field probably accompanied the
desired oscillating magnetic field.

Since in the NMR scenarios the laboratory frame $J_{n}$ is time-independent, a
finite DNSP requires either ‘non-secular’ terms in the exchange tensor, such
as $I_{x}s_{z}$,[49, 55, 54, 57] or Rabi-driving also the nuclear spin [56],
where the ‘secular’ exchange term $I_{z}s_{z}$ allows for spin flips (as in
the standard Hartmann-Hahn scenario [47]).

Concluding, there are two sources of the time dependence of the transformed
hyperfine tensor $J^{\prime}_{n}$. One is the non-collinearity of the spin
quantization directions and is due to the micro-magnet-induced magnetic field
gradients. The second is due to the time-dependent spatial oscillations of the
electron induced by the EDSR drive. We refer to the two sources as the two
mechanisms of the DNSP. Neither is present in the standard NMR scenario, while
at least one is necessary for a finite DNSP in a quantum dot in the coherent
regime of EDSR.

###  IV.2 Identification of the secondary resonance

Figure 2: Angles and directions relevant for EDSR. The electron ground-state
spin direction $\mathbf{z}_{e}$ is along or opposite to the magnetic field
$\mathbf{B}_{e}$, depending on the electron $g$-factor sign. In the frame
rotating with the electron Larmor frequency, the EDSR field is along the
vector ${\mathbf{o}_{e}}$ and makes an angle $\pi/2-\gamma$ with the axis of
the precession. For the nuclear ground-state spin direction $\mathbf{z}_{n}$,
apart from relating to the magnetic field by the $g$ factor, it also changes
with the position within the dot due to the micro magnet. The rotation
$R_{\mathbf{z}_{e}\to\mathbf{z}_{n}}$ that rotates the axis $\mathbf{z}_{e}$
into the axis $\mathbf{z}_{n}$ is parameterized by two Euler angles:
$\delta^{\prime}$ for the initial rotation around $\mathbf{z_{e}}$, and
$\delta$ for the final rotation around the rotated axis
${\mathbf{y}_{e}}^{\prime}$.

After explaining the physical origins of the effects and their differences
from the Hartmann-Hahn scenario of NMR, we now proceed with straightforward
manipulations of Eq. (18). The technical reason for employing the
transformation $U$ was to move all time-dependence into the last term of
$H^{\prime}$. Since it is the smallest term, it can be treated perturbatively.
To this end, we first diagonalize the unperturbed part by introducing the
following unit vectors and angles

$\displaystyle{\mathbf{y}_{e}}$
$\displaystyle=R_{\mathbf{z}_{e},\phi_{\mathrm{rf}}}\cdot\mathbf{b},$ (22a)
$\displaystyle{\mathbf{x}_{e}}$
$\displaystyle={\mathbf{y}_{e}}\times\mathbf{z}_{e},$ (22b)
$\displaystyle{\mathbf{o}_{e}}$
$\displaystyle=\mathbf{z}_{e}\sin\gamma+{\mathbf{y}_{e}}\cos\gamma,$ (22c)
$\displaystyle\sin\gamma$ $\displaystyle=-\frac{\omega_{\Delta}}{\omega_{R}},$
(22d) $\displaystyle\cos\gamma$
$\displaystyle=\frac{\omega_{RR}}{\omega_{R}}.$ (22e)

Also, $\omega_{R}=\sqrt{\omega_{RR}^{2}+\omega_{\Delta}^{2}}$ is the
(positive) Rabi frequency and $R_{\mathbf{n},\alpha}$ is a $3\times 3$ matrix
corresponding to a rotation around vector $\mathbf{n}$ by angle $\alpha$. The
axes and angles are shown in Fig. 2. The Hamiltonian becomes

$H^{\prime}=-(\hbar\omega_{n}-\hbar\omega_{\mathrm{rf}})\mathbf{I}\cdot\mathbf{z}_{n}-\hbar\omega_{R}\mathbf{s}\cdot{\mathbf{o}_{e}}+\delta\mathbf{I}\cdot
J_{n}^{\prime}(t)\cdot\mathbf{s}.$ (23)

The unperturbed part of the Hamiltonian (the first two terms) has eigenstates
with the nuclear and electron spins parallel or antiparallel to the vectors
$\mathbf{z}_{n}$ and ${\mathbf{o}_{e}}$. We denote them as $|sj\rangle$,

$H^{\prime}(J^{\prime}_{n}=0)|sj\rangle=E_{sj}|sj\rangle,$ (24)

where $s$ and $j$ denote the spin eigenvalues: $s\in\\{+1/2,-1/2\\}$ and
$j\in\\{+I,+I-1,\ldots,-I\\}$ with a general integer or half-integer value for
$I$. The corresponding energy is 121212Concerning unperturbed energies, we
thus include only the collective Overhauser field from all nuclei acting on
the electron [entering into $\hbar\omega_{R}$ through Eq. (10)] and neglect
the Overhauser field and the Knight field stemming from the last term in Eq.
(23). Ref. [56] deals with the scenario where the diagonal part of the
hyperfine interaction is strong and needs to be included in the unperturbed
energies.

$E_{sj}=-(\hbar\omega_{n}-\hbar\omega_{\mathrm{rf}})j-\hbar\omega_{R}s,$ (25)

The energy difference between a pair of eigenstates is

$E_{sj}-E_{s^{\prime}j^{\prime}}=(\hbar\omega_{n}-\hbar\omega_{\mathrm{rf}})(j^{\prime}-j)+\hbar\omega_{R}(s^{\prime}-s).$
(26)

The DNSP (and the Hartmann-Hahn effect) arises if a pair of these eigenstates
is Rabi-driven through the time-dependent term in Eq. (23) on resonance with
the energy difference Eq. (26).131313In contrast, in Refs. [22, 46] the energy
mismatch is assumed to be compensated by an additional agent, such as an
applied source-drain voltage. In Ref. [45] (Ref. [59]), it is the finite
linewidth of the electron (hole) spin. Since we are interested in transitions
that change the nuclear spin, $j^{\prime}\neq j$, the large energy difference
$\hbar\omega_{\mathrm{rf}}$ must be compensated by a time-dependent term
oscillating at a similar frequency. The matrix elements of $J_{n}^{\prime}$
are polynomial functions of exponentials $\exp(\pm i\omega_{\mathrm{rf}}t)$
and, therefore, contain only integer multiples of $\omega_{\mathrm{rf}}$ as
frequencies. The integer one multiple can compensate the driving frequency in
Eq. (26) and what remains is

$\hbar\omega_{n}(j^{\prime}-j)+\hbar\omega_{R}(s^{\prime}-s).$

This difference can become zero only if the electron also flips,
$s=-s^{\prime}$, and we get that a quasi-resonant pair fulfills

$s+j=s^{\prime}+j^{\prime}.$ (27)

Concluding, the only states which can become quasi-resonant are (we explicitly
denote the spin quantization directions in subscripts as a reminder)

$|s_{{\mathbf{o}_{e}}}=1/2,j_{\mathbf{z}_{n}}\rangle\leftrightarrow|s_{{\mathbf{o}_{e}}}=-1/2,(j+1)_{\mathbf{z}_{n}}\rangle,$
(28)

and that happens if the Hartmann-Hahn-like condition,

$\hbar\omega_{n}\approx\hbar\omega_{R},$ (29)

is fulfilled.

###  IV.3 Secondary Rabi oscillations

We depict the result of the preceding analysis by the following Hamiltonian
for the electron-nuclear spin pair,

$\left(\begin{tabular}[]{c|cccc}$H_{s^{\prime}j^{\prime},sj}$&$\uparrow\uparrow$&$\uparrow\downarrow$&$\downarrow\uparrow$&$\downarrow\downarrow$\\\
\hline\cr$\uparrow\uparrow$&$E_{\uparrow\uparrow}$&$\cdot$&$\cdot$&$\cdot$\\\
$\uparrow\downarrow$&$\cdot$&$E_{\uparrow\downarrow}$&$Y^{\dagger}$&$\cdot$\\\
$\downarrow\uparrow$&$\cdot$&$Y$&$E_{\downarrow\uparrow}$&$\cdot$\\\
$\downarrow\downarrow$&$\cdot$&$\cdot$&$\cdot$&$E_{\uparrow\uparrow}$\\\
\end{tabular}\right).$ (30)

We have used a pictorial notation for the spin states, $\uparrow$ and
$\downarrow$ for the electron states $+1/2$ and $-1/2$, and for nuclear states
$j+1$ and $j$. In addition to the state energies, given by Eq. (25), one needs
only one matrix element,

$Y=\langle\downarrow\uparrow|\delta\mathbf{I}\cdot
J_{n}^{\prime}(t)\cdot\mathbf{s}|\uparrow\downarrow\rangle\equiv\langle\downarrow\uparrow|I_{+}s_{-}|\uparrow\downarrow\rangle
X\equiv\mathcal{I}_{+}X,$ (31)

for the only pair of states which might become quasi-resonant. Here, we have
introduced the spin ladder operators $s_{\pm}=s_{x}\pm is_{y}$ and
$I_{\pm}=I_{x}\pm iI_{y}$. All other states are off-resonant, with the
Hamiltonian matrix elements negligible with respect to the energy differences.
These negligible off-resonant elements are denoted by dots in Eq. (30).
Therefore, one can focus on the state pair
$\\{\uparrow\downarrow,\downarrow\uparrow\\}$ as an effective two-level system
displaying Rabi oscillations.141414Note that these are ‘secondary’ Rabi
oscillations, different from the Rabi oscillations of the electron spin
itself. The ‘primary’ electron spin Rabi oscillations are taken into
account—in the basis corresponding to Eq. (30)—through the energies only. The
appearance of the ‘secondary’ Rabi oscillations in a frame where the ‘primary’
oscillations are already trivial is the essence of the Hartmann-Hahn effect,
see Eqs. (49)–(50) in Ref. [47]. In Eq. (31), we have introduced the
abbreviations $X$ and $\mathcal{I}_{+}$, as parts of the matrix element $Y$
that we calculate below separately.

Figure 3: Bloch sphere for the secondary Rabi oscillations. The electron-
nuclear states $|\\!\\!\uparrow\downarrow\rangle$ and
$|\\!\\!\downarrow\uparrow\rangle$ define the north and south pole of the
Bloch sphere. The in-plane axes are chosen so that the energy vector is in the
yz plane, making an angle $\pi/2-\Gamma$ with the $z$ axis. This angle is
defined by the matrix element $|Y|$ and the energy difference
$\hbar\omega_{n}-\hbar\omega_{R}$. Finally, $\mathbf{p}(0)$ is the initial
polarization of the system.

The corresponding $2\times 2$ block of the Hamiltonian can be then treated by
the textbook method for the Rabi problem. We define a Bloch sphere spanning
the two orthogonal states $\\{\uparrow\downarrow,\downarrow\uparrow\\}$ which
we place on the sphere $z$ axis. There are two parameters important for the
Rabi oscillations: the energy difference of the states, which is
$E_{\uparrow\downarrow}-E_{\downarrow\uparrow}=\hbar\omega_{n}-\hbar\omega_{R}$,
and the magnitude of the matrix element $|Y|$. The phase of $Y$ defines only
where to put the in-plane axes, x and y, of the Bloch sphere, and is not
relevant in the following. The two parameters define the (positive) Rabi
frequency

$\hbar{\omega_{R}^{\mathrm{hh}}}=\sqrt{|Y|^{2}+\left|\hbar\omega_{n}-\hbar\omega_{R}\right|^{2}},$
(32)

and the angle $\Gamma$ which will turn out useful,

$\displaystyle\sin\Gamma$
$\displaystyle=\frac{\hbar\omega_{n}-\hbar\omega_{R}}{\hbar{\omega_{R}^{\mathrm{hh}}}},$
(33a) $\displaystyle\cos\Gamma$
$\displaystyle=\frac{|Y|}{\hbar{\omega_{R}^{\mathrm{hh}}}}.$ (33b)

These quantities are shown in Fig. 3.

Now we come to a somewhat subtle point concerning the initial state, that is,
the system state at the time when the electron enters the dot or its EDSR
driving begins. The phase relation of the $\uparrow\downarrow$ and
$\downarrow\uparrow$ components of this initial state has contributions not
only from the phase difference of the electron spin up and down state, which
is controlled, but also from a similar phase difference on the nuclear spin,
which is not controlled. Alternatively viewed, the Hamiltonian in Eq. (30)
induces entanglement between the electron and nuclear spin. This entanglement
is lost repeatedly as the electron is repeatedly reinitialized through the
reservoir or otherwise. As a consequence, on average (over the experiment
cycle repetitions) the initial state in the Bloch sphere in Fig. 3 can only
have a non-zero component along the sphere $z$ axis.151515In the language of
Ref. [60], in our scenario we have ‘cross-polarization’, but no ’coherence
transfer’. Our assumption means that we do not consider that the nuclear spin
precession and the moments when the electron EDSR rotation starts are
synchronized over many cycles. Such long-time synchronization is essential for
nuclear auto-focusing [61, 52]. We will thus describe this initial state by a
density matrix, parameterized by a vector $\mathbf{p}(t)$, which is at time
$t=0$ aligned with the $z$ axis, and its length might be smaller than one.

The dynamics of this ‘polarization’ vector is a simple precession and can be
expressed, for example, by

$\mathbf{p}(t)=R^{-1}_{\mathbf{x},\pi/2-\Gamma}\cdot
R_{\mathbf{z},{\omega_{R}^{\mathrm{hh}}}t}\cdot
R_{\mathbf{x},\pi/2-\Gamma}\cdot\mathbf{p}(0),$ (34)

where $\mathbf{x}$ and $\mathbf{z}$ are unit vectors along the Bloch-sphere
axes. In addition, it is only the $z$ component that is relevant, as we have
just discussed. It is then straightforward to evaluate the previous equation
for that component arriving at

$p_{z}(t)=p_{z}(0)\left(\sin^{2}\Gamma+\cos^{2}\Gamma\cos{\omega_{R}^{\mathrm{hh}}}t\right).$
(35)

This result is the first main ingredient of the DNSP polarization rate.

###  IV.4 Calculation of the matrix element $\mathcal{I}_{+}$

We now look at the initial polarization $p_{z}(0)$. As explained, it is
contributed by the initial polarization of both the electron and the nucleus.
The conversion from these two polarizations to $p_{z}(0)$ is not completely
trivial because we consider a general nuclear spin $I$ and because the
polarizations need to be weighted by the spin-dependent transition matrix
element $\mathcal{I}_{+}$. The calculation is shown in App. C and gives

$\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}=I\times\Big{(}\alpha_{I}(p_{n})p_{e}-p_{n}\Big{)}.$
(36)

Here, $p_{e}$ is the initial polarization of the electron spin _along the axis
${\mathbf{o}_{e}}$_ and $p_{n}$ is the polarization of the nuclear spin along
the external magnetic field. Both of these polarizations are normalized so
that the maximal possible polarization corresponds to $p=1$. Finally,
$\alpha_{I}(p_{n})$ is a factor of order one, which we calculate in App. C,
see Eq. (82b). We get $\alpha_{I}=(2/3)(I+1)$ for $p_{n}\ll 1$ and
$\alpha_{I}=1$ for $1-p_{n}\ll 1$.

Equation (36) states that the electron polarization $p_{e}$ is the source of
the nuclear polarization $p_{n}$. As the latter develops a finite value, the
rate diminishes. For nuclear spin $I=1/2$ one has $\alpha_{I}=1$ for any
$p_{n}$, and the rate is proportional to the difference $p_{e}-p_{n}$, a
natural result. The proportionality factor $\alpha_{I}$ differs from one for
nuclear spin $I>1/2$. In any case, in the majority of experiments the steady
state nuclear polarization will be reached once the DNSP rate is balanced by
additional decay channels, such as nuclear diffusion, rather than due to the
DNSP rate dropping to zero at $p_{n}=p_{e}\alpha_{I}$. Therefore, the second
term in the bracket in Eq. (36) can usually be dropped.

Let us now elucidate the electron polarization $p_{e}$, considering two
typical experiments. In the first, the initial electron state is along the
external magnetic field. Once the driving is turned on, the electron performs
Rabi oscillations. This choice is the standard EDSR and means the initial
electron polarization is equal to
$\mathbf{z}_{e}\cdot{\mathbf{o}_{e}}=\sin\gamma$. In the second, the electron
is ‘spin-locked’, meaning its spin is along ${\mathbf{o}_{e}}$ and the initial
polarization is one.161616In this case the system has to be initialized either
adiabatically changing the driving frequency [47] or using a phase shift in
the driving pulse [55]. Even if it is the former, we are not concerned with
the transition period needed to spin-lock the electron. We assume that the
transition period is shorter than the time during which the electron remains
spin-locked with a constant Rabi frequency. Finally, a more complicated
initial polarization when the electron is driven off resonance was considered
in Ref. [57]. Summarizing, we get

$\displaystyle p_{e}$ $\displaystyle=\sin\gamma$ $\displaystyle\mathrm{EDSR},$
(37a) $\displaystyle p_{e}$ $\displaystyle=1$
$\displaystyle\mathrm{spin\,locking}.$ (37b)

While the first choice corresponds to the standard EDSR, the advantage of the
second one (concerning possible DNSP) is that the lifetime of the electron
spin is longer in the spin-locked state, compared to the lifetime of the Rabi
oscillations [62]. Finally, we note that in both scenarios one can invert the
polarization $p_{e}\to-p_{e}$, by preparing the electron in the excited state,
rather than in the ground state.

###  IV.5 Calculation of the matrix element $X$

We now turn to $X$, the second component of the matrix element $Y$. Writing
$\delta\mathbf{I}\cdot{J_{n}^{\prime}}(t)\cdot\mathbf{s}$ in the interaction
picture

$\langle sj|\exp(iE_{sj}t/\hbar)\delta\mathbf{I}\cdot
J_{n}^{\prime}(t)\cdot\mathbf{s}\exp(-iE_{s^{\prime}j^{\prime}}t/\hbar)|s^{\prime}j^{\prime}\rangle,$
(38)

one can see that the resonant component of $J^{\prime}_{n}(t)$ is the one of
frequency $2s\times\omega_{\mathrm{rf}}$. Introducing the Fourier components
in this matrix,

$J_{n}^{\prime}(t)=\sum_{k\in\mathbb{Z}}\exp(ik\omega_{\mathrm{rf}}t){J_{n}^{\prime}}^{(k)},$
(39)

the resonant matrix element in Eq. (38) would be
${J_{n}^{\prime}}(t)\approx{J_{n}^{\prime}}^{(2s)}\exp(i2s\omega_{\mathrm{rf}}t)$.
Keeping only this term, essentially the rotating wave approximation, the
calculation of the matrix element is a straightforward algebraic exercise and
we delegate it to App. D. The result, after the spatial average over the dot
coordinates, is

$\langle|X|^{2}\rangle=X_{\mathrm{df}}^{2}+X_{\mathrm{sh}}^{2}+2\xi
X_{\mathrm{df}}X_{\mathrm{sh}},$ (40a) where $\displaystyle X_{\mathrm{df}}$
$\displaystyle=\frac{J}{4}\frac{\nabla_{\perp}Bl}{B}\cos\gamma,$ (40b)
$\displaystyle X_{\mathrm{sh}}$
$\displaystyle=\frac{J}{4}\frac{d}{l}(1+\sin\gamma),$ (40c) $\displaystyle\xi$
$\displaystyle=\cos(\delta^{\prime}+\phi_{\mathrm{rf}})\cos(\phi),$ (40d)

with $J$ being the average $\langle J_{n}\rangle$, $\nabla_{\perp}B$ being the
magnitude of the gradient of the transverse components of the magnetic field,
and the angles $\delta^{\prime}$ and $\phi$ express the mutual orientation of
the magnetic field gradient and the dot displacement (see App.D for details).
As in experiments these directions are difficult to control or even to know,
instead of going into its rather tedious analysis, we drop the interference
term from Eq.(40a). We retain only the first two terms:171717The latter
mechanism was considered in several previous works on DNSP in quantum dots
[45, 22, 46, 50] starting with Ref. [63]. As we explain in App. G, our Eq.
(40c) can be thought of as a generalization of these previous works.
$X_{\mathrm{df}}$ is due to a deflection (thus ‘df’) of the spin quantization
axes of the electron and the nucleus, and requires a finite gradient of the
transverse magnetic field component. $X_{\mathrm{sh}}$ is due to the time
dependence of the electron-nuclear spin coupling constant, in turn due to the
time dependence of $|\Psi(\mathbf{r}_{n})|^{2}$, in turn due to the physical
shifts (thus ‘sh’) of the quantum dot electron with respect to the crystal
lattice. Equation (40) is the second main ingredient for the calculation of
the DNSP rates.

###  IV.6 Evaluation of the DNSP rate

We now have all ingredients needed to evaluate the DNSP rate. We define the
individual nuclear spin polarization rate by

$\Gamma_{n}=\frac{\langle\overline{p_{z}(0)-p_{z}(t)}\rangle}{t},$ (41)

where the bar denotes the statistical average over the nuclear spin
distribution and the angle brackets the average over the dot coordinates. The
overall sign has been chosen to define a positive polarization rate as the
decrease of $p_{z}$, that is a transition of nuclear spin from $\downarrow$
towards $\uparrow$. In other words, a positive polarization rate means that
nuclear spins are pumped into their energy ground state, being along or
opposite to the external field depending on the nuclear $g$-factor sign. Using
Eqs. (33b) and (35) gives

$\Gamma_{n}=\frac{1}{\hbar^{2}}\,\langle\overline{p_{z}(0)|X\mathcal{I}_{+}|^{2}}\rangle\frac{1-\cos{\omega_{R}^{\mathrm{hh}}}t}{({\omega_{R}^{\mathrm{hh}}})^{2}t}.$
(42)

Next, we approximate the averaging over the nuclear spins by evaluating it
separately for the matrix element $X^{2}$ and the rest,

$\Gamma_{n}=\frac{1}{\hbar^{2}}\,\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}\,\,\langle|X^{2}|\rangle\frac{1-\cos{\omega_{R}^{\mathrm{hh}}}t}{({\omega_{R}^{\mathrm{hh}}})^{2}t}.$
(43)

The two needed results are given in Eqs. (36) and (40a).

We have arrived at a rate for an ‘average’ nuclear spin, which is not really a
rate: it contains time, since it originates from coherent precession expressed
by Eq. (35). We convert it to a time-independent rate181818As a remark, this
time-dependence was kept in Ref. [54], resulting in a non-trivial time-
dependence of the polarization. For example, a polarization overshoot seen in
the data in Fig. 3 therein could be explained with it. by considering the
limit $t\to\infty$ upon which the last factor in Eq. (43) becomes a delta
function of a finite width given by the matrix element $|Y|$. Since for our
parameters the latter is several orders of magnitude smaller than other energy
smearings that we consider below, we neglect it,

$\frac{1-\cos{\omega_{R}^{\mathrm{hh}}}t}{({\omega_{R}^{\mathrm{hh}}})^{2}t}\to\pi\delta(\omega_{R}-\omega_{n}).$
(44)

We now define the total polarization rate

$\Gamma_{i,\mathrm{tot}}=\sum_{n\in
i}\Gamma_{n}=N_{i}\int\mathrm{d}\omega_{n}\,g(\omega_{n})\Gamma_{n}(\omega_{n}),$
(45)

introducing the nuclear frequency density $g(\omega)$ as the fraction of
$i$-isotope nuclei with Larmor frequency $\omega$ out of their total number
$N_{i}$. The function is derived in App. A. We get

$\Gamma_{i,\mathrm{tot}}=N_{i}\frac{\pi}{\hbar^{2}}\,\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}\,\,\langle|X^{2}|\rangle
g(\omega_{R}).$ (46)

Note the crucial role of the micro magnet, setting the width of the
distribution $g(\omega)$: the larger the gradient, the more dispersed the
Larmor frequencies of the nuclei in the dot area, and the wider the resonance.
Here, the resonance means the electron Rabi frequency $\omega_{R}$ hitting the
peak of the function $g$, which is located at the Larmor frequency of the
nuclei in the dot center.

###  IV.7 Final form of the DNSP rate and its discussion

We now put together the pieces to present the rate in a user-friendly form. In
the course of derivation, we have used several approximations, which are
expected to bring an error of order one. Therefore, we neglect small terms, in
order to arrive at a simple formula with an appealing physical interpretation:

$\partial_{t}p_{i}=\frac{\pi}{\hbar^{2}}\left(X_{\mathrm{df}}^{2}+X_{\mathrm{sh}}^{2}\right)\left(\alpha_{I}p_{e}-p_{i}\right)G_{\Sigma}\left(\omega_{R}-\omega_{i}\right),$
(47a) where $\displaystyle X_{\mathrm{df}}$
$\displaystyle=\frac{A_{i}}{4N_{\mathrm{tot}}}\frac{l\nabla_{\perp}B}{B}\cos\gamma,$
(47b) $\displaystyle X_{\mathrm{sh}}$
$\displaystyle=\frac{A_{i}}{4N_{\mathrm{tot}}}\frac{d}{l}(1+\sin\gamma),$
(47c) $\displaystyle p_{e}$
$\displaystyle=\pm\left\\{\begin{tabular}[]{c@{\hspace{2cm}}r}$\sin\gamma$\hfil\hskip
56.9055pt&for EDSR,\\\ 1\hfil\hskip 56.9055pt&for spin
locking,\end{tabular}\right.$ (47f) $\displaystyle G_{\Sigma}(x)$
$\displaystyle=\frac{1}{\sqrt{2\pi}\Sigma}\exp\left(-\frac{x^{2}}{2\Sigma^{2}}\right),$
(47g) $\displaystyle\Sigma_{\mathrm{\mu M}}$
$\displaystyle=\omega_{i}\frac{l\nabla_{||}B}{2B},$ (47h)
$\displaystyle\alpha_{I}$
$\displaystyle=\left\\{\begin{tabular}[]{c@{\hspace{2cm}}r}$\frac{2}{3}(I_{i}+1)$\hfil\hskip
56.9055pt&for $p_{i}\approx 0$,\\\ 1\hfil\hskip 56.9055pt&for $p_{i}\approx
1$,\end{tabular}\right.$ (47k) $\displaystyle\tan\gamma$
$\displaystyle=\frac{\omega_{e}-\omega_{\mathrm{rf}}}{\omega_{RR}}.$ (47l)

Here, the quantities dependent on the atomic isotope have the subscript $i$,
$l$ is the dot in-plane confinement length, typically tens of nanometers, $d$
is the dot displacement magnitude given by Eq. (11), typically below a
nanometer. The plus sign for $p_{e}$ applies if the electron is initially in
the ground state of the static field in the laboratory frame (for EDSR) or the
rotating frame (for spin-locking). If initially the electron spin is in the
excited state, the minus sign applies. Finally, $\nabla_{||}B$ is the
magnitude of the longitudinal (along the vector $\langle\mathbf{B}\rangle$)
component of the magnetic-field gradient, and $\nabla_{\perp}B$ is the
magnitude of the gradient of the magnetic field transverse components. For the
moment, we assume that $\Sigma=\Sigma_{\mathrm{\mu M}}$; however, below we
list additional sources contributing to $\Sigma$ in Eq. (47a) beyond Eq.
(47h).

Let us make a few comments on the DNSP rate given in Eq. (47), our main
result.

  * 1.

Equation (47a) gives the rate of polarization of isotope $i$. It can be
converted to the total ‘spin-injection’ rate by
$\Gamma_{i,\mathrm{tot}}=I_{i}N_{i}\partial_{t}p_{i}$. Since the hyperfine
interaction is spin preserving, this total rate of spin injected into the
nuclei is compensated by the opposite change of the electron spin (component
along the external magnetic field).

  * 3.

The nuclear polarization direction is defined as the positive rate
corresponding to pumping-in the nuclear spin energy ground state (along the
magnetic field if the nuclear $g$ factor is positive).

  * 5.

Neglecting the saturation effect, meaning dropping $p_{i}$ from the right-hand
side of Eq. (47a), the DNSP rate has a characteristic shape as a function of
the detuning from the electron Rabi resonance, parameterized by $\gamma$ here.
Namely, since $\omega_{R}(\gamma)=\omega_{R}(-\gamma)$, the DNSP rate is
antisymmetric in $\gamma$ in EDSR and symmetric in spin-locking experiments if
the ‘deflection’ mechanism dominates. The ‘shaking’ mechanism makes the
profile strongly asymmetric in both cases, through the factor
$(1+\sin\gamma)$. The shape of the DNSP rate as a function of $\gamma$ can
then hint at the dominant mechanism.

  * 7.

In experiments with a single dot, the DNSP will be typically done by repeating
a cycle including the electron spin initialization, driving, and, perhaps,
measurement. In this case, one should renormalize to the rate observed over
the laboratory time by
$\Gamma\to\Gamma\times(T_{\mathrm{pulse}}/T_{\mathrm{cycle}})$, reflecting
that the cycle contains ‘dead time’ with respect to the DNSP.

  * 9.

During a single cycle, Eq. (47a) is valid only up to time $T_{\mathrm{pulse}}$
such that $\Gamma_{\mathrm{tot}}T_{\mathrm{pulse}}\lesssim 1$, since the
electron spin can not change by more than a single full flip.191919Maximizing
the portion of the electron spin transferred to nuclei over one cycle was done
in Ref. [64].

  * 11.

Since the total spin of nuclei is difficult to measure directly, it is useful
to convert the nuclear polarization into quantities directly observable
through the electron. In App. B we express the effects of the DNSP given in
Eq. (47) as the change of the electron Larmor frequency, due to the change of
the Overhauser field,

$\partial_{t}\left(g_{e}\mu_{B}B_{\mathrm{Ov}}\right)=\sum_{i}\phi_{i}A_{i}I_{i}\partial_{t}p_{i},$
(48)

and as the change of the detuning,

$\partial_{t}f_{\Delta}=-\frac{1}{2\pi\hbar}\mathrm{sgn}\left(g_{e}\right)\sum_{i}\phi_{i}|A_{i}|I_{i}\partial_{t}p_{i}.$
(49)

  * 13.

In the far-off-resonance limit, corresponding to $\gamma\to\pm\pi/2$ in our
notation, one of the adopted assumptions is not fulfilled, see Eq. (50)
below.202020The far-off-resonance limit was considered in Ref. [57]. While we
believe that Eq. (47) can still be used for qualitative estimates, it might
break down in certain limits, one example given in App. G.

  * 15.

The micro magnet was essential for several elements: the primary Rabi
oscillations of the electron, the deflection of the quantization axes of the
electron and the nucleus, and the dispersion of the nuclear Larmor frequencies
across the dot. In the next section we argue that there are intrinsic sources
for the latter two, so that they are present in comparable magnitudes in
experiments without a micro magnet. The analysis here then applies also if the
micro magnet, as the source of the Rabi oscillations, is replaced by the
intrinsic spin-orbit interaction. In other words, it applies also for holes,
as long as Eq. (1) is still applicable, see App. I. If it is not, meaning the
spin-orbit length is smaller than the size of the dot, we expect DNSP with
non-trivial spatial textures analogous to those predicted in Ref. [45].

  * 17.

Considering the nuclear spins in isolation, as we have done at the outset of
the derivation of Eq. (47), is quite a cavalier approximation. We believe that
it suffices for what we aim at, being a rough estimation of the DNSP rate.
Another motivation to adopt it is the fact that the full problem—of an
electron spin relaxing into an interacting dipole-dipole coupled nuclear
system—is too difficult: While a formal expression for the rate can be found
in the literature (see Eq. (2.36) in Ref. [65], Eq. (4.1) in Ref. [66], or Eq.
(13) in [67]), its evaluation is not easy, see the discussion in the
introduction of Ref. [65] and in Ref. [66].

After deriving the DNSP rate within our model, we now generalize the resulting
formula to grasp effects important in real-world experiments.

##  V Model limitations and extensions

The above DNSP effects rely on fulfilling the Hartmann-Hahn condition, Eq.
(29). Specifically, the rotating wave approximation that we adopted in Sec.
IV.3 in describing the secondary Rabi oscillations assumes that among the four
energies

$\hbar\omega_{n},\,\hbar\omega_{R},\,\hbar\omega_{n}+\hbar\omega_{R},\,\hbar\omega_{n}-\hbar\omega_{R},$
(50)

the last is by far the smallest. We now look with what precision these
energies (or frequencies) and their differences are defined.

Concerning the nuclear Larmor frequencies, we have considered their smearing
across the quantum dot due to the micro magnet, arriving at a Gaussian density
(47g) with the dispersion (47h). The parameters given in the caption of Fig. 1
give the frequency dispersion of several tens of
$2\pi\times$kHz.212121Specifically, for a longitudinal gradient of 0.3
mT/$\mu$m and the dot lateral size $l=34$ nm we get $\Sigma_{\mathrm{\mu
M}}(^{29}\mathrm{Si})=2\pi\times 43$ kHz, $\Sigma_{\mathrm{\mu
M}}(^{69}\mathrm{Ga})=2\pi\times 52$ kHz, $\Sigma_{\mathrm{\mu
M}}(^{71}\mathrm{Ga})=2\pi\times 66$ kHz, $\Sigma_{\mathrm{\mu
M}}(^{75}\mathrm{Ga})=2\pi\times 37$ kHz. This value should be compared to
additional frequency smearing sources:222222In the NMR literature, the
dispersion of nuclear energies due to nuclear dipole-dipole interactions is
often taken as Gaussian. For example, see Eq. (A20) in Ref. [68]. Therefore,
those numbers are directly comparable to our $\Sigma_{\mathrm{\mu M}}$. On the
one hand, the bulk value for both the intrinsic nuclear linewidth deduced from
the $T_{2}$ times and the local field (dipolar and other) from other nuclei
look negligible.232323Ref. [69] found $\Sigma_{\mathrm{T_{2}}}\lesssim
2\pi\times 1$ kHz. Slightly larger values for 75As and 71Ga in lattice-matched
dots are collected from other references in Ref. [70]. Refs. [71, 72] give the
nuclear local field in GaAs as up to a few Gauss (it is anisotropic),
corresponding to $\Sigma_{\mathrm{dip}}\sim 2\pi\times$ (a few) kHz. On the
other hand, in a nanostructure the inhomogeneous strain and electric fields
amplify line widths: the quadrupole splitting $\Sigma_{\mathrm{Q}}\gtrsim
2\pi\times 10$ kHz [73] or the Knight field from the electron
$\Sigma_{\mathrm{K}}$ of a similar magnitude are typical (these values are for
GaAs).242424For our parameters, we estimate $\Sigma_{\mathrm{K}}\sim J\lesssim
2\pi\times 10$ kHz. More precisely, for quantum-dot parameters $l_{z}=10$ nm
and $l=34$ nm, the electron-frequency shift upon a single nuclear spin flip,
$J_{n}/\hbar$, is equal to $2\pi\times 6$ kHz for 69Ga, $2\pi\times 7.5$ kHz
for 71Ga, $2\pi\times 7$ kHz for 75As; and for $l_{z}=6$ nm and $l=20$ nm, it
is $2\pi\times(-1.7)$ kHz for 29Si. In self-assembled dots, the Knight fields
are much larger, and the single-nuclear-flip electron-frequency shift of 200
kHz could be detected in Ref. [43]. The total frequency span of a given
isotope might crawl to 100 kHz.252525See Fig. 3a in [74] or Fig. 2 in Ref.
[75], showing the line profile of ${}^{75}\mathrm{As}$ at high magnetic
fields. All these sources can be included in our formula by simply adding the
corresponding variances, redefining the parameter $\Sigma$ in Eq. (47g) as
follows

$\Sigma^{2}\to\Sigma_{\mathrm{\mu
M}}^{2}+\Sigma_{\mathrm{T_{2}}}^{2}+\Sigma_{\mathrm{dip}}^{2}+\Sigma_{\mathrm{Q}}^{2}+\Sigma_{\mathrm{K}}^{2}.$
(51)

As an important consequence, one _expects the discussed DNSP effects even in
samples without a micro magnet_ : The longitudinal magnetic field gradient is
effectively replaced by the sources given on the right-hand side of Eq. (51)
without the first term which then equals zero. Similarly, some of these terms
contribute also to the deflection of the quantization axis of nuclear spins,
that is, an effective transverse gradient. The quasi-static dipole field of
other nuclei parameterized by $\Sigma_{\mathrm{T_{2}^{*}}}^{2}$ is isotropic
and can be thus taken as an effective contribution to the gradient
$\nabla_{\perp}B$ in Eq. (47b). The quadrupolar fields also contribute, though
they are anisotropic so that the contributing part depends on the direction of
the magnetic field and the details of the atomic electric field
gradients.262626In experiments with self-assembled quantum dots, the
quadrupolar fields are thought to dominate the DNSP effects [76]. One
important consequence of considering quadrupolar interaction explicitly (which
we do not do here), is that it allows for double spin-flip transitions,
$\Delta I_{z}=\pm 2$, in addition to single-flip ones, $\Delta I_{z}=\pm 1$.
The multiple resonance peaks, corresponding to Raman-transition detuning equal
to once _and twice_ the nuclear Zeeman energy, were observed in Refs. [41, 40,
42]. Finally, the Knight field from the electron is fast oscillating which
averages out its components perpendicular to the external magnetic field. The
remaining component is along the external magnetic field and does not give any
deflection. In sum, for experiments without a micro magnet the effective
transverse gradient entering Eq. (47b) should be assigned a value according to
a conversion formula

$\frac{l\nabla_{\perp}B}{B}\to\frac{\Sigma}{\omega_{i}},$ (52)

with $\Sigma$ somewhat smaller than the one given by Eq. (51).

We now turn to the frequency of the electron as another source of uncertainty
in Eq. (29). Copying the formula here again, the electron Rabi frequency is
$\omega_{R}=\sqrt{(\omega_{RR})^{2}+(\omega_{\mathrm{rf}}-\omega_{e})^{2}}$.
First, during the driving the Overhauser field will diffuse, changing the
electron Larmor frequency $\omega_{e}$. However, for pulses of order $\mu$s,
we find that the resulting shift is smaller than a few $2\pi\times$kHz and
thus negligible for the discussion here.272727For $T_{\mathrm{pulse}}=1$
$\mu$s, we estimate the diffusion-induced variance of the Overhauser field,
$\Sigma_{B}$, of $2\pi\times 8$ kHz from the measurements of Ref. [36],
$2\pi\times 7$ kHz from Ref. [34], or $2\pi\times 6$ kHz from Ref. [77]
(values for GaAs). More importantly, within a finite time interval $T$, no
frequency can be defined with uncertainty much below $\delta\omega\sim
1/T$.282828The numerical prefactor $c$ to use in the relation
$\delta\omega=c\times 1/T$ is not obvious. We define it by demanding
$\int_{-\delta\omega}^{\delta\omega}f(\omega)\mathrm{d}\omega=1/2$, with
$f(\omega)$ being the spectral density. For $f$ equal to a Lorenzian, such as
Eq. (53), one has $\delta\omega=\Sigma$ and thus $c=1$. For $f$ equal to the
left-hand side of Eq. (44) with $t=T$, we get $\delta\omega\approx\pi/2\times
1/T$, a value that we adopt in plots. A Rabi pulse applied for 1 $\mu$s gives
$\delta\omega\sim 2\pi\times 160$ kHz. This smearing should be assigned to the
equality sign in Eq. (29), rather than to any individual frequency, but let us
interpret it as an effective electron lifetime. In general, one considers it
together with the lifetime of Rabi oscillations, or the Rabi decay time
$T_{2}^{\mathrm{Rabi}}$,292929We use the notation of Ref. [3]. The Rabi decay
$T_{2}^{\mathrm{Rabi}}$ is contributed by the decay and decoherence times in
the rotated frame, often denoted by $T_{1\rho}$ and $T_{2\rho}$ [78], the
former introduced by Ref. [62] denoted therein as $T_{2e}$. adding
$(\pi/2)\times 1/T_{\mathrm{pulse}}$ and $1/T_{2}^{\mathrm{Rabi}}$ in square.
Nevertheless, since the latter is negligible in our scenario, we define
$\Sigma_{\mathrm{p}}=(\pi/2)\times 1/T_{\mathrm{pulse}}$.303030Previous works
on DNSP arising from ESR in quantum dots [22, 46, 45] considered that
$1/T_{2}^{\mathrm{Rabi}}$ as just described dominates all other time-decay or
frequency-smearing scales. These works do not even consider the nuclear
hyperfine energy. This approximation was probably motivated by early
experiments [10, 5] where only a few Rabi oscillations were discernible. More
recently, Rabi oscillations of single spins of much higher quality were
achieved: the decay time $T_{2}^{\mathrm{Rabi}}$ was larger than the Rabi
oscillation period by the factor 42.5 in Ref. [36] (GaAs), 70 in Ref. [79]
(natural Si) and 444 in Ref. [80] (isotopically purified Si). In other words,
for current experiments, it might be reasonable to assume
$T_{\mathrm{pulse}}\ll T_{2}^{\mathrm{Rabi}}$. An important difference to the
sources in Eq. (51) discussed in the previous paragraph is that this type of
smearing, essentially originating from the Heisenberg uncertainty relation,
leads to a Lorenzian, rather than a Gaussian, spectral density

$F_{\Sigma}(\omega)=\frac{1}{\pi}\frac{\Sigma}{(\omega-\omega_{R})^{2}+\Sigma^{2}}.$
(53)

This smearing could be included in the main result, Eq. (47a), by replacing
the spectral density in Eq. (47g) by the convolution

$G_{\Sigma}(\omega_{R})\to\left(G_{\Sigma}\star
F_{\Sigma_{\mathrm{p}}}\right)(\omega_{R}).$ (54)

However, we will not use Eq. (54). Since the reasoning which lead to both Eq.
(47g) and Eq. (53) was only qualitative, dwelling on an exact expression in
Eq. (54) is not meaningful. Instead, we simply add the finite-lifetime
smearing $\Sigma_{\mathrm{p}}$ into the list in Eq. (51) and use that as the
width of the spectral density function entering Eq. (47) with either the
Gaussian or the Lorenzian profile.

To complete the list of smearing mechanisms, note that due to the nuclear and
electrical noise, in an experiment the electron detuning frequency varies with
time and thus can be known and controlled only approximately. In experiments
employing estimation and feedback, similar to the one producing the data in
Fig. 1, the resulting uncertainty was $2\pi\times$288 kHz in Ref. [36],
several hundreds of $2\pi\times$kHz in Ref. [77] and several times
$2\pi\times$78 kHz (the frequency bin) in Ref. [34]. This uncertainty is yet
another source of averaging: The experimentally measured polarization rate
corresponds to

$\langle\partial_{t}p_{i}\rangle(\omega_{\Delta})=\int_{-\infty}^{\infty}\mathrm{d}\omega_{\mathrm{err}}\,G_{\Sigma_{\mathrm{err}}}(\omega_{\mathrm{err}})\,\partial_{t}p_{i}(\omega_{\Delta}+\omega_{\mathrm{err}}),$
(55)

where $\Sigma_{\mathrm{err}}$ is the precision with which the detuning angular
frequency can be fixed during the collection of data assigned to a single
point on the curve such as plotted in Fig. 1. This averaging is different from
the previous two, since now it is not only the spectral density that is
smeared, but also the angle $\gamma$ dependency that is averaged. Therefore,
it would suppress the anti-symmetric-in-$\gamma$ parts of the polarization
rates, which can be identified easily by looking at Eqs. (47b)–(47f).

To conclude, there are three different types of averaging that need to be done
with Eq. (47a): a Gaussian and a Lorentzian smearing of the spectral function,
and a Gaussian averaging of the whole formula. Roughly, we replace them by
adding all the smearing sources to $\Sigma$ used in (47g).

With the polarization rate derived and analyzed in detail, we next move to
examining system dynamics in the presence of DNSP pumping.

##  VI Polarization-rate profile, system dynamics, and feedback

In this section, we look at three topics. First, we illustrate the
polarization-rate magnitude expected in a typical quantum dot, and discuss the
rate inversion-symmetry with respect to the zero detuning $f_{\Delta}=0$.
Second, we examine polarization-rate feedback induced by changes in the
detuning aiming at a substantial nuclear polarization. Third, we analyze the
effects of the feedback on suppressing or enhancing detuning fluctuations,
which influence qubit gate fidelities.

###  VI.1 Polarization rate profile

Figure 4: The polarization rate as a function of the electron detuning
frequency for 75As. In the upper panel $f_{RR}=f_{i}/2$, in the middle panel
$f_{RR}=f_{i}$, and in the lower panel $f_{RR}=2f_{i}$. Further parameters are
as in Fig. 1 except for $T_{\mathrm{pulse}}=10$ $\mu$s.

We illustrate the behavior of the polarization rate derived in Eq. (47) by
plotting it for the arsenic isotope as a function of the detuning in Fig. 4.
Analogous plots for other nuclei of GaAs, and for a Si dot where the rate is
orders of magnitude smaller, are in App. F. Figure 4a shows the rate in the
regime where the electron Rabi frequency at resonance is smaller than the
nuclear Larmor frequency. The rate has a resonant peak at a finite detuning,
where the electron Rabi and nuclear Larmor frequencies become equal. At this
resonance the rate can reach large values, depending on the resonance width,
which has been discussed in Sec. V. The deflection mechanism corresponds to a
rate with a definite left-right symmetry in the figure, symmetric for a lock-
in initial state and antisymmetric for an initial state along the magnetic
field.313131In other words, the EDSR-scenario curves cross zero at zero
detuning, due to the factor $p_{e}=\pm\sin\gamma$. Polarization rates with
this profile are called ’cooling functions’ in Refs. [40, 42]. In those
experiments, the polarization is explained as due to asymmetry in the density
of final states [59, 81]. The shaking mechanism corresponds to a strongly
asymmetric rate, with appreciable values at negative detunings only.

Figure 4c shows the case with the Rabi frequency at zero detuning larger than
the nuclear Larmor frequency. In this case, the condition of the Hartmann-Hahn
resonance can not be reached for any detuning. While the symmetry properties
of the rate components discussed in the previous paragraph still hold, there
is no resonance peak and the rates are much smaller overall. This difference,
between the resonant and non-resonant regime, is the larger the larger is the
ratio $f_{RR}/f_{i}$. Finally, at large detuning the rates fall off as
$1/f_{\Delta}^{2}$, which is the same in the upper panel, though hard to see
there because of the resonant peak.

Figure 4b shows the crossover case $f_{RR}=f_{i}$. Here, the two resonance
peaks visible in the upper panel merge into one. Compared to those two
resonances, the merged peak is broader and (for spin locking) has a somewhat
anomalous shape (it has a flat top). This property can be understood by noting
that the derivative $\partial f_{R}/\partial f_{\Delta}$ becomes zero at zero
detuning.

###  VI.2 Feedback

Equation (49) hints at feedback effects: The detuning frequency $f_{\Delta}$
changes if the nuclear polarization changes, since the electron feels it as
the Overhauser field. However, the polarization rates themselves strongly
depend on the detuning. Such mutual dependence of the nuclear polarization
rate and its effects, the accumulated nuclear polarization, has been studied
at length (see the second paragraph of the introduction and the references
therein).

Motivated by those works, we now look at the feedback effects in our system.
We start by pointing out one crucial difference. Here, the DNSP polarization
is a resonance phenomenon, so that the dependence of the polarization rate on
the electron detuning might become (close to resonance) much more sensitive
than the dependence in the Pauli spin blockade setups [24]. While this fact
will make building up large polarizations more difficult, it might allow for
more efficient Overhauser field stabilization and the associated dephasing
suppression.

To appreciate this sensitivity, we copy here Eq. (77) derived in App. B [it
also follows from Eq. (49)]

$\frac{\partial f_{\Delta}}{\partial
p_{i}}=-\frac{\mathrm{sgn}(g_{e})}{2\pi\hbar}\phi_{i}|A_{i}|I_{i}.$ (56)

This equation relates changes in the nuclear polarization $p_{i}$ to changes
in the electron detuning _at a fixed value of the driving frequency_.
Evaluating the constants on the right-hand side, we get 25 MHz in natural
silicon (it would be sixty times less in isotopically-purified 800 ppm
silicon), and from about 7 to 17 GHz for the three isotopes in GaAs.
Therefore, especially in the latter material, a tiny change in the nuclear
polarization—say a few of 0.01%—can bring the system into and out of the
resonance, turning on and off the DNSP rate.

Figure 5: System dynamics under DNSP and decay. The parameters are the same as
in Fig. 4 except for $f_{RR}=5$ MHz. (a) The two axes give the system state
coordinates: the electron detuning on the horizontal axis and the nuclear
polarization on the vertical axis. The vectors show the direction and rate at
which the system moves from a given configuration. The colored arrows depict
the DNSP rate for $i=^{75}$As, the black arrows denote the spin decay into
$p_{i}=0$. At a fixed driving frequency the system can move along a line since
all vectors throughout the plot are parallel. The position of the line is
fixed by the detuning at zero nuclear polarization (empty circle), here fixed
to $-7$ MHz. (b) The polarization [’pump’; the right-hand side of Eq. (47a)]
and decay [$Rp_{i}$, the right-hand side of Eq. (57)] rates at fixed driving
frequency. The equilibrium is where the two rates are equal, denoted by the
filled circle.
In (a) the arrows are scaled for visibility: While a larger arrow means a
larger rate, the proportionality is not linear for a given color and not to
scale between different colors. The arrows’ map is only illustrative. The rate
magnitudes are quantitative in (b).

To shed light on the possible system dynamics, we plot the DNSP rates in Fig.
5 in a two-dimensional plot. We assume the ’spin locking’ scenario, see Eq.
(47f), where the rates are somewhat larger than for the ’EDSR’ choice.323232A
feedback exploiting the EDSR scenario was implemented in Ref. [44]. The
horizontal axis is the detuning, the vertical the nuclear polarization. The
colored arrows show the polarization rate: the arrow length scales with the
rate magnitude and the arrow direction shows which way the system evolves at a
fixed driving frequency. The black arrows represent nuclear spin-polarization
decay, due to diffusion or other means, according to

$\partial_{t}p_{i}=-Rp_{i}.$ (57)

The decay constant $R$ depends on the material nuclear spin diffusion
constant, the dot geometry, and possibly on the isotope. Since these
dependencies might be complicated,333333For example, Ref. [82] converts the
observed Overhauser field dynamics into the effective material diffusion
constant and finds that its value changes strongly with the magnetic field. it
is more practical to extract the decay scale $R$ from experimental data rather
than to calculate it from first principles. Typical decay times of nuclear
polarization in dots is from seconds (see Ref. [82] or the estimates of
parameter $\kappa$ in App. H) to minutes (see Fig. 3 in Ref. [36] or Fig. 3e
in Ref. [83]).

One can understand the system behavior from Fig. 5a. As a simple example, it
shows the rate for 75As isotope in a GaAs dot with the driving frequency fixed
to a certain value corresponding to the detuning $-7$ MHz at zero nuclear
polarization. This state is denoted by the empty circle in Fig. 5. With the
driving frequency fixed, the system can move only along the blue line. It will
reach a steady state at a finite positive polarization $p=p_{eq}$ where the
polarization and decay rates are equal (they are shown in Fig. 5b). The system
will stay at such finite polarization as long as random (thermal) fluctuations
do not take it out of the window where the DNSP rate is sizable. The larger
the value of the equilibrium polarization $p_{eq}$, the stronger the forces on
the system at the equilibrium and the smaller the fluctuations around the
steady state [22, 50].343434The decrease of fluctuations when the forces
become larger can be also understood from the model in App. H: Eq. (116)
states that the fluctuations $\sigma_{\Omega}^{2}$ are proportional to the
inverse of the decay rate $2/\kappa$. On the other hand, also the more
volatile the steady state becomes and the more easily it can be kicked off by
thermal fluctuation into $p=0$. In other words, if $p_{eq}$ is large enough,
the system will be bistable. What is large enough is decided by the width of
the resonance, in turn given also by the inverse of the micro-magnet gradient
$B_{||}$ and additional sources according to the discussion around Eq. (51).

One can consider more complicated evolutions when the driving frequency is
changed. A change in the driving frequency translates into a horizontal shift
of the blue line. When the system state is represented by the filled circle in
the figure, a sudden change of the driving will move it together with the blue
line horizontally, that is, keeping the current value of the polarization. In
changes that are more adiabatic, the system state will tend to follow the
local equilibrium position on the blue line. A simple scenario would be a slow
increase of the rf-frequency, starting at a negative detuning
$f_{\Delta}=f_{\mathrm{rf}}-f_{e}$. The polarization would steadily increase
until the equilibrium polarization would become too large to be sustained. The
required speed of change of the driving frequency can be read off from Fig.
5b, or directly from a plot like Fig. 4: the optimal speed to built a large
polarization is a value somewhat smaller than the polarization rate at the
peak, which is a few hundreds of MHz/s for these parameters.

###  VI.3 Restoring force

To elaborate on the previous section, we next consider the electron spin
coherently driven by EDSR with the goal of performing a qubit gate. One
typical situation is that the electron is driven at zero detuning and starts
polarized along the external field. It differs from the previous by having now
$p_{e}=\pm\sin\gamma$. We are interested in how the arising DNSP polarization
affects gate precision. Specifically, we analyze the DNSP influence on the
stability of the desired condition $f_{\Delta}=0$. Using Eqs. (47) and (49),
we get

$\begin{split}\partial_{t}f_{\Delta}^{*}&=\mathrm{sgn}(p_{e}^{*})\mathrm{sgn}(g_{e})\frac{f_{\Delta}^{*}}{f_{RR}}\left(X_{\mathrm{df}}^{*2}+X_{\mathrm{sh}}^{*2}\right)\frac{1}{6\hbar^{3}}\\\
&\qquad\times\sum_{i}\phi_{i}|A_{i}|I_{i}(I_{i}+1)G_{\Sigma}\left(\omega_{RR}-\omega_{i}\right).\end{split}$
(58)

To arrive at these formulas, we have used Eq. (37a), expanded Eq. (47a) in the
limit around $\gamma\approx 0$, and, to simplify, dropped the polarization
$p_{i}$ from the right-hand side and used $\alpha_{I}$ for the small $p_{n}$
limit. The star as the superscript denotes a relation to the limit $\gamma\to
0$. Specifically, for the matrix elements $X_{\mathrm{sh}}$ and
$X_{\mathrm{df}}$ the star means that they are evaluated using Eqs. (47b) and
(47c) with $\gamma=0$. Also, we have added the initial state specification as
$\mathrm{sgn}(p_{e}^{*})$ with the value $+1$ for the ground state and $-1$
for the excited state. Finally, we also note that except of $g_{e}$,
$p_{e}^{*}$, and $f_{\Delta}$, quantities in the expression are positive.

Equation (58) describes a simple feedback, since the rate of change of the
detuning is proportional to the detuning value. Whether the feedback is
negative (fluctuations suppressed) or positive (fluctuations amplified) is
decided by the overall sign, the product of signs of the electron $g$ factor
$g_{e}$ and the initial state $p_{e}^{*}$. This latter product can be
contracted to ’electron spin initially along $\mathbf{B}_{e}$’ being the sign
-1 (negative feedback) and ’electron spin initially opposite to
$\mathbf{B}_{e}$’ being the sign +1 (positive feedback).353535The fact that
the feedback switches from positive (‘resonance seeking’) to negative
(‘resonance avoiding ) upon inverting the electron spin was pointed out in
Ref. [84]. Let us first discuss the first alternative.

A negative feedback means that driving the electron spin _stabilizes_ the
desired condition $f_{\Delta}=0$. To quantify this effect, we write Eq. (58)
in the form

$\partial_{t}f_{\Delta}^{*}=-\Gamma^{*}f_{\Delta}^{*},$ (59)

introducing $\Gamma^{*}$ as the feedback strength with the units of inverse
time. To assess how efficient the stabilization is, we judge it against the
intrinsic thermal fluctuations of the nuclei. However, the comparison is not
straightforward, since these thermal fluctuations proceed as a diffusion of
the Overhauser field, characterized by a diffusion constant, which is not a
rate. To bridge this gap, in App. H we describe this diffusion by a bounded
random walk model, which contains two parameters: the diffusion constant
$D_{\Omega}$ and a time $\kappa$ related to the restoring force that keeps the
Overhauser-field fluctuations bounded.

The behavior of the system can be then understood as follows: Let us assume
that the detuning is set to the desired value $f_{\Delta}=0$. At this value,
the polarization rate is zero. The detuning will diffuse away from the desired
condition according to the diffusion constant $D_{\Omega}$. This short-time
diffusion speed is not affected by the DNSP and the feedback. Without any
feedback, the Overhauser field will reach the long-time variance
$\sigma_{\Omega}^{2}=D_{\Omega}\kappa/2$. In App. H we show that the restoring
force in this process can be represented in a form identical to Eq. (59) upon
identifying the constant $\Gamma$ with $1/\kappa$. Thus, one can assign an
intrinsic restoring force $\Gamma_{0}\equiv 1/\kappa_{\mathrm{th}}$ to the
thermal diffusion. The DNSP feedback increases the restoring force by adding
$\Gamma^{*}\equiv 1/\kappa_{\mathrm{fb}}$ to the intrinsic component. The
fluctuations are then described by the variance

$\sigma_{\Omega}^{*2}=D_{\Omega}\left(\frac{1}{\kappa_{\mathrm{th}}}+\frac{1}{\kappa_{\mathrm{fb}}}\right)^{-1}.$
(60)

To quantify the efficiency, one should compare the intrinsic rate $\Gamma_{0}$
to the one due to feedback $\Gamma^{*}$. If $\Gamma^{*}\ll\Gamma_{0}$, the
feedback is negligible. If $\Gamma^{*}\gg\Gamma_{0}$, the feedback
substantially decreases the magnitude of the fluctuations, cutting the
resulting variance by factor $\Gamma^{*}/\Gamma_{0}$.

Figure 6: Stabilization by feedback in GaAs. The EDSR-driven electron starts
in its ground state and precesses around an in-plane axis of the Bloch sphere,
meaning $f_{\Delta}=0$. The plot shows the restoring rate $\Gamma^{*}$ as a
function of the Rabi frequency at resonance for the three isotopes. We took
$T_{\mathrm{burst}}=T_{\mathrm{cycle}}=1$ $\mu$s, zero additional broadening,
and the remaining parameters as in Fig. 1. The value in the box is the
expected range for the intrinsic restoring force
$\Gamma_{0}=\kappa_{\mathrm{th}}^{-1}$. The analogous figure for an electron
qubit in Si is in App. F.2 and for a hole qubit in SiGe in App. I.

We plot the quantity $\Gamma^{*}$ in Fig. 6. It shows that the DNSP-induced
feedback might be indeed substantial, with $\Gamma^{*}$ larger than
$\Gamma_{0}$ by up to two or three orders of magnitude for our parameters at
the resonance with the arsenic isotope. The effect on the gate fidelity is
more complicated, since the feedback depends on the initial state. While for
some input states the feedback improves the fidelity by stabilizing the
detuning, the effects are opposite for other input states. Concerning gate
fidelities, it seems advisable to keep the system away from the Hartmann-Hahn
resonance.

###  VI.4 Closing remarks

We note that the simplified picture presented in the above by discussing a
single isotope is complicated in GaAs by having three different isotopes with
different resonance frequencies. Still, the DNSP rates depend, through the
actual value of the detuning, only on the sum of the corresponding Overhauser
fields. The simplest-looking scheme to stabilize the total Overhauser field is
to use the Hartmann-Hahn resonance of a single isotope with the most efficient
polarization, being 75As in our estimates.

Let us also reiterate a point crucial for both observing and exploiting the
DNSP rates discussed in this article. As already stressed several times, these
rates are resonance phenomena, sensitive to the electron detuning, in turn to
its Larmor frequency. A change in the frequency by a few MHz can substantially
change the polarization rates. Therefore, adjusting the driving frequency to
the instantaneous value of the electron Larmor frequency is essential. It can
be possibly done by periodic estimation of this frequency [34, 36]. Another
possibility is to use chirps of the driving frequency[85, 64].

Let us conclude by saying that there is room for more investigations of
feedback effects based on Hartmann-Hahn DNSP in gated quantum dots.

##  VII Conclusions

In this article, we have investigated dynamical nuclear spin polarization
arising in a quantum dot with a single electron whose spin is electrically
driven to perform Rabi oscillations. We considered the coherent regime where
many Rabi oscillations happen before the electron leaves the dot or its spin
decoheres. In this regime, the electron spin can polarize nuclear spins in the
quantum dot volume through an analog to the Hartmann-Hahn effect known from
NMR [47]. This is a resonance phenomenon, occurring when the electron Rabi
frequency becomes equal to the nuclear Larmor frequency.

We have derived the corresponding nuclear-spin polarization rate under general
conditions in Sec. IV so that the main result, Eq. (47), covers both GaAs and
Si dots, and, with slight adjustments,363636We give the main result, Eq. (47),
assuming isotropic hyperfine tensor, $\propto\mathbf{s}\cdot\mathbf{I}$. It
remains unchanged for ‘secular’ hyperfine tensor, $s_{z}I_{z}$. If additional,
‘non-secular’, terms are present, often the case for holes, they will generate
additional terms in Eq. (94), which need to be reflected in Eq. (47) using
Tab. 2. Also, since micromagnets are not needed for hole spin qubits [86, 87,
88], the micromagnet-related parameters entering Eq. (47) need to be
reinterpreted as discussed in Sec. V, see especially Eqs. (51) and (52). even
Ge or Si hole dots.

When converted to changes in the electron detuning from the Rabi resonance,
the nuclear-spin polarization rates in GaAs reach tens to hundreds of MHz/s.
The theory fares well with a preliminary measurement in a GaAs sample
presented in Fig. 1. In Si, the rates are orders of magnitude smaller. While
we do not present data for Si, our theory predicts rates of order tens of
kHz/s.

We have identified two essential differences to the standard Hartmann-Hahn
scenario: 1) The micro-magnet magnetic-field gradient slightly deflects the
spin-quantization axes of the electron and nuclei. 2) The electric driving
slightly wiggles the electron with respect to the atomic lattice. These two
effects correspond to two different mechanisms of polarization. In Sec. V, we
have reasoned that the polarization will be present even in samples without a
micromagnet, and we have provided estimates with which a polarization rate can
be assigned to this scenario.

Finally, we have analyzed the feedback in the system. It stems from the fact
that the polarization rate is sensitive to the electron detuning from the Rabi
resonance, which in turn is sensitive to the accumulated nuclear polarization
through the Overhauser field. In Sec. VI, we have looked at the possibility of
reaching a sizable nuclear polarization and the consequences of the Hartmann-
Hahn resonance on the fidelity of a gate implemented as coherent Rabi
precession. Concerning the first, the achievable nuclear polarization is
ultimately set by how sharp the resonance can be made, in turn dependent on
the electron spin coherence and the micro-magnet gradient. If used as active
feedback, we estimate that exploiting the resonance can decrease the
fluctuations of the Overhauser field by two or more orders of magnitude (in
GaAs). Concerning the gate fidelities, we have found that it is improved for
some input states and worsened for others. We do not evaluate the fidelities,
and remain at the advice of avoiding the resonance when implementing quantum
gates on an electron or hole spin qubit.

###### Acknowledgements.

PS would like to thank Minoru Kawamura for a useful discussion. We thank for
the financial support from CREST JST grant No. JPMJCR1675, JST Moonshot R&D
grant No. JPMJMS226B-1, JST PRESTO grant No. JPMJPR2017, JSPS KAKENHI grant
No. 18H01819 and from the Swiss National Science Foundation and NCCR SPIN
grant No. 51NF40-180604.

##  Appendix A Density of nuclear Larmor frequencies

Figure 7: Cumulative distribution of the effective number of nuclei. The
horizontal axis shows both the real space coordinate $x$, here along the
gradient of the longitudinal component of the magnetic field, and the nuclear
Larmor frequency. The frequency increases monotonically with $x$ and in the
dot center it equals $\omega_{i}$.

In this section, we introduce the distribution $g$ used in Eq. (45) to replace
the summation over discrete nuclei. The quantity being summed contains a
factor $v_{0}^{2}|\Psi_{n}|^{4}$, arising from the square of the right-hand
side of Eq. (4). Here, we have denoted
$\Psi_{n}\equiv\Psi(\mathbf{r}_{n},z_{n})$. Therefore, let us consider

$\sum_{n\in\mathrm{set}}v_{0}^{2}|\Psi_{n}|^{4}\times\mathrm{constants},$ (61)

where the ‘set’ defines which nuclei are included. In our case, it is nuclei
of isotope $i$ with a given Larmor frequency. Also, ‘constants’ are further
terms that can depend on the isotope, but not on the nucleus spatial position.
Since such constants only propagate through all the formulas below, we omit
them. We rewrite the sum as

$\frac{1}{N_{\mathrm{tot}}^{2}}\sum_{n\in\mathrm{set}}N_{\mathrm{tot}}^{2}v_{0}^{2}|\Psi_{n}|^{4}.$
(62)

taking out a dimensionless factor $N_{\mathrm{tot}}^{-2}$. In line with the
existing literature, we move this factor into the matrix elements $X$, see
Eqs. (47b)-(47c). These matrix elements then contain the ‘average’ hyperfine
strength $A_{i}/N_{\mathrm{tot}}$. The dividing factor, written suggestively
as $N_{\mathrm{tot}}$, is interpreted as the total (counting all isotopes)
effective number of atomic nuclei within the dot, defining it by

$N_{\mathrm{tot}}=\frac{1}{v_{0}\int\mathrm{d}\mathbf{r}\,\mathrm{d}z|\Psi(\mathbf{r},z)|^{4}}.$
(63)

With this rescaling, the sum that we are interested in is

$\sum_{n\in\mathrm{set}}N_{\mathrm{tot}}^{2}v_{0}^{2}|\Psi_{n}|^{4}.$ (64)

Since the nuclear Larmor frequency is a smooth function of the nuclear
position, we replace the discrete summation by integration in space with the
three-dimensional volume element $\mathrm{d}V$. As we only include the isotope
$i$, the volume density of nuclei is $\phi_{i}/v_{0}$. We get

$\int_{D}\mathrm{d}V\frac{\phi_{i}}{v_{0}}N_{\mathrm{tot}}^{2}v_{0}^{2}|\Psi|^{4},$
(65)

where the restriction $n\in\mathrm{set}$ has been expressed as a volume $D$.
We define $N_{i}\equiv\phi_{i}N_{\mathrm{tot}}$ as the effective number of
isotope-$i$ nuclei, and use Eq. (63) to finally get

$\mathrm{d}N_{i}=N_{i}\frac{\mathrm{d}V|\Psi|^{4}}{\int\mathrm{d}V|\Psi|^{4}},$
(66)

as the effective number of nuclei of isotope $i$ within a volume element
$\mathrm{d}V$. In this expression, the denominator normalizes the density
$\mathrm{d}VN_{i}|\Psi|^{4}$ into a dimensionless quantity. If integrated over
all space, it gives the effective number of isotope-$i$ nuclei in the dot.

We now consider the desired restriction on the nuclei included in the sum,
being a given value of their Larmor frequency. The latter is proportional to
the magnitude of the magnetic field at the position of the corresponding
nucleus, $B_{n}$. In the lowest order of the magnetic-field gradients and
neglecting the shifts along the $z$ coordinate, this magnitude varies linearly
over the dot in-plane coordinates,

$B(x_{n})\approx B(x_{0})+(x_{n}-x_{0})\nabla_{||}B,$ (67)

where we choose the in-plane coordinates such that $x$ is along the gradient
of the magnetic field longitudinal component (the component along the
direction of the magnetic field at the dot center; see also App. D) and
$\nabla_{||}B$ is the magnitude of this gradient.

The restriction on the nuclear Larmor frequency is then a restriction on the
$x$-coordinate, and we can integrate out the remaining two coordinates $y$ and
$z$,

$\mathrm{d}N_{i}=N_{i}\frac{\sqrt{2}}{l\sqrt{\pi}}\exp\left(-\frac{2(x-x_{0})^{2}}{l^{2}}\right)\mathrm{d}x,$
(68)

where we have used the Gaussian form for the in-plane wave function, Eq. (1).

The desired density can be now obtained from the cumulative distribution (see
Fig. 7 for an illustration)

$\begin{split}\int_{-\infty}^{\omega}g(\omega)\mathrm{d}\omega&=\int_{-\infty}^{x(\omega)}\frac{\mathrm{d}N_{i}}{N_{i}}\\\
&=\int_{-\infty}^{x(\omega)}\frac{\sqrt{2}}{l\sqrt{\pi}}\exp\left(-\frac{2(x^{\prime}-x_{0})^{2}}{l^{2}}\right)\mathrm{d}x^{\prime},\end{split}$
(69)

where $x(\omega)$ is the coordinate at which the Larmor frequency is $\omega$.
It can be obtained from the relation

$\frac{\omega-\omega_{i}}{\omega_{i}}=\Big{(}x(\omega)-x(\omega_{i})\Big{)}\frac{\nabla_{||}B}{B},$
(70)

where $\omega_{i}$ is the (isotope-dependent) nuclear Larmor frequency at the
dot center, $x(\omega_{i})=x_{0}$.

Differentiating Eq. (69) with respect to $\omega$, and using Eq. (70), we get

$g_{i}(\omega)=\frac{1}{\omega_{i}}\frac{B}{l\nabla_{||}B}\frac{\sqrt{2}}{\sqrt{\pi}}\exp\left[-2\left(\frac{\omega-\omega_{i}}{\omega_{i}}\frac{B}{l\nabla_{||}B}\right)^{2}\right].$
(71)

The first term sets the scale, the rest is a dimensionless peak profile
centered at $\omega=\omega_{i}$. It encodes the resonance character of the
problem: since the nuclear $g$ factors differ for different isotopes, they
become resonant at different values of the electron Rabi frequency
$\omega_{R}$. The width of resonance is a fraction of the nuclear Larmor
frequency proportional to the gradient of the magnetic-field longitudinal
component.

##  Appendix B DNSP rate expressed as the detuning change

Here, we give the electron Larmor frequency including the contribution from
the nuclear polarization. While the formulas might look too straightforward
even for an appendix, they might be useful when considering materials with
different signs of the $g$ factors.

The electron spin couples to the external magnetic field and the effective
field arising from polarized nuclei, called also the Overhauser field,

$H_{e}^{Z}=g_{e}\mu_{B}\mathbf{B}\cdot\mathbf{s}+\sum_{i}\phi_{i}A_{i}I_{i}p_{i}\mathbf{z}_{n}\cdot\mathbf{s}.$
(72)

We used the definition of the polarization $p_{i}$ to be along the unit vector
$\mathbf{z}_{n}=\mathrm{sgn}(g_{n})\mathbf{B}/B$. With this, and using also
$A_{i}=\mathrm{sgn}(g_{i})|A_{i}|$, we can write the above Hamiltonian as

$H_{e}^{Z}=\mathrm{sgn}(g_{e})|g_{e}|\mu_{B}\mathbf{B}\cdot\mathbf{s}\left(1+\sum_{i}\phi_{i}|A_{i}|I_{i}p_{i}\frac{\mathrm{sgn}(g_{e})}{|g_{e}|\mu_{B}B}\right).$
(73)

The electron Larmor frequency is the magnitude of the vector multiplying the
electron spin operator,

$\hbar\omega_{e}=|g_{e}|\mu_{B}B\left|1+\sum_{i}\phi_{i}|A_{i}|I_{i}p_{i}\frac{\mathrm{sgn}(g_{e})}{|g_{e}|\mu_{B}B}\right|.$
(74)

Most often, the nuclear polarization is not so large as to make the Overhauser
field bigger than the external field. Then, the second term inside the
absolute value is smaller in magnitude than the first, making their sum
positive and the absolute value sign unnecessary,

$\hbar\omega_{e}=|g_{e}|\mu_{B}B+\mathrm{sgn}(g_{e})\sum_{i}\phi_{i}|A_{i}|I_{i}p_{i}.$
(75)

We can now covert the DNSP polarization rate into the rate of change of the
electron Larmor frequency and the detuning,

$\partial_{t}f_{\Delta}=-\partial_{t}f_{e}=-\frac{\mathrm{sgn}(g_{e})}{2\pi\hbar}\sum_{i}\phi_{i}|A_{i}|I_{i}\partial_{t}p_{i},$
(76)

the first equation following from our definition
$f_{\Delta}=f_{\mathrm{rf}}-f_{e}$. Finally, we note the relation between the
change of the electron detuning with respect to the change in the nuclear
polarization,

$\frac{\partial f_{\Delta}}{\partial
p_{i}}=-\frac{\mathrm{sgn}(g_{e})}{2\pi\hbar}\phi_{i}|A_{i}|I_{i}.$ (77)

It becomes useful when considering possible feedback in the system.

##  Appendix C Derivation of Eq. (36)

Figure 8: Spin polarization transitions. The diagram shows the states of a
system composed of an electron spin and a nuclear spin, the latter illustrated
for $I=3/2$. The electron spin can be either up or down, with corresponding
probabilities $p^{e}_{\uparrow}$ and $p^{e}_{\downarrow}$. The nuclear spin
can be in one of the four states, with corresponding probabilities
$p^{n}_{j}$. The matrix element $\mathcal{I}_{+}$ connects states as denoted
by the blue lines. A transition increasing the nuclear polarization
corresponds to going along one of the blue lines upwards, from left to right.

Here, we derive Eq. (36). The line over the left-hand side of that equation
denotes the average over the probability distributions, or density matrices,
of the electron and the nuclear spin. Figure 8 helps to visualize the
transitions, and shows why the matrix element $\mathcal{I}_{+}$ should be
averaged together with (and not independently to) the polarization $p_{z}$.

To perform the calculation, one needs to quantify the probabilities of the
basis states $|sj\rangle$. As explained in the main text, we consider them
separable into the corresponding probabilities for the electron spin $s$ and
the nuclear spin $j$, $p_{sj}=p^{e}_{s}p^{n}_{j}$. With the electron spin
having only two states available, their probabilities can be expressed through
a single number, let us denote it by $p_{e}$, because of the normalization
$p^{e}_{\uparrow}+p^{e}_{\downarrow}=1$. Namely,

$\displaystyle p^{e}_{\uparrow}=\frac{1}{2}\left(1+p_{e}\right),$ (78a)
$\displaystyle p^{e}_{\downarrow}=\frac{1}{2}\left(1-p_{e}\right).$ (78b)

These relations then define $p_{e}$ as the initial electron-spin-polarization
along the axis ${\mathbf{o}_{e}}$, and lead to Eq. (47f).

On the other hand, there might be more than two nuclear spin states in
general. Still, we define the nuclear polarization by

$p_{n}\equiv\frac{\langle
I_{z}\rangle}{I}=\frac{1}{I}\sum_{j=-I}^{I}jp^{n}_{j}.$ (79)

This single number, together with the normalization $\sum_{j}p^{n}_{j}=1$, is
not enough to specify the probabilities $p_{j}^{n}$ uniquely.373737For
example, the occupations of the four sublevels of spin 3/2 got far from
thermal distribution under the feedback employed in Ref. [42]. Nevertheless,
starting with

$\begin{split}\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}=\sum_{j}&p_{\uparrow
j}|\langle\downarrow,j+1|s_{-}I_{+}|j\uparrow\rangle|^{2}\\\ &-p_{\downarrow
j}|\langle\uparrow,j-1|s_{+}I_{-}|j\downarrow\rangle|^{2},\end{split}$ (80)

it is a few-line algebra to get

$\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}=-\langle
I_{z}\rangle+p_{e}\left[I(I+1)-\langle I_{z}^{2}\rangle\right].$ (81)

Eq. (80) expresses the rate (proportionality factor) for building the nuclear
polarization as the difference of the rates for transitions increasing the
value of spin $j$ and the rate decreasing it, see Fig. 8. Using the definition
of $p_{n}$ we can then write

$\displaystyle\overline{p_{z}(0)|\mathcal{I}_{+}|^{2}}$
$\displaystyle=I\times\left(p_{e}\alpha_{I}(p_{n})-p_{n}\right),$ (82a)
$\displaystyle\alpha_{I}(p_{n})$ $\displaystyle\equiv(I+1)-\langle
I_{z}^{2}\rangle/I.$ (82b)

For small nuclear polarization, one has $\langle
I_{z}^{2}\rangle=I(I+1)/3+O(p_{n}^{2})$. For the opposite limit, $p_{n}\to 1$,
we got $\langle I_{z}^{2}\rangle=I^{2}-O[(1-p_{n})]$. Therefore,

$\displaystyle\alpha_{I}$ $\displaystyle=\frac{2}{3}(I+1),\quad$
$\displaystyle\mathrm{for}\,\,p_{n}\to 0,$ (83a) $\displaystyle\alpha_{I}$
$\displaystyle=1,\quad$ $\displaystyle\mathrm{for}\,\,p_{n}\to 1.$ (83b)

At intermediate polarization, $\alpha$ will be somewhere between these two
limiting values. For spin one-half the two limiting values are the same and
$\alpha=1$ for any $p_{n}$.

##  Appendix D Derivation of Eq. (40)

Our goal is to calculate the transition matrix element

$Y=\langle
s^{\prime},j^{\prime}|\delta\mathbf{I}\cdot{J_{n}^{\prime}}^{(-2s)}\cdot\mathbf{s}|s,j\rangle.$
(84)

Here, the spin indexes are related by Eq. (27), the corresponding quantization
axes are ${\mathbf{o}_{e}}$ and $\mathbf{z}_{n}$, see Eq. (28), and the time-
dependent tensor ${J_{n}^{\prime}}$ is defined in Eqs. (17), (19), and (39).
Using these relations, and choosing $s=1/2$, we write the matrix element in a
more concrete form

$Y=\langle\downarrow,j+1|\delta\mathbf{I}\cdot{J_{n}^{\prime}}^{(-1)}\cdot\mathbf{s}|\uparrow,j\rangle.$
(85)

We now do two straightforward transformations. First, we express the spin-
operator vectors in coordinates aligned with their quantization axes. For
example, the electron spin $\uparrow$ is an eigenstate of the operator

$\sigma_{{\mathbf{o}_{e}}}\equiv\boldsymbol{\sigma}\cdot{\mathbf{o}_{e}}=\boldsymbol{\sigma}\cdot
R_{\mathbf{z}_{e}\to{\mathbf{o}_{e}}}[\mathbf{z}_{e}]=\left(R^{-1}_{\mathbf{z}_{e}\to{\mathbf{o}_{e}}}[\boldsymbol{\sigma}]\right)\cdot\mathbf{z}_{e},$
(86)

where we denote $R_{\mathbf{n}\to\mathbf{m}}$ a rotation operator taking unit
vector $\mathbf{n}$ to unit vector $\mathbf{m}$. Second, we introduce ‘ladder’
operators for spins; for example,

$\left(\begin{tabular}[]{c}$\sigma_{x}$\\\ $\sigma_{y}$\\\
$\sigma_{z}$\end{tabular}\right)=\left(\begin{tabular}[]{ccc}$1/2$&$1/2$&0\\\
$-\mathrm{i}/2$&$\mathrm{i}/2$&0\\\
0&0&1\end{tabular}\right)\cdot\left(\begin{tabular}[]{c}$\sigma_{+}$\\\
$\sigma_{-}$\\\ $\sigma_{z}$\end{tabular}\right).$ (87)

With these, the operator of interest can be written

$\delta\mathbf{I}\cdot{J_{n}^{\prime}}^{(-1)}\cdot\mathbf{s}=\delta\mathbf{I}_{\mathbf{z}_{n},L}\cdot\left(L^{\mathrm{T}}R^{\mathrm{T}}_{\mathbf{z}_{e}\to\mathbf{z}_{n}}{J_{n}^{\prime}}^{(-1)}R_{\mathbf{z}_{e}\to{\mathbf{o}_{e}}}L\right)\cdot\mathbf{\sigma}_{{\mathbf{o}_{e}},L},$
(88)

where $L$ is the three by three matrix in Eq. (87) and the subscript
$(\mathbf{n},L)$ on the spin-operator vector states that the vector components
are in the ladder operators basis in the coordinate frame with its third axis
along $\mathbf{n}$. The advantage of such transformation is that in this basis
we can treat the spin quantum numbers $s,j$ as representing the ‘usual’ basis
with the spin quantization axis along ‘z’. Also, as the only possibly nonzero
component of the polarization $\langle\mathbf{I}\rangle$ is z, we can drop the
polarization, $\delta\mathbf{I}\to\mathbf{I}$, and get the matrix element in
Eq. (85) as

$\begin{split}Y&=\langle\downarrow,j+1|I_{+}[J_{n}(t)M(t)]^{(-1)}_{+-}s_{-}|\uparrow,j\rangle\equiv
X\mathcal{I}_{+},\end{split}$ (89)

where we used Eq. (19) and Eq. (15) with the time dependence according to Eq.
(11), to express the element through the following short hands,

$\displaystyle\mathcal{I}_{+}$
$\displaystyle=\langle\downarrow,j+1|I_{+}s_{-}|\uparrow,j\rangle,$ (90)
$\displaystyle X$ $\displaystyle=[J_{n}(t)M(t)]^{(-1)}_{+-},$ (91)
$\displaystyle J_{n}(t)$
$\displaystyle=J_{n}\left(1-2\frac{\mathbf{d}(t)\cdot(\mathbf{r}-\mathbf{r}_{0})}{l^{2}}\cos(\omega_{\mathrm{rf}}t-\phi_{\mathrm{rf}})\right),$
(92) $\displaystyle M(t)$
$\displaystyle=L^{\mathrm{T}}R^{\mathrm{T}}_{\mathbf{z}_{e}\to\mathbf{z}_{n}}R_{\mathbf{z}_{n},\omega_{\mathrm{rf}}t}R_{\mathbf{z}_{e},\omega_{\mathrm{rf}}t}^{-1}R_{\mathbf{z}_{e}\to{\mathbf{o}_{e}}}L.$
(93)

In the last line, we used an alternative notation for rotations, putting
$R_{\mathbf{n},\alpha}$ for the matrix implementing rotation around unit
vector $\mathbf{n}$ by angle $\alpha$.

Fourier | matrix elements in ladder basis
---|---
index $k$ | $M_{++}^{(k)}$ | $M_{+-}^{(k)}$ | $M_{-+}^{(k)}$ | $M_{--}^{(k)}$
$\,\,\,\,\,0$ | $\frac{e^{\mathrm{i}\delta^{\prime}}(\sin\gamma-1)(1+\cos\delta)}{8}$ | $\frac{e^{\mathrm{i}\delta^{\prime}}(\sin\gamma+1)(1+\cos\delta)}{8}$ | $\frac{e^{-\mathrm{i}\delta^{\prime}}(\sin\gamma+1)(1+\cos\delta)}{8}$ | $\frac{e^{-\mathrm{i}\delta^{\prime}}(\sin\gamma-1)(1+\cos\delta)}{8}$
$-1$ | $\frac{\cos\gamma\sin\delta}{4}$ | $\frac{\cos\gamma\sin\delta}{4}$ | 0 | 0
$+1$ | 0 | 0 | $\frac{\cos\gamma\sin\delta}{4}$ | $\frac{\cos\gamma\sin\delta}{4}$
$-2$ | $\frac{e^{-\mathrm{i}\delta^{\prime}}(\sin\gamma+1)(\cos\delta-1)}{8}$ | $\frac{e^{-\mathrm{i}\delta^{\prime}}(\sin\gamma-1)(\cos\delta-1)}{8}$ | 0 | 0
$+2$ | 0 | 0 | $\frac{e^{\mathrm{i}\delta^{\prime}}(\sin\gamma-1)(\cos\delta-1)}{8}$ | $\frac{e^{\mathrm{i}\delta^{\prime}}(\sin\gamma+1)(\cos\delta-1)}{8}$

Table 1:  Matrix elements of $M(t)$ defined in Eq. (93). We do not give the
elements $M_{z\cdot}$ and $M_{\cdot z}$ as they do not couple resonant states.
The elements have symmetry
$M^{(k)}_{ff^{\prime}}=(M^{(-k)}_{\overline{f},\overline{f}^{\prime}})^{*}$,
which we interpret as the amplitude for a spin-spin transition at energy
quantum $k$ being complex conjugate of a reverse transition at opposite
energy. Here, the index inversion is defined by $\overline{+}=-$,
$\overline{-}=+$, and $\overline{z}=z$.

Since $J_{n}(t)$ contains only $-1$, $0$, and $+1$ Fourier components, to get
the component $\pm 1$ of the product $J_{n}(t)M(t)$, we need the Fourier
components of $M(t)$ up to $\pm 2$. They are given in Tab. 1 for the parts of
interest. The desired matrix element is

$\begin{split}X&\equiv[J_{n}(t)M(t)]^{(-1)}_{+-}\\\
&=J_{n}^{(-1)}M^{(0)}_{+-}+J_{n}^{(0)}M^{(-1)}_{+-}+J_{n}^{(1)}M^{(-2)}_{+-},\end{split}$
(94)

and the three terms are, respectively,

$\displaystyle-$ $\displaystyle
J_{n}\frac{1+\cos\delta}{8}e^{\mathrm{i}(\phi_{\mathrm{rf}}+\delta^{\prime})}(1+\sin\gamma)\frac{\mathbf{d}(t)\cdot(\mathbf{r}-\mathbf{r}_{0})}{l^{2}},$
(95a) $\displaystyle J_{n}\frac{\sin\delta}{4}\cos\gamma,$ (95b)
$\displaystyle-$ $\displaystyle
J_{n}\frac{1-\cos\delta}{8}e^{-\mathrm{i}(\phi_{\mathrm{rf}}+\delta^{\prime})}(1-\sin\gamma)\frac{\mathbf{d}(t)\cdot(\mathbf{r}-\mathbf{r}_{0})}{l^{2}},$
(95c)

where $\delta^{\prime}$ and $\delta$ are the two Euler angles of the rotation
$R_{\mathbf{z}_{e}\to\mathbf{z}_{n}}=R_{\mathbf{z}_{e},\delta^{\prime}}\circ
R_{\mathbf{y}_{e},\delta}$, see Fig. 2.

For realistic micro-magnet gradients and quantum dot sizes, the change of the
magnetic field across the quantum dot is small compared to the magnetic field
magnitude. The angle $\delta$ is then close to either 0, when
$\mathrm{sgn}(g_{e}g_{i})=-1$ (the case of both Si and GaAs conduction band),
or $\pi$, when $\mathrm{sgn}(g_{e}g_{i})=+1$. Out of the two terms in Eq.
(95a) and Eq. (95c), these two scenarios imply that the second or the first
can be neglected, respectively. The matrix elements in Eq. (95) show that
these two scenarios map to each other upon inverting the sign of $\gamma$.
Therefore, the relative sign of the electron and nuclear $g$ factors implies
no essential difference for the arising DNSP rate magnitude.

| Matrix element value
---|---
$M_{+-}^{(0)}$ | $\frac{1+\sin\gamma}{8}\Big{(}(J_{xx}\cos\delta+J_{yy}+iJ_{xy}\cos\delta-iJ_{yx})\cos\delta^{\prime}-(J_{zx}+iJ_{zy})\sin\delta+(iJ_{xx}-J_{yx}+J_{yx}\cos\delta+iJ_{yy}\cos\delta)\sin\delta^{\prime}\Big{)}$
$M_{+-}^{(-1)}$ | $-\frac{\cos\gamma}{4}\Big{(}(J_{xz}\cos\delta-iJ_{yz})\cos\delta^{\prime}-J_{zz}\sin\delta+(J_{yz}\cos\delta+iJ_{xz})\sin\delta^{\prime}\Big{)}$
$M_{+-}^{(+1)}$ | $0$
$M_{+-}^{(-2)}$ | $\frac{-1+\sin\gamma}{8}\Big{(}(J_{xx}\cos\delta-J_{yy}-iJ_{xy}\cos\delta-iJ_{yx})\cos\delta^{\prime}-(J_{zx}-iJ_{zy})\sin\delta+(iJ_{xx}+J_{xy}+J_{yx}\cos\delta-iJ_{yy}\cos\delta)\sin\delta^{\prime}\Big{)}$
$M_{+-}^{(+2)}$ | $0$
Table 2:  Matrix elements of $M(t)$ defined in Eq. (93) for a general
hyperfine interaction, given by $I_{i}s_{j}J_{ij}$. We give only the elements
$M_{+-}$. The symmetry
$M^{(k)}_{ff^{\prime}}=(M^{(-k)}_{\overline{f},\overline{f}^{\prime}})^{*}$
still holds.

On the other hand, the micro-magnet gradient makes the angles
$\delta,\delta^{\prime}$ dependent on the position within the dot,
complicating the analysis. We consider a simplified scenario. The restriction
is insignificant for the results presented in this article, but simplifies the
notation and calculations. Namely, the gradient of the magnetic field at the
dot position is given by the tensor $\nabla_{i}\langle B_{j}\rangle$.383838The
derivative is with respect to the dot center $\mathbf{r}_{0}$. For in-plane
displacements, the six derivatives, $\nabla_{x_{0}}\langle\mathbf{B}\rangle$
and $\nabla_{y_{0}}\langle\mathbf{B}\rangle$, enter the problem. We split them
to the gradient of the field along its direction (also denoted as the field
longitudinal component),
$\boldsymbol{\nabla}(\langle\mathbf{B}\rangle\cdot\mathbf{z}_{e})$, and the
two gradients of the two remaining transverse components. The former is
important for the resonance width, see Eq. (71) in App. A. The latter can be
represented by a two by two matrix, schematically denoted by
$\boldsymbol{\nabla}(\langle\mathbf{B}\rangle\times\mathbf{z}_{e})$. Our
simplified scenario corresponds to neglecting the smaller-in-magnitude of the
two singular values $w_{1}$ and $w_{2}$ of this matrix. Assuming $w_{1}$ is
the larger one, the component of the magnetic field perpendicular to
$\mathbf{z}_{e}$ is given by
$(\mathbf{r}-\mathbf{r}_{0})\cdot\mathbf{u}_{1}w_{1}\mathbf{v}_{1}$, where
$\mathbf{u}_{i}$ and $\mathbf{v}_{i}$ are the unit vectors from the singular
value decomposition.393939We use the notation of Chapter 2.9 of Ref. [89]. See
therein for details on the singular value decomposition. More important than
their values, we note that in this case the angle $\delta^{\prime}$ is fixed,
given by the direction of the vector $\mathbf{v}_{1}$, while $\delta$ is
position dependent, given by404040The plus sign applies if the axes
$\mathbf{z}_{e}$ and $\mathbf{z}_{n}$ are close to parallel ($\delta\approx
0$) and minus sign if they are close to antiparallel $(\delta\approx\pi)$. If
it is the minus sign here, it inverts the relative sign of the interference
term in Eq. (40d). To ease the notation, we include this possible minus sign
by redefining $\delta^{\prime}$, adding $\pi$ to it.

$\begin{split}\pm\sin\delta\approx\tan\delta&=\frac{((\mathbf{r}-\mathbf{r}_{0})\cdot\boldsymbol{\nabla})(\langle\mathbf{B}\rangle\times\mathbf{z}_{e})\cdot\mathbf{v}_{1}}{\langle\mathbf{B}\rangle}\\\
&\approx\frac{(\mathbf{r}-\mathbf{r}_{0})\cdot\mathbf{u}_{1}w_{1}}{\langle\mathbf{B}\rangle}.\end{split}$
(96)

In the main text, we use a shorthand notation $\nabla_{\perp}B\equiv
w_{1}=|\boldsymbol{\nabla}(\langle\mathbf{B}\rangle\times\mathbf{z}_{e})\cdot\mathbf{v}_{1}|$
as the gradient size of the transverse component of the magnetic field, and
$\nabla_{||}B\equiv|\boldsymbol{\nabla}(\langle\mathbf{B}\rangle\cdot\mathbf{z}_{e})|$
as the gradient size of the longitudinal component. Also, we denote $\phi$ as
the angle of vectors $\mathbf{d}$ and $\mathbf{v}_{1}$.

With the geometry clarified, let us go back to Eq. (94). It is a sum of three
terms. To be specific, let us take the $\delta\approx 0$ scenario. Each term
contains a small factor: in the first, it is the dot shift compared to its
size $O(\mathbf{d}/l)$, in the second the deflection angle $O(\delta)$, and in
the third there are both $O(\mathbf{d}/l)$ and $O(\delta^{2})$. As already
noted, the third term can be neglected with respect to the first. (If
$\delta\approx\pi$, the roles of the first and third terms would be swapped).
The transition amplitude is thus a sum of two terms. The complex factor
$\exp[\mathrm{i}(\phi_{\mathrm{rf}}+\delta^{\prime})]$ makes the two terms
interfere in the matrix element squared magnitude $|X|^{2}$. Once again, we
are interested in the average of this expression over the dot. Equation (40)
follows after a short algebra using Eq. (94), Eq. (96), and the following
averages,

$\displaystyle\langle\mathbf{r}-\mathbf{r}_{0}\rangle=\boldsymbol{0},$
$\displaystyle\langle(\mathbf{a}\cdot(\mathbf{r}-\mathbf{r}_{0}))(\mathbf{b}\cdot(\mathbf{r}-\mathbf{r}_{0}))\rangle=(\mathbf{a}\cdot\mathbf{b})l^{2}.$

We conclude with a comment to the interference strength $\xi$ given in Eq.
(40d): It is a product of two cosines. If neither of the two arguments is
known, one could replace them by their average using
$\xi\to\sqrt{\langle\xi^{2}\rangle}=1/2$, where the average
$\langle\cdot\rangle$ is the integral over the unknown angles with a uniform
prior probability distribution. In other words, the interference is somewhat
suppressed by the misalignment of the essentially random directions of vectors
$\mathbf{d}$, $\mathbf{v}_{1}$, and the angle $\phi_{\mathrm{rf}}$. Within our
precision here, we simply neglect the interference.

##  Appendix E Anisotropic hyperfine interaction

We now extend the previous appendix by considering a more general form of the
hyperfine Hamiltonian $H_{\mathrm{hf}}$ in Eq. (4), with the spin-spin
interaction not necessarily isotropic. It would result in the hyperfine
coupling in Eq. (15) becoming a second-rank tensor, with Cartesian components
denoted here as $J_{xx}$, $J_{xy}$, and so on. The relevant matrix elements
for a general hyperfine tensor, calculated from Eq. (93), are given in Tab. 2.

We are motivated by the possible application of our formulas to hole qubits
(see App. I). Before that, we look at what tensor matrix elements are required
for a non-zero DNSP, making the connection to the existing literature. From
Tab. 2, we find that without the axes deflection, neither the isotropic
exchange ($J_{xx}=J_{yy}=J_{zz}$ as the only non-zero matrix elements) nor the
‘secular exchange’ ($J_{zz}$ the only non-zero matrix element; the name is
used in the NMR literature), induce transitions. Whereas the former applies
for dipole-dipole interactions in liquid solutions [55], the latter form of
spin-spin coupling originating in dipole-dipole interaction is typically
considered in the solid state [68, 67].414141However, there are also isotropic
interactions: In its derivation of the polarization rate, Ref. [47] considered
the electron-mediated nuclear-nuclear exchange as such an isotropic
interaction.

We thus have the following analogy to the NMR and the Hartmann-Hahn effect:
While there both spins are driven, here it is only one of them (the electron).
Since driving a spin effectively deflects its energy quantization axis from
the direction of the magnetic field, driving also the second nuclear spins in
NMR is analogous to having here a non-zero deflection angle $\delta$ due to
the micro magnet.424242Sec. IV.C of Ref. [47] contains a discussion of the
case where the second spin is not driven (as here) and invokes an exchange
tensor with components such as $J_{zx}$ and $J_{zy}$ needed to produce finite
non-zero matrix element for the polarization transition. The same anisotropic
terms were considered also in Refs. [55, 54].

In the literature on self-assembled quantum dots, the ‘non-secular’ (the NMR
name) hyperfine interaction terms, such as $J_{xz}$, are called ‘non-
collinear’. While in Ref. [81, 59] such terms are assigned to the effects of
the light-hole–heavy-hole mixing on the hole hyperfine tensor (see App. I), in
the majority of the works in that field the ‘non-collinearity’ is understood
as due to the quadrupolar fields [76] (they were considered as the DNSP source
in Ref. [90] and coined as ‘non-collinear’ hyperfine tensor in Ref. [84],
including the quadrupolar effects perturbatively). If due to quadrupolar
fields, the ‘non-collinearity’ then qualitatively corresponds, in our work, to
the combined effect of an isotopic electron-nuclear hyperfine interaction and
the deflection of the quantization axes.

##  Appendix F Additional plots.

We present additional plots analogous to Figs. 4 and 6 of the main text.

###  F.1 Plots analogous to Fig. 4.

Figure 9: Polarization rate as a function of the electron detuning frequency
for gallium isotopes. The plot is analogous to Fig. 4, and all parameters are
the same as there. Figure 10: Polarization rate as a function of the electron
detuning frequency for 29Si in a silicon quantum dot. The plot is analogous to
Fig. 4 and the parameters are as there except for: pulse time
$T_{\mathrm{pulse}}=100$ $\mu$s, cycle time $T_{\mathrm{cycle}}=200$ $\mu$s,
dot in-plane size $l=20$ nm, additional smearing $2\pi\times 25$ kHz.

Figure 9 shows DNSP rates for the two gallium isotopes of GaAs, using the same
parameters as in Fig. 4. Figure 10 is a similar plot for the 29Si isotope of a
Si dot. There, some parameters are slightly changed, reflecting that silicon
dots are typically smaller and have better coherence because of nuclear-
induced dephasing being smaller than in GaAs. For these parameters, the DNSP
rates in Si are about four orders of magnitude smaller than in GaAs.

###  F.2 Plots analogous to Fig. 6.

Figure 11: Stabilization by feedback in an electron qubit in Si. The plot is
analogous to Fig. 6 (see its caption for a plot description) and adopts the
same parameters except for the material parameters of Si and a smaller dot,
$l=20$ nm.

To illustrate the magnitude of the DNSP polarization rate in silicon, we plot
the restoring rate $\Gamma^{*}$ in Fig. 11. Comparing to Fig. 6, we observe
that in Si the polarization is several orders of magnitude smaller than in
GaAs. It is also much smaller than the intrinsic restoring force $\Gamma_{0}$
due to thermal diffusion. In this respect, the EDSR-induced dynamics of
nuclear spins is minor.

##  Appendix G Laird mechanism

We now consider a different resonance, not of the Hartmann-Hahn type. We
include it for completeness, and because both the calculation and the
corresponding experiment are related to the ones we have considered. Namely,
we now assume that for some reason, the electrical driving is not effective in
driving the electron spin. While the dot is electrically driven as before,
there is no EDSR (disregarding nuclei). The most straightforward scenario
would be a dot without a micromagnet.

Still, the electron spin can be transferred to the nuclei, so that DNSP
arises. However, it happens at a different resonance condition, namely when
the driving frequency equals the difference of the electron and nuclear Larmor
frequencies.434343Such slight detunings which lead to spin-selective electron-
nuclear flip-flops were considered in Ref. [50]. As we explain here, the DNSP
arises as a backaction of the torque that a random transverse component of the
Overhauser field exerts on the electron.444444Several previous works [91, 53,
92] analyzed the DNSP arising from a periodically reset electron spin,
treating the electron and Overhauser fields as classical vectors precessing
around each other. That a semiclassical model contains all the relevant
physics is confirmed by the fact that it also explains the observed nuclei-
induced electron-spin dephasing [93]. When the resonance condition is
fulfilled, this torque results in the electron EDSR and the nuclear
polarization. The effect was experimentally demonstrated by Laird et al. [63].

To estimate the strength of this mechanism, and compare it to the ones from
the main text, we now make an analogous derivation of the DNSP rate for this
scenario. Since the micro magnet is not relevant now, we drop it from the
problem. The Hamiltonian in Eq. (16) simplifies considerably,

$H=-\hbar\omega_{n}\mathbf{I}_{n}\cdot\mathbf{z}_{n}-\hbar\omega_{e}\mathbf{s}\cdot\mathbf{z}_{e}+J_{n}(t)\delta\mathbf{I}_{n}\cdot\mathbf{s}.$
(97)

The two vectors $\mathbf{z}_{n}$ and $\mathbf{z}_{e}$ are now parallel (or
antiparallel). We go into a frame rotating with the nuclear spin,

$H^{\prime}=-\left[\hbar\omega_{e}+\mathrm{sgn}(g_{e}g_{n})\hbar\omega_{n}\right]s_{z}+J_{n}(t)\delta\mathbf{I}_{n}\cdot\mathbf{s}.$
(98)

In the last term, we keep only the transverse (in spin) and varying (in time)
component,

$\begin{split}H^{\prime}&=-\left[\hbar\omega_{e}+\mathrm{sgn}(g_{e}g_{n})\hbar\omega_{n}\right]s_{z}+[J_{n}(t)-J_{n}(0)]\mathbf{I}_{n\perp}\cdot\mathbf{s}.\end{split}$
(99)

We sum over all nuclei and consider the electron dynamics described by

$\displaystyle
H_{e}=-(\hbar\omega_{\mathrm{rf}}-\hbar\omega_{\Delta}^{L})s_{z}$
$\displaystyle\qquad\qquad+2\cos(\omega_{\mathrm{rf}}t-\phi_{\mathrm{rf}})\hbar\boldsymbol{\omega}_{RR}^{L}\cdot\mathbf{s},$
(100a)
$\displaystyle\hbar\omega_{\Delta}^{L}=\hbar\omega_{\mathrm{rf}}-\left[\hbar\omega_{e}+\mathrm{sgn}(g_{e}g_{n})\hbar\omega_{n}\right],$
(100b)
$\displaystyle\hbar\boldsymbol{\omega}_{RR}^{L}=\frac{1}{2}\sum_{n}v_{0}A_{n}(\mathbf{d}\cdot\boldsymbol{\nabla})|\Psi_{n}|^{2}\mathbf{I}_{n,_{\perp}},$
(100c)

Thus, we obtained a Rabi Hamiltonian $H_{e}$ with the detuning
$\hbar\omega_{\Delta}^{L}$ and the driving field
$\hbar\boldsymbol{\omega}_{RR}^{L}$. In this section, we introduce several
quantities analogous to the ones in the main text, denoting them by the
superscript $L$ for ‘Laird’.

We estimate the typical value of the transverse field, averaging over the dot

$\begin{split}&\langle\left(\hbar\boldsymbol{\omega}_{RR}^{L}\right)^{2}\rangle\\\
&\quad=\frac{1}{4}\langle\sum_{n,m}v_{0}^{2}A_{i}^{2}4(\mathbf{d}\cdot\mathbf{r}_{n0})(\mathbf{d}\cdot\mathbf{r}_{m0})l^{-4}|\Psi_{n}|^{4}\mathbf{I}_{n,_{\perp}}\cdot\mathbf{I}_{m,_{\perp}}\rangle\\\
&\quad=2\frac{I(I+1)}{3}(d/l)^{2}\phi_{i}A_{i}^{2}v_{0}/V_{D},\end{split}$
(101)

with the short hand $\mathbf{r}_{n0}=\mathbf{r}_{m}-\mathbf{r}_{0}$. In the
averaging, we assume that the polarization $p_{i}$ is small and use
$\langle\mathbf{I}_{n,_{\perp}}\cdot\mathbf{I}_{m,_{\perp}}\rangle=\delta_{n,m}(2/3)I(I+1)$
corresponding to unpolarized nuclei.

Figure 12: Rabi oscillations in the Laird mechanism. The schematic defines the
Rabi angle $\gamma^{L}$ in terms of the detuning and the matrix element given
in Eq. (102). This figure is analogous to Fig. 3.

The transverse field corresponds to the Rabi precession angle

$\displaystyle\sin\gamma^{L}$
$\displaystyle=-\frac{\hbar\omega_{\Delta}^{L}}{\hbar\omega_{R}^{L}},$ (102a)
$\displaystyle\cos\gamma^{L}$
$\displaystyle=\frac{\hbar\omega_{RR}^{L}}{\hbar\omega_{R}^{L}},$ (102b)
$\displaystyle\hbar\omega_{R}^{L}$
$\displaystyle=\sqrt{\left(\hbar\omega_{RR}^{L}\right)^{2}+\left(\hbar\omega_{\Delta}^{L}\right)^{2}}.$
(102c)

The electron spin z-component evolves according to

$\begin{split}s_{z}(t)&=s_{z}(0)\left[\sin^{2}\gamma^{L}+\cos^{2}\gamma^{L}\cos\omega_{R}^{L}t\right]\\\
&=s_{z}(0)\left[1+\cos^{2}\gamma^{L}\left(\cos\omega_{R}^{L}t-1\right)\right],\end{split}$
(103)

an equation analogous to Eq.(35).

Finally, since the z-component of the total spin of the system is conserved,
the change of the electron spin equals the opposite change of the spin of the
nuclei,

$I_{z}(t)-I_{z}(0)=s_{z}(0)-s_{z}(t)=\frac{p_{e}}{2}\cos^{2}\gamma^{L}\left(1-\cos\omega_{R}^{L}t\right),$
(104)

where we have put $s_{z}(0)=p_{e}/2$ in line with the notation in Eq. (78).
With the total duration of the driving being $T_{\mathrm{pulse}}$ (we drop the
subscript and use $T$ in the following two equations to improve readability),
this change is equivalent to a rate (of polarization the total nuclear spin of
isotope $i$)

$\Gamma_{i}^{L}=\frac{p_{e}}{2}\cos^{2}\gamma^{L}\frac{1-\cos\omega_{R}^{L}T}{T},$
(105)

in turn equivalent to the polarization rate

$\begin{split}\partial_{t}p_{i}&=\frac{1}{I_{i}\phi_{i}N_{\mathrm{tot}}}\Gamma^{L}_{i}\\\
&=\frac{p_{e}}{2}\frac{1}{I_{i}\phi_{i}N_{\mathrm{tot}}}\frac{\left(\hbar\omega_{RR}^{L}\right)^{2}}{\left(\hbar\omega_{R}^{L}\right)^{2}}\frac{1-\cos\omega_{R}^{L}T}{T}\\\
&=\frac{p_{e}}{2}\frac{1}{\phi_{i}N_{\mathrm{tot}}}\frac{(2/3)(I_{i}+1)(d/l)^{2}\phi_{i}A_{i}^{2}}{\hbar^{2}N_{\mathrm{tot}}}\frac{1-\cos\omega_{R}^{L}T}{\left(\omega_{R}^{L}\right)^{2}T}\\\
&=p_{e}\frac{(I_{i}+1)}{3\hbar^{2}}\frac{A_{i}^{2}}{N_{\mathrm{tot}}^{2}}\frac{d^{2}}{l^{2}}\frac{1-\cos\omega_{R}^{L}T}{\left(\omega_{R}^{L}\right)^{2}T}.\end{split}$
(106)

In analogy to Eq. (44), we interpret the last factor as a (Lorenzian-shaped)
spectral density: in the limit of a long evolution time $T\to\infty$ and a
small hyperfine matrix element $\omega_{RR}^{L}\to 0$, it becomes a delta
function

$\pi\hbar\delta\Big{(}\hbar\omega_{e}+\mathrm{sgn}(g_{e}g_{n})\hbar\omega_{n}-\hbar\omega_{\mathrm{rf}}\Big{)},$
(107)

imposing the conservation of the energy transfer between the electron spin, a
nuclear spin, and a microwave photon. With this interpretation, we cast the
rate in line with the notation of Eq. (47a),

$\partial_{t}p_{i}=\frac{\pi}{\hbar^{2}}X_{L}^{2}\alpha_{I}p_{e}G_{\Sigma_{L}}(\omega_{\Delta}^{L}),$
(108a) where $\displaystyle X_{L}$
$\displaystyle=\frac{A_{i}}{N_{\mathrm{tot}}}\frac{d}{l},$ (108b)
$\displaystyle\Sigma_{L}^{2}$
$\displaystyle=\frac{2I_{i}(I_{i}+1)}{3}\frac{d^{2}}{l^{2}}\frac{\phi_{i}A_{i}^{2}}{N_{\mathrm{tot}}}.$
(108c)

We have arrived at a formula analogous to Eq. (47). It is interesting to note
that, up to an additional factor of 2 in the matrix element $X_{L}$, Eq. (108)
corresponds to Eq. (47a) including only the ’shaking’ mechanism in the limit
$f_{RR}\to 0$ with $\gamma\to\pi/2$. The remaining differences are natural:
First, since we assumed unpolarized nuclei, the rate in (108a) contains the
factor from Eq. (36) evaluated at $p_{n}=0$. Second, the width of the spectral
function now refers only to the thermal distribution of the Overhauser field.
The latter is similar to the values seen in Sec. V.454545Using $d=0.5$ nm,
$l_{z}=10$ nm, and $l=34$ nm, we get $\Sigma_{L}(^{29}\mathrm{Si})=2\pi\times
34$ kHz, $\Sigma_{L}(^{69}\mathrm{Ga})=2\pi\times 135$ kHz,
$\Sigma_{L}(^{71}\mathrm{Ga})=2\pi\times 140$ kHz,
$\Sigma_{L}(^{75}\mathrm{Ga})=2\pi\times 207$ kHz. However, one also needs to
point out substantial differences:

First, the mechanism considered in this section originates in the (reaction)
torque that the electron spin exerts in response to the (action) torque from
nuclei inducing the electron Rabi rotation. In the main text, this torque was
due to the micromagnet and had nothing to do with nuclei. While a stochastic
gradient from the Overhauser field will coexist with the one due to a micro
magnet, they will have a random mutual orientation (alternatively: random
phase). If it is the micro-magnet gradient that dominates, the random phase
suppresses the ‘Laird’ polarization rate and makes it zero on average in
experiments with micro magnets.

Second, the derivation here applies in the incoherent regime, otherwise the
time-dependent factor in Eq. (106) should not be converted to a delta
function, but kept as oscillating, leading to an oscillating nuclear
polarization. Taking the opposite view, trying to use Eq. (47) in the far-off
resonant regime, we do not expect to recover Eq. (108a) from Eq. (47a) upon
taking the limit $f_{RR}\to 0$. Namely, the assumption that the last term in
Eq. 50 is the smallest is not fulfilled far from the resonance and explains
the unnatural result $(1+\sin\gamma)/2\to\theta(\gamma)$ in the limit
$f_{RR}\to 0$. To correct for this deficiency, one would need to keep both in-
phase and out-of-phase frequency components, for example using the technique
of Ref. [94]. However, since we are interested primarily in DNSP arising in
dots with high-quality single-qubit operations, we do not pursue the off-
resonant regime, and the connection between Eq. (108a) and Eq. (47a), further.

The most important conclusion of this section is that the DNSP arising as the
backaction of the electron ‘primary’ Rabi oscillation on the nuclear spins,
that is, the ‘Laird’ mechanism, can be neglected if nuclei are not the
dominant source of the primary Rabi oscillations, that is, in experiments
employing micro magnets or spin-orbit coupling. The nuclear contribution to
the ‘primary’ Rabi oscillations of the electron spin was neglected in the main
text, attributing it to the micromagnet entirely. While nuclei also
contribute, the corresponding DNSP rate is going to be much smaller than Eq.
(108a), the latter comparable to one of the mechanisms included in Eq. (47a).

##  Appendix H Effective parameters of bounded diffusion

The Overhauser field acting on the electron spin in a quantum dot fluctuates
because of diffusive thermal fluctuations of nuclear spins mediated by dipolar
nuclear spin-spin interactions. The diffusion results in the Overhauser-field
variance growing linearly over short times and saturating at long times: The
long-time average (probability distribution) of the Overhauser-field
components is a Gaussian with a finite variance centered at zero. A simple
model of such stochastic quantity is a random walk with a harmonic restoring
force [95, 96]. Using the notation of Ref. [97], with
$K(\omega_{0},\omega,\delta t)$ being the conditional probability of the
electron Larmor frequency having value $\omega$ at time $t$ if it had value
$\omega_{0}$ at time $t_{0}=t-\delta t$, the model gives

$\displaystyle K(\omega_{0},\omega,\delta t)=\frac{1}{\sqrt{2\pi\sigma_{\delta
t}^{2}}}\exp\left[-\frac{(\omega-\omega_{0}e^{-\delta
t/\kappa})^{2}}{2\sigma_{\delta t}^{2}}\right],$ (109a) where $\sigma_{\delta
t}^{2}=\sigma_{\Omega}^{2}(1-e^{-2\delta t/\kappa}).$ (109b)

Hence, the model has two parameters, $\sigma_{\Omega}^{2}$ and $\kappa$. The
first parameter is the variance long-time saturation value. For the Overhauser
field contribution to the electron Larmor frequency,

$\sigma_{\Omega}^{2}=\frac{1}{\hbar^{2}}\langle\left(\delta\hbar\omega_{e}\right)^{2}\rangle,$
(110)

it is, by a calculation analogous to Eq. (101),

$\begin{split}\sigma_{\Omega}^{2}&=\frac{1}{\hbar^{2}}\langle\sum_{n,m}v_{0}^{2}A_{n}A_{m}|\Psi_{n}|^{2}|\Psi_{m}|^{2}\mathbf{I}_{n,z}\cdot\mathbf{I}_{m,z}\rangle\\\