Dynamical nuclear spin polarization in a quantum dot with an electron spin driven by electric dipole spin resonance

On top of a homogeneous field of a solenoid coil, a micromagnet 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 Appendix 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.

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 micromagnet 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 that will be useful in the derivations below.

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 paper 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 Appendix B for the derivation).

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 micromagnet-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 Table L in Appendix L.

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 ‘nonsecular’ 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 noncollinearity of the spin quantization directions and is due to the micromagnet-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.

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). Reference [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 is¹⁴¹⁴14This step can be understood as going into a rotating frame effectively undoing the rotation due to the second term of Eq. (17). We include an alternative derivation of the polarization rate using such a frame in Appendix K, see Eq. (158).

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

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 that 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.

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

↑↑E↑↑⋅⋅⋅↑↓⋅E↑↓Y†⋅↓↑⋅YE↓↑⋅↓↓⋅⋅⋅E↑↑).\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).

		H⁢s′j′,⁢sj	↑↑	↑↓	↓↑	↓↓		(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 that 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.¹⁵¹⁵15Note 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) and (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.

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.¹⁶¹⁶16In 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 a long-time synchronization is essential for nuclear autofocusing [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.

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 Appendix 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 Appendix 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.¹⁷¹⁷17In 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.

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 Appendix 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:¹⁸¹⁸18The 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 Appendix 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.

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 rate¹⁹¹⁹19As 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 Appendix 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 micromagnet, 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.

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.²⁰²⁰20Maximizing 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 Appendix 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.²¹²¹21The 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 Appendix G.

• 15.

  The micromagnet 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 micromagnet. The analysis here then applies also if the micromagnet, 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 Appendix I. If it is not, meaning the spin-orbit length is smaller than the size of the dot, we expect DNSP with nontrivial 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.

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 Appendix 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.³²³²32In 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 nonresonant 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.

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 Appendix 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.

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.³³³³33A 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,³⁴³⁴34For 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 Appendix 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].³⁵³⁵35The decrease of fluctuations when the forces become larger can be also understood from the model in Appendix 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 micromagnet 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.

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.