Nuclear Spin Dynamics in Double Quantum Dots: Multi-stability, Dynamical Polarization, Criticality and Entanglement

Abstract

In the previous Chapter we have investigated the transient creation of nuclear coherence as a result of electron transport through a single quantum dot.

3.11 Appendix to Chap. 3

3.1.1 3.11.1 Spin-Blockade Regime

In this Appendix, for completeness we explicitly derive inequalities involving the chemical potentials $\mu _{L\left( R\right) }$ of the left and right lead, respectively, as well as the Coulomb energies introduced in Eq. (3.7) that need to be satisfied in order to tune the DQD into the desired Pauli-blockade regime in which at maximum two electrons reside on the DQD. For simplicity, Zeeman splittings are neglected for the moment as they typically constitute a much smaller energy scale compared to the Coulomb energies. Still, an extension to include them is straight-forward. Then, the bare energies $E_{\left( m,n\right) }$ for a state with $\left( m,n\right)$ charge configuration can easily be read off from the Anderson Hamiltonian $H_{S}$. In particular, we obtain

$$\begin{aligned} E_{\left( 1,1\right) }= & {} \,\varepsilon _{L}+\varepsilon _{R}+U_{LR},\end{aligned}$$

(3.94)

$$\begin{aligned} E_{\left( 2,1\right) }= & {} \,2\varepsilon _{L}+\varepsilon _{R}+U_{L}+2U_{LR},\end{aligned}$$

(3.95)

$$\begin{aligned} E_{\left( 1,2\right) }= & {} \,\varepsilon _{L}+2\varepsilon _{R}+U_{R}+2U_{LR},\end{aligned}$$

(3.96)

$$\begin{aligned} E_{\left( 0,2\right) }= & {} \,2\varepsilon _{R}+U_{R},\end{aligned}$$

(3.97)

$$\begin{aligned} E_{\left( 2,0\right) }= & {} \,2\varepsilon _{L}+U_{L}. \end{aligned}$$

(3.98)

In order to exclude the occupation of $\left( 2,1\right)$ and $\left( 1,2\right)$ states if the DQD is in a $\left( 1,1\right)$ charge configuration the left chemical potential must fulfill the inequality $\mu _{L}<E_{\left( 2,1\right) }-E_{\left( 1,1\right) }=\varepsilon _{L}+U_{L}+U_{LR}$. An analog condition needs to be satisfied for the right chemical potential $\mu _{R}$ so that we can write in total

$$\begin{aligned} \mu _{i}<\varepsilon _{i}+U_{i}+U_{LR}. \end{aligned}$$

(3.99)

The same requirement should hold if the DQD is in a $\left( 0,2\right)$ or $\left( 2,0\right)$ charge configuration which leads to

$$\begin{aligned} \mu _{i}<\varepsilon _{i}+2U_{LR}. \end{aligned}$$

(3.100)

At the same time, the chemical potentials $\mu _{i}$ are tuned sufficiently high so that an electron is added to the DQD from the leads whenever only a single electron resides in the DQD. For example, this results in $\mu _{L}>E_{\left( 1,1\right) }-\varepsilon _{R}=\varepsilon _{L}+U_{LR}$. An analog condition needs to hold for the right lead which gives

$$\begin{aligned} \mu _{i}>\varepsilon _{i}+U_{LR}. \end{aligned}$$

(3.101)

In particular this inequality guarantees that the right dot is always occupied, since $\mu _{R}>\varepsilon _{R}$. Moreover, localized singlet states cannot populated directly if $\mu _{i}<\varepsilon _{i}+U_{i}$ holds. Since $U_{LR}<U_{i}$, the conditions to realize the desired two-electron regime can be summarized as

$$\begin{aligned} \varepsilon _{i}+U_{LR}<\mu _{i}<\varepsilon _{i}+2U_{LR}. \end{aligned}$$

(3.102)

By applying a large bias that approximately compensates the charging energy of the two electrons residing on the right dot, that is $\varepsilon _{L}\approx \varepsilon _{R}+U_{R}-U_{LR}$, the occupation of a localized singlet with charge configuration $\left( 2,0\right)$ can typically be neglected [111, 113]. In this regime, only states with the charge configurations $\left( 0,1\right)$, $\left( 1,0\right)$, $\left( 1,1\right)$ and $\left( 0,2\right)$ are relevant. Also, due to the large bias, admixing within the one-electron manifold is strongly suppressed—for typical parameters we estimate $t/\left( \varepsilon _{L}-\varepsilon _{R}\right) \approx 10^{-2}$—such that the relevant single electron states that participate in the transport cycle in the spin-blockade regime are the two lowest ones $\left| 0,\sigma \right\rangle =d_{R\sigma }^{\dagger }\left| 0\right\rangle$ with $\left( 0,1\right)$ charge configuration [62].

3.1.2 3.11.2 Quantum Master Equation in Spin-Blockade Regime

Following the essential steps presented in Ref. [46], we now derive an effective master equation for the DQD system which experiences irreversible dynamics via the electron’s coupling to the reservoirs in the leads. We start out from the von Neumann equation for the global density matrix given in Eq. (3.18). It turns out to be convenient to decompose $\mathcal {H}$ as

$$\begin{aligned} \mathcal {H}=H_{0}+H_{1}+H_{T}, \end{aligned}$$

(3.103)

with $H_{0}=H_{S}+H_{B}$ and $H_{1}=V_{\mathrm {HF}}+H_{t}$. We define the superoperator P as

$$\begin{aligned} P\varrho =\mathsf {Tr}_{\mathsf {B}}\left[ \varrho \right] \otimes \rho _{B}^{0}. \end{aligned}$$

(3.104)

It acts on the total system’s density matrix $\varrho$ and projects the environment onto their respective thermal equilibrium states, labeled as $\rho _{B}^{0}$. The map P satisfies $P^{2}=P$ and is therefore called a projector. By deriving a closed equation for the projection $P\varrho$ and tracing out the unobserved reservoir degrees of freedom, we arrive at the Nakajima-Zwanzig master equation for the system’s density matrix

$$\begin{aligned} \dot{\rho }= & {} \, \left[ \mathcal {L}_{S}+\mathcal {L}_{1}\right] \rho \\&+\int _{0}^{t}d\tau \mathsf {Tr}_{\mathsf {B}}\left[ \mathcal {L}_{T}e^{\left( {{\mathcal {L}}}_{0}+\mathcal {L}_{T}+\mathcal {L}_{1}\right) \tau }\mathcal {L}_{T}\rho \left( t-\tau \right) \otimes \rho _{B}^{0}\right] .\nonumber \end{aligned}$$

(3.105)

where the Liouville superoperators are defined as usual via $\mathcal {L}_{\alpha }\cdot =-i\left[ H_{\alpha },\cdot \right]$. Next, we introduce two approximations: First, in the weak coupling limit, we neglect all orders higher than two in $\mathcal {L}_{T}$. This is well known as the Born approximation. Accordingly, we neglect $\mathcal {L}_{T}$ in the exponential of the integrand. Second, we apply the approximation of independent rates of variations [60] which can be justified self-consistently, if the bath correlation time $\tau _{c}$ is short compared to the typical timescales associated with the system’s internal interactions, that is $g_{\mathrm {hf}}\tau _{c}\ll 1$ and $t\tau _{c}\ll 1$, and if $H_{1}$ can be treated as a perturbation with respect to $H_{0}$. In our system, the latter is justified as $H_{0}$ incorporates the large Coulomb energy scales which energetically separate the manifold with two electrons on the DQD from the lower manifold with only one electron residing in the DQD, whereas $H_{1}$ induces couplings within these manifolds only. In this limit, the master equation then reduces to

$$\begin{aligned} \dot{\rho }= & {} \,\left[ \mathcal {L}_{S}+\mathcal {L}_{1}\right] \rho \\&+\int _{0}^{t}d\tau \mathsf {Tr}_{\mathsf {B}}\left[ \mathcal {L}_{T}e^{{{\mathcal {L}}}_{0}\tau }\mathcal {L}_{T}\rho \left( t-\tau \right) \otimes \rho _{B}^{0}\right] .\nonumber \end{aligned}$$

(3.106)

In the next step, we write out the tunnel Hamiltonian $H_{T}$ in terms of the relevant spin-eigenstates. Here, we single out one term explicitly, but all others follow along the lines. We get

$$\begin{aligned} \dot{\rho }= & {} \,\cdots +\sum _{\sigma }\int _{0}^{t}d\tau \mathcal {C}\left( \tau \right) \left| 0,\sigma \right\rangle \left\langle S_{02}\right| \\&\left[ e^{-iH_{0}\tau }\rho \left( t-\tau \right) e^{iH_{0}\tau }\right] \left| S_{02}\right\rangle \left\langle 0,\sigma \right| ,\nonumber \end{aligned}$$

(3.107)

where

$$\begin{aligned} \mathcal {C}\left( \tau \right) =\int _{0}^{\infty }d\varepsilon J\left( \varepsilon \right) e^{i\left( \Delta E-\varepsilon \right) \tau }, \end{aligned}$$

(3.108)

and $J\left( \varepsilon \right) =\left| T_{R}\right| ^{2}n_{R}\left( \varepsilon \right) \left[ 1-f_{R}\left( \varepsilon \right) \right]$ is the spectral density of the right lead, with $n_{R}\left( \varepsilon \right)$ being the density of states per spin of the right lead; $f_{\alpha }\left( \varepsilon \right)$ denotes the Fermi function of lead $\alpha =L,R$ and $\Delta E$ is the energy splitting between the two levels involved, i.e., for the term explicitly shown above $\Delta E=\varepsilon _{R}+U_{R}$. The correlation time of the bath $\tau _{c}$ is determined by the decay of the memory-kernel $\mathcal {C}\left( \tau \right)$. The Markov approximation is valid if the spectral density $J\left( \varepsilon \right)$ is flat on the scale of all the effects that we have neglected in the previous steps. Typically, the effective density of states $D\left( \varepsilon \right) =\left| T_{R}\right| ^{2}n_{R}\left( \varepsilon \right)$ is weakly energy dependent so that this argument is mainly concerned with the Fermi functions of the left (right) lead $f_{L\left( R\right) }\left( \varepsilon \right)$, respectively. Therefore, if $f_{i}\left( \varepsilon \right)$ is flat on the scale of $\sim t$, $\sim g_{\mathrm {hf}}$ and the dissipative decay rates $\sim \Gamma$, it can be evaluated at $\Delta E$ and a Markovian treatment is valid [46]. In summary, this results in

$$\begin{aligned} \dot{\rho }=\,\cdots +\Gamma _{R}\sum _{\sigma }\mathcal {D}\left[ \left| 0,\sigma \right\rangle \left\langle S_{02}\right| \right] \rho , \end{aligned}$$

(3.109)

where $\Gamma _{R}$ is the typical sequential tunneling rate $\Gamma _{R}=2\pi \left| T_{R}\right| ^{2}n_{R}\left( \Delta E\right) \left[ 1-f_{R}\left( \Delta E\right) \right]$ describing direct hopping at leading order in the dot-lead coupling [46, 64].

Pauli blockade.—The derivation above allows for a clear understanding of the Pauli-spin blockade in which only the level $\left| S_{02}\right\rangle$ can decay into the right lead whereas all two electron states with $\left( 1,1\right)$ charge configuration are stable. If the $\left| S_{02}\right\rangle$ level decays, an energy of $\Delta E_{2}=E_{\left( 0,2\right) }-\varepsilon _{R}=\varepsilon _{R}+U_{R}$ is released on the DQD which has to be absorbed by the right reservoir due to energy conservation arguments. On the contrary, if one of the $\left( 1,1\right)$ levels were to decay to the right lead, an energy of $\Delta E_{1}=E_{\left( 1,1\right) }-\varepsilon _{L}=\varepsilon _{R}+U_{LR}$ would dissipate into the continuum. Therefore, the DQD is operated in the Pauli blockade regime if $f_{R}\left( \Delta E_{2}\right) =0$ and $f_{R}\left( \Delta E_{1}\right) =1$ is satisfied. Experimentally, this can be realized easily as $\Delta E_{2}$ scales with the on-site Coulomb energy $\Delta E_{2}\sim U_{R}$, whereas $\Delta E_{1}$ scales only with the interdot Coulomb energy $\Delta E_{1}\sim U_{LR}$.

Taking into account all relevant dissipative processes within the Pauli-blockade regime and assuming the Fermi function of the left lead $f_{L}\left( \varepsilon \right)$ to be sufficiently flat, the full quantum master equation for the DQD reads

$$\begin{aligned} \dot{\rho }= & {} -i\left[ H_{S}+H_{1},\rho \right] +\Gamma _{R}\sum _{\sigma }\mathcal {D}\left[ \left| 0,\sigma \right\rangle \left\langle S_{02}\right| \right] \rho \nonumber \\&+\,\Gamma _{L}\left\{ \mathcal {D}\left[ \left| T_{+}\right\rangle \left\langle 0,\Uparrow \right| \right] \rho +\mathcal {D}\left[ \left| \Downarrow \Uparrow \right\rangle \left\langle 0,\Uparrow \right| \right] \rho \right\} \nonumber \\&+\,\Gamma _{L}\left\{ \mathcal {D}\left[ \left| T_{-}\right\rangle \left\langle 0,\Downarrow \right| \right] \rho +\mathcal {D}\left[ \left| \Uparrow \Downarrow \right\rangle \left\langle 0,\Downarrow \right| \right] \rho \right\} , \end{aligned}$$

(3.110)

where the rate $\Gamma _{R}\sim \left[ 1-f_{R}\left( \Delta E_{2}\right) \right]$ describes the decay of the localized singlet $\left| S_{02}\right\rangle$ into the right lead, while the second and third line represent subsequent recharging of the DQD with the corresponding rate $\Gamma _{L}\propto \left| T_{L}\right| ^{2}$.⁵

We can obtain a simplified description for the regime in which on relevant timescales the DQD is always populated by two electrons. This holds for sufficiently strong recharging of the DQD which can be implemented experimentally by making the left tunnel barrier $T_{L}$ more transparent than the right one $T_{R}$ [46, 61, 62]. In this limit, we can eliminate the intermediate stage in the sequential tunneling process $\left( 0,2\right) \rightarrow \left( 0,1\right) \rightarrow \left( 1,1\right)$ and parametrize $H_{S}+H_{1}$ in the two-electron regime as $H_{\mathrm {el}}+H_{\mathrm {ff}}+H_{\mathrm {zz}}$. Then, we arrive at the effective master equation

$$\begin{aligned} \dot{\rho }=-i\left[ H_{\mathrm {el}},\rho \right] +\mathcal {K}_{\Gamma }\rho +\mathcal {V}\rho , \end{aligned}$$

(3.111)

where the dissipator

$$\begin{aligned} \mathcal {K}_{\Gamma }\rho =\Gamma \sum _{x\in \left( 1,1\right) }\mathcal {D}\left[ \left| x\right\rangle \left\langle S_{02}\right| \right] \rho \end{aligned}$$

(3.112)

models electron transport through the DQD; the sum runs over all four electronic bare levels with $\left( 1,1\right)$ charge configuration, i.e., $\left| \sigma ,\sigma '\right\rangle$ for $\sigma ,\sigma '=\Uparrow ,\Downarrow$: Thus, in the limit of interest, the $\left( 1,1\right)$ charge states are reloaded with an effective rate $\Gamma =\Gamma _{R}/2$ via the decay of the localized singlet $\left| S_{02}\right\rangle$ [61, 62].

Transport dissipator in eigenbasis of $H_{\mathrm {el}}$.—The electronic transport dissipator $\mathcal {K}_{\Gamma }$ as stated in Eq. (3.112) describes electron transport in the bare basis of the two-orbital Anderson Hamiltonian which does not correspond to the eigenbasis of $H_{\mathrm {el}}$ due to the presence of the interdot tunnel coupling $H_{t}$; in deriving Eq. (3.112) admixing due to $H_{t}$ has been neglected based on the approximation of independent rates of variation [60]. It is valid if $t\tau _{c}\ll 1$ where $\tau _{c}\approx 10^{-15}\,{{\mathrm {s}}}$ specifies the bath correlation time [46]. Performing a basis transformation $\tilde{\rho }=V^{\dagger }\rho V$ which diagonalizes the electronic Hamiltonian $\tilde{H}_{\mathrm {el}}=V^{\dagger }H_{\mathrm {el}}V=\mathrm {diag}\left( \omega _{0},-\omega _{0},\varepsilon _{1},\varepsilon _{2},\varepsilon _{3}\right)$ and neglecting terms rotating at a frequency of $\varepsilon _{l}-\varepsilon _{k}$ for $k\ne l$, the electronic transport dissipator takes on the form⁶

$$\begin{aligned} \mathcal {K}_{\Gamma }\tilde{\rho }= & {} \sum _{k,\nu =\pm }\Gamma _{k}\mathcal {D}\left[ \left| T_{\nu }\right\rangle \left\langle \lambda _{k}\right| \right] \tilde{\rho }\\&+\sum _{k,j}\Gamma _{k\rightarrow j}\mathcal {D}\left[ \left| \lambda _{j}\right\rangle \left\langle \lambda _{k}\right| \right] \tilde{\rho },\nonumber \end{aligned}$$

(3.113)

where $\Gamma _{k}=\kappa _{k}^{2}\Gamma$ and $\Gamma _{k\rightarrow j}=\Gamma _{k}[1-\left| \kappa _{j}\right| ^{2}]$. Since only $\left( 1,1\right)$ states can be refilled, the rate at which the level $\left| \lambda _{j}\right\rangle$ is populated is proportional to $\sim [1-\left| \kappa _{j}\right| ^{2}]$; compare Ref. [61]. While the first line in Eq. (3.113) models the decay from the dressed energy eigenstates $\left| \lambda _{k}\right\rangle$ back to the Pauli-blocked triplet subspace $\left| T_{\nu }\right\rangle \left( {{\nu }=\pm }\right)$ with an effective rate according to their overlap with the localized singlet, the second line refers to decay and dephasing processes acting entirely within the ‘fast’ subspace spanned by $\left\{ \left| \lambda _{k}\right\rangle \right\}$. Intuitively, they should not affect the nuclear dynamics that take place on a much longer timescale. This intuitive picture is corroborated by exact diagonalization results: Leaving the HF interaction $\mathcal {V}$ aside for the moment, we compare the dynamics $\dot{\rho }=\mathcal {K}_{0}\rho$ generated by the full electronic Liouvillian

$$\begin{aligned} \mathcal {K}_{0}\rho= & {} -i\left[ H_{\mathrm {el}},\rho \right] +\mathcal {K}_{\Gamma }\rho \end{aligned}$$

(3.114)

$$\begin{aligned}&+\,\mathcal {K}_{{{\pm }}}\rho +\mathcal {L}_{\mathrm {deph}}\rho , \nonumber \\ \mathcal {K}_{{{\pm }}}\rho= & {} \Gamma _{\pm }\sum _{\nu =\pm }\mathcal {D}\left[ \left| T_{\bar{\nu }}\right\rangle \left\langle T_{\nu }\right| \right] \rho \\&+\,\Gamma _{\pm }\sum _{\nu =\pm }\left[ \mathcal {D}\left[ \left| T_{\nu }\right\rangle \left\langle T_{0}\right| \right] \rho +\mathcal {D}\left[ \left| T_{0}\right\rangle \left\langle T_{\nu }\right| \right] \rho \right] \nonumber \end{aligned}$$

(3.115)

Fig. 3.17

Electronic asymptotic decay rate $\mathrm {ADR}_{\mathrm {el}}$ and fidelity $\mathcal {F}_{\mathrm {el}}$ for the purely electronic Lindblad dynamics: The results obtained for the full dissipator given in Eq. (3.114) (circles) are in good agreement with the results we get for the simplified description as stated in Eq. (3.116) (squares). The blue and red curves correspond to $\Gamma =25\,\upmu \mathrm {eV}$, $\Gamma _{\pm }=0.25\,\upmu \mathrm {eV}$, $\Gamma _{\mathrm {deph}}=0.5\,\upmu \mathrm {eV}$ and $\Gamma =25\,\upmu \mathrm {eV}$, $\Gamma _{\pm }=0.3\,\upmu \mathrm {eV}$, $\Gamma _{\mathrm {deph}}=0$, respectively. Inset The fidelity $\mathcal {F}_{\mathrm {el}}$ as a figure of merit for the similarity between the quasi-steady-state solutions $\rho _{\mathrm {ss}}^{\mathrm {el}}$ and $\tilde{\rho }_{\mathrm {ss}}^{\mathrm {el}}$, respectively. Other numerical parameters are: $t=20\,\upmu \mathrm {eV}$, $\varepsilon =30\,\upmu \mathrm {eV}$ and $\omega _{0}=0$

formulated in terms of the five undressed, bare levels $\left\{ \left| \sigma ,\sigma '\right\rangle ,\left| S_{02}\right\rangle \right\}$ to the following Liouvillian

$$\begin{aligned} \mathcal {L}_{0}\tilde{\rho }= & {} -i\left[ \tilde{H}_{\mathrm {el}},\tilde{\rho }\right] +\mathcal {L}_{\Gamma }\tilde{\rho }\nonumber \\&+\,\mathcal {L}_{\pm }\tilde{\rho }+\mathcal {L}_{\mathrm {deph}}\tilde{\rho }, \end{aligned}$$

(3.116)

which is based on the simplified form as stated in Eq. (3.113).⁷ Here, we have also disregarded all dissipative processes acting entirely within the fast subspace, that is all terms of the form $\mathcal {D}\left[ \left| \lambda _{j}\right\rangle \left\langle \lambda _{k}\right| \right]$; see the second line in Eq. (3.113). First, as shown in Fig. 3.17, we have checked numerically that both $\mathcal {K}_{0}$ and $\mathcal {L}_{0}$ feature very similar electronic quasisteady states, fulfilling $\mathcal {K}_{0}\left[ \rho _{\mathrm {ss}}^{\mathrm {el}}\right] =0$ and $\mathcal {L}_{0}\left[ \tilde{\rho }_{\mathrm {ss}}^{\mathrm {el}}\right] =0$, respectively, with a Uhlmann fidelity [115] $\mathcal {F}_{\mathrm {el}}\left( \rho _{\mathrm {ss}}^{\mathrm {el}},\tilde{\rho }_{\mathrm {ss}}^{\mathrm {el}}\right) =\left\| \sqrt{\rho _{\mathrm {ss}}^{\mathrm {el}}}\sqrt{\tilde{\rho }_{\mathrm {ss}}^{\mathrm {el}}}\right\| _{\mathrm {tr}}$ exceeding $99\,\%$; here, $\left\| \cdot \right\| _{\mathrm {tr}}$ is the trace norm, the sum of the singular values. Second, we examine the electronic asymptotic decay rate $\mathrm {ADR}_{\mathrm {el}}$, corresponding to the eigenvalue with the largest real part different from zero, which quantifies the typical timescale on which the electronic subsystem reaches its quasi-steady state [70]. In other words, the $\mathrm {ADR}_{\mathrm {el}}$ gives the spectral gap of the electronic Liouvillian $\mathcal {K}_{0}\left( \mathcal {L}_{0}\right)$ setting the inverse relaxation time towards the steady state and therefore characterizes the long-time behaviour of the electronic system. The two models produce very similar results: Depending on the particular choice of parameters, the electronic $\mathrm {ADR}_{\mathrm {el}}$ is set either by the eigenvectors $\left| \lambda _{2}\right\rangle \left\langle T_{\pm }\right|$, $\left| T_{+}\right\rangle \left\langle T_{-}\right|$ and $\left| T_{+}\right\rangle \left\langle T_{+}\right| -\left| T_{-}\right\rangle \left\langle T_{-}\right|$ which explains the kinks observed in Fig. 3.17 as changes of the eigenvectors determining the $\mathrm {ADR}_{\mathrm {el}}$. In summary, both the electronic quasisteady state $\left( \rho _{\mathrm {ss}}^{\mathrm {el}}\approx \tilde{\rho }_{\mathrm {ss}}^{\mathrm {el}}\right)$ and the electronic asymptotic decay rate $\mathrm {ADR}_{\mathrm {el}}$ are well captured by the approximative Liouvillian given in Eq. (3.116). Further arguments justifying this approximation are provided in Appendix 3.11.3.

3.1.3 3.11.3 Transport-Mediated Transitions in Fast Electronic Subspace

In this Appendix, we provide analytical arguments why one can drop the second line in Eq. (3.113) and keep only the first one to account for a description of electron transport in the eigenbasis of $H_{\mathrm {el}}$. The second line, given by

$$\begin{aligned} \mathcal {L}_{\mathrm {fast}}\rho =\sum _{k,j}\Gamma _{k\rightarrow j}\mathcal {D}\left[ \left| \lambda _{j}\right\rangle \left\langle \lambda _{k}\right| \right] \rho , \end{aligned}$$

(3.117)

describes transport-mediated transitions in the fast subspace $\left\{ \left| \lambda _{k}\right\rangle \right\}$. The transition rate $\Gamma _{k\rightarrow j}=\kappa _{k}^{2}\left[ 1-\kappa _{j}^{2}\right] \Gamma$ refers to a transport-mediated decay process from $\left| \lambda _{k}\right\rangle$ to $\left| \lambda _{j}\right\rangle$. Here, we show that $\mathcal {L}_{\mathrm {fast}}$ simply amounts to an effective dephasing mechanism which can be absorbed into a redefinition of the effective transport rate $\Gamma$.

The only way our model is affected by $\mathcal {L}_{\mathrm {fast}}$ is that it adds another dephasing channel for the coherences $\left| \lambda _{k}\right\rangle \left\langle T_{\pm }\right|$ which are created by the hyperfine flip-flop dynamics; see Appendix 3.11.9. In fact, we have

$$\begin{aligned} \mathcal {L}_{\mathrm {fast}}\left[ \left| \lambda _{k}\right\rangle \left\langle T_{\pm }\right| \right]= & {} -\Gamma _{\mathrm {fast},k}\left| \lambda _{k}\right\rangle \left\langle T_{\pm }\right| ,\end{aligned}$$

(3.118)

$$\begin{aligned} \Gamma _{\mathrm {fast},k}= & {} \frac{1}{2}\sum _{j}\Gamma _{k\rightarrow j}. \end{aligned}$$

(3.119)

Due to the normalization condition $\sum _{j}\kappa _{j}^{2}=1$, the new effective dephasing rate $\Gamma _{\mathrm {fast},k}$ is readily found to coincide with the effective transport rate $\Gamma _{k}$, that is $\Gamma _{\mathrm {fast},k}=\Gamma _{k}=\kappa _{k}^{2}\Gamma$. This equality is readily understood since all four $\left( 1,1\right)$ levels are populated equally. While $\Gamma _{k}$ describes the decay to the two Pauli-blocked triplet levels, $\Gamma _{\mathrm {fast},k}$ accounts for the remaining transitions within the $\left( 1,1\right)$ sector. Therefore, when accounting for $\mathcal {L}_{\mathrm {fast}}$, the total effective dephasing rates $\tilde{\Gamma }_{k}$ needs to be modified as $\tilde{\Gamma }_{k}\rightarrow \tilde{\Gamma }_{k}+\Gamma _{k}=2\Gamma _{k}+3\Gamma _{\pm }+\Gamma _{\mathrm {deph}}/4$. The factor of 2 is readily absorbed into our model by a simple redefinition of the overall transport rate $\Gamma \rightarrow 2\Gamma$.

3.1.4 3.11.4 Electronic Lifting of Pauli-Blockade

This Appendix provides a detailed analysis of purely electronic mechanisms which can lift the Pauli-blockade without affecting directly the nuclear spins. Apart from cotunneling processes discussed in the main text, here we analyze virtual spin exchange processes and spin-orbital effects [32, 47]. It is shown, that these mechanisms, though microscopically distinct, phenomenologically amount to effective incoherent mixing and pure dephasing processes within the (1, 1) subspace which, for the sake of theoretical generality, are subsumed under the term in Eq. (3.1).

Let us also note that electron spin resonance (ESR) techniques in combination with dephasing could be treated on a similar footing. As recently shown in Ref. [37], in the presence of a gradient $\Delta$, ESR techniques can be used to drive the electronic system into the entangled steady state $\left| -\right\rangle =\left( \left| T_{+}\right\rangle -\left| T_{-}\right\rangle \right) /\sqrt{2}$. Magnetic noise may then be employed to engineer the desired electronic quasisteady state.

Spin Exchange with the Leads

In the Pauli-blockade regime the $\left( 1,1\right)$ triplet states $\left| T_{\pm }\right\rangle$ do not decay directly, but—apart from the cotunneling processes described in the main text—they may exchange electrons with the reservoirs in the leads via higher-order virtual processes [32, 47]. We now turn to these virtual, spin-exchange processes which can be analyzed along the lines of the interdot cotunneling effects. Again, for concreteness we fix the initial state of the DQD to be $\left| T_{+}\right\rangle$ and, based on the approximation of independent rates of variation [60], explain the physics in terms of the electronic bare states. The spin-blocked level $\left| T_{+}\right\rangle$ can virtually exchange an electron spin with the left lead yielding an incoherent coupling with the state $\left| \Downarrow \Uparrow \right\rangle$; this process is mediated by the intermediate singly occupied DQD level $\left| 0,\Uparrow \right\rangle$ where no electron resides on the left dot. Then, from $\left| \Downarrow \Uparrow \right\rangle$ the system may decay back to the $\left( 1,1\right)$ subspace via the localized singlet $\left| S_{02}\right\rangle$. Therefore, for this analysis, in Fig. 3.4 we simply have to replace $\left| T_{+}(0,2)\right\rangle$ and $\Gamma _{\mathrm {ct}}$ by $\left| 0,\Uparrow \right\rangle$ and $\Gamma _{\mathrm {se}}$, respectively. Along the lines of our previous analysis of cotunneling within the DQD, the bottleneck of the overall process is set by the first step, labeled as $\Gamma _{\mathrm {se}}$. The main purpose of this Appendix is an estimate for the rate $\Gamma _{\mathrm {se}}$.

The effective spin-exchange rate can be calculated in a “golden rule” approach in which transitions for different initial and final reservoir states are weighted according to the respective Fermi distribution functions and added incoherently [116]; for more details, see Refs. [117, 118]. Up to second order in $H_{T}$, the cotunneling rate $\Gamma _{\mathrm {se}}$ for the process $\left| T_{+}\right\rangle \rightsquigarrow \left| \Downarrow \Uparrow \right\rangle$ is then found to be

$$\begin{aligned} \Gamma _{\mathrm {se}}= & {} \,2\pi n_{L}^{2}\left| T_{L}\right| ^{4}\int _{\mu _{L}}^{\mu _{L}+\Delta }d\varepsilon \frac{1}{\left( \varepsilon -\delta _{+}\right) ^{2}}\nonumber \\\approx & {} \frac{\Gamma _{L}^{2}}{2\pi }\frac{\Delta }{\left( \mu _{L}-\delta _{+}\right) ^{2}}. \end{aligned}$$

(3.120)

Here, $n_{L}$ is the left lead density of states at the Fermi energy, $\mu _{L}$ is the chemical potential of the left lead, $\Delta =E_{T_{+}}-E_{\Downarrow \Uparrow }$ is the energy released on the DQD (which gets absorbed by the reservoir) and $\delta _{+}=E_{T_{+}}-E_{0\Uparrow }={{\varepsilon }}_{L\uparrow }+U_{LR}$ refers to the energy difference between a doubly and singly occupied DQD in the intermediate virtual state. Moreover, $\Gamma _{L}$ refers to the first-order sequential tunneling rates $\Gamma _{L}=2\pi n_{L}\left| T_{L}\right| ^{2}$ for the left $\left( L\right)$ lead. Note that in the limit $T\rightarrow 0$ the DQD cannot be excited; accordingly, for $\Delta >0$, the transition $\left| T_{\pm }\right\rangle \rightsquigarrow \left| \Uparrow \Downarrow \right\rangle$ is forbidden due to energy conservation [64]. As expected, $\Gamma _{\mathrm {se}}$ is proportional to $\sim \left| T_{L}\right| ^{4}$, but suppressed by the energy penalty $\Delta _{\mathrm {se}}^{+}=\mu _{L}-\delta _{+}$ which characterizes the violation of the two-electron condition in Eq. (3.102) in the virtual intermediate step. Notably, this can easily be tuned electrostatically via the chemical potential $\mu _{L}$. Comparing the parameter dependence $\Gamma _{\mathrm {se}}\sim \left| T_{L}\right| ^{4}$ to $\Gamma _{\mathrm {ct}}\sim t^{2}\left| T_{L}\right| ^{2}$ shows that, in contrast to the cotunneling processes $\Gamma _{\mathrm {ct}}$, $\Gamma _{\mathrm {se}}$ is independent of the interdot tunneling parameter t. Moreover, it can be made efficient by tuning properly the energy penalty $\Delta _{\mathrm {se}}^{+}$ and the tunnel coupling to the reservoir $T_{L}$. A similar analysis can be carried out for example for the effective decay process $\left| T_{-}\right\rangle \rightsquigarrow \left| \Downarrow \Uparrow \right\rangle$ by spin-exchange with the right reservoir. The corresponding rates are the same if $\Gamma _{L}/\Delta _{\mathrm {se}}^{+}=\Gamma _{R}/\Delta _{\mathrm {se}}^{-}$, where $\Delta _{{{\mathrm {se}}}}^{-}=\mu _{R}-\left( \varepsilon _{R\downarrow }+U_{LR}\right)$, is satisfied. Taking the energy penalty as $\Delta _{\mathrm {se}}\approx \Delta _{\mathrm {st}}$, a comparison of $\Gamma _{\mathrm {se}}$ to interdot cotunneling transitions (as discussed in the main text) gives $\Gamma _{\mathrm {ct}}/\Gamma _{\mathrm {se}}\approx 2\pi t^2/(\Gamma \Delta )$. Thus, for $\Gamma \approx 2\pi t$ and $t\approx \Delta$ (as considered in this work), we get approximately $\Gamma _{\mathrm {ct}}\approx \Gamma _{\mathrm {se}}$.

The effective spin-exchange rate $\Gamma _{\mathrm {se}}$ can be made very efficient in the high gradient regime. For example, to obtain $\Gamma _{\mathrm {se}}\approx 1\,\upmu \mathrm {eV}$ when $\Delta \approx 40\,\upmu \mathrm {eV}$, we estimate the required characteristic energy penalty to be $\Delta _{\mathrm {se}}\approx 200\,\upmu \mathrm {eV}$. As stated in the main text, for an energy penalty of $\sim \!500\,\upmu \mathrm {eV}$ and for $\Gamma _{L}\approx 100\,\upmu \mathrm {eV}$, we estimate $\Gamma _{\mathrm {se}}\approx 0.25\,\upmu \mathrm {eV}$, making $\Gamma _{\mathrm {se}}$ fast compared to typical nuclear timescales; note that for less transparent barriers with $\Gamma _{L}\approx 1\,\upmu \mathrm {eV}$, $\Gamma _{\mathrm {se}}$ is four orders of magnitude smaller, in agreement with values given in Ref. [47]. Moreover, as apparent from Eq. (3.120), in the low gradient regime $\Gamma _{\mathrm {se}}\sim \Delta$ is suppressed due to a vanishing phase space of reservoir electrons that can contribute to this process without violating energy conservation. To remedy this, one can lower the energy penalty $\Delta _{\mathrm {se}}$; however, if $\Delta _{\mathrm {se}}$ becomes comparable to $\Gamma$, this leads to a violation of the Markov approximation and tunes the system away from the sequential tunneling regime. Note that the factor $\Delta$ appears in Eq. (3.120) as we consider explicitly the inelastic transition $\left| T_{+}\right\rangle \rightsquigarrow \left| \Downarrow \Uparrow \right\rangle$. In a more general analysis, $\Delta$ should be replaced by the energy separation $\Delta E$ (which is released by the DQD into the reservoir) for the particular transition at hand [64].

Here, we have considered spin-exchange via singly-occupied levels in the virtual intermediate stage only; they are detuned by the characteristic energy penalty $\delta =\left| \mu _{i}-\left( \varepsilon _{i}+U_{LR}\right) \right|$ for $i=L,R$. In principle, spin exchange with the leads can also occur via electronic levels with $\left( 1,2\right)$ or $\left( 2,1\right)$ charge configuration. However, here the characteristic energy penalty can be estimated as $\delta =\left| \varepsilon _{i}+U_{i}+U_{LR}-\mu _{i}\right|$ which can be significantly bigger due to the appearance of the on-site Coulomb energies $U_{i}$ in this expression. Therefore, they have been disregarded in the analysis above.

Spin Orbit Interaction

For the triplet states $\left| T_{\pm }\right\rangle$ interdot tunneling is suppressed due to Pauli spin blockade, but—apart from HF interaction with the nuclear spins—it can be mediated by spin-orbit interaction which does not conserve the electronic spin. In contrast to hyperfine mediated lifting of the spin blockade, spin-orbital effects provide another purely electronic alternative to escape the spin blockade, i.e., without affecting the nuclear spins. They describe interdot hopping accompanied by a spin rotation thereby coupling the triplet states $\left| T_{\pm }\right\rangle$ with single occupation of each dot to the singlet state $\left| S_{02}\right\rangle$ with double occupation of the right dot. Therefore, following Refs. [62, 111, 113, 119, 120], spin-orbital effects can be described phenomenologically in terms of the Hamiltonian

$$\begin{aligned} H_{\mathrm {so}}=t_{\mathrm {so}}\left( \left| T_{+}\right\rangle \left\langle S_{02}\right| +\left| T_{-}\right\rangle \left\langle S_{02}\right| +\mathrm {h.c.}\right) , \end{aligned}$$

(3.121)

where the coupling parameter $t_{\mathrm {so}}$ in general depends on the orientation of the the DQD with respect to the crystallographic axes. Typical values of $t_{\mathrm {so}}$ can be estimated as $t_{\mathrm {so}}\approx \left( d/l_{\mathrm {so}}\right) t$, where t is the usual spin-conserving tunnel coupling, d the interdot distance and $l_{\mathrm {so}}$ the material-specific spin-orbit length ($l_{\mathrm {so}}\approx 1{-}10\,\upmu \mathrm {m}$ for GaAs); this estimate is in good agreement with the exact equation given in Ref. [113] and yields $t_{\mathrm {so}}\approx \left( 0.01{-}0.1\right) t$.

In Eq. (3.121) we have disregarded the spin-orbit coupling for the triplet $\left| T_{0}\right\rangle =\frac{1}{\sqrt{2}}(\left| \Uparrow \Downarrow \right\rangle$ $+\left| \Downarrow \Uparrow \right\rangle )$. It may be taken into account by introducing the modified interdot tunneling Hamiltonian $H_{t}\rightarrow H'_{t}$ with $H'_{t}=t_{\uparrow \downarrow }\left| \Uparrow \Downarrow \right\rangle \left\langle S_{02}\right| -t_{\downarrow \uparrow }\left| \Downarrow \Uparrow \right\rangle \left\langle S_{02}\right| +\mathrm {h.c.},$ where the tunneling parameters $t_{\uparrow \downarrow }$ and $t_{\downarrow \uparrow }$ are approximately given by $t_{\uparrow \downarrow \left( \downarrow \uparrow \right) }=t\pm t_{\mathrm {so}}/\sqrt{2}\approx t,$ since the second term marks only a small modification of the order of 5 %. While $\left| T_{0}\right\rangle$ is dark under tunneling in the singlet subspace, that is $H_{t}\left| T_{0}\right\rangle =0$, similarly the slightly modified (unnormalized) state $\left| T_{0}'\right\rangle =t_{\downarrow \uparrow }\left| \Uparrow \Downarrow \right\rangle +t_{\uparrow \downarrow }\left| \Downarrow \Uparrow \right\rangle$ is dark under $H'_{t}$. Since this effect does not lead to any qualitative changes, it is disregarded.

Fig. 3.18

Phenomenological treatment of spin-orbital effects in the spin-blockade regime. Scheme of the simplified electronic system: The triplet states $\left| T_{\pm }\right\rangle$ are coherently coupled to the local singlet $\left| S_{02}\right\rangle$ by spin-orbit interaction. Via coupling to the leads, the DQD is discharged and recharged again with an effective rate $\Gamma$. The triplet states may experience a Zeeman splitting $\omega _{0}$. The parameter $\varepsilon$ specifies the interdot energy offset. Since $\Gamma ,\varepsilon \gg t_{\mathrm {so}}$, the local singlet $\left| S_{02}\right\rangle$ can be eliminated adiabatically yielding effective dissipative processes of strength $\Gamma _{\mathrm {so}}$ (green dashed arrows)

Phenomenological treatment.—In the following, we first focus on the effects generated by $H_{\mathrm {so}}$ within the three-level subspace $\left\{ \left| T_{\pm }\right\rangle ,\left| S_{02}\right\rangle \right\}$. Within this reduced level scheme, the dynamics $\dot{\rho }=\mathcal {L}_{\mathrm {rd}}\rho$ are governed by the Liouvillian

$$\begin{aligned} \mathcal {L}_{\mathrm {rd}}\rho= & {} -i\left[ \mathcal {H}_{\mathrm {rd}},\rho \right] +\Gamma \sum _{\nu =\pm }\mathcal {D}\left[ \left| T_{\nu }\right\rangle \left\langle S_{02}\right| \right] \rho \end{aligned}$$

(3.122)

where the relevant Hamiltonian within this subspace is

$$\begin{aligned} \mathcal {H}_{\mathrm {rd}}=\omega _{0}\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| -\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) -\varepsilon \left| S_{02}\right\rangle \left\langle S_{02}\right| +H_{\mathrm {so}}. \end{aligned}$$

(3.123)

This situation is schematized in Fig. 3.18. The external Zeeman splitting $\omega _{0}$ is assumed to be small compared to the interdot detuning $\varepsilon$ yielding approximately equal detunings between the triplet states $\left| T_{\pm }\right\rangle$ and $\left| S_{02}\right\rangle$. In particular, we consider the regime $t_{\mathrm {so}}\ll \varepsilon ,\Gamma$, with the corresponding separation of timescales allowing for an alternative, effective description of spin-orbital effects. Since the short-lived singlet state $\left| S_{02}\right\rangle$ is populated negligibly throughout the dynamics, it can be eliminated adiabatically using standard techniques. The symmetric superposition $\left| -\right\rangle =\left( \left| T_{+}\right\rangle -\left| T_{-}\right\rangle \right) /\sqrt{2}$ is a dark state with respect to the spin-orbit Hamiltonian $H_{\mathrm {so}}$. Therefore, it is instructive to formulate the resulting effective master equation in terms of the symmetric superposition states $\left| \pm \right\rangle =\left( \left| T_{+}\right\rangle \pm \left| T_{-}\right\rangle \right) /\sqrt{2}$. Within the two-dimensional subspace spanned by the symmetric superpositions $\left| \pm \right\rangle$, the effective dynamics is given by

$$\begin{aligned} \dot{\rho }= & {} +\,i\omega _{0}\left[ \left| -\right\rangle \left\langle +\right| +\left| +\right\rangle \left\langle -\right| ,\rho \right] \\&-\,i\Omega _{\mathrm {so}}\left[ \left| +\right\rangle \left\langle +\right| -\left| -\right\rangle \left\langle -\right| ,\rho \right] \nonumber \\&+\,2\Gamma _{\mathrm {so}}\mathcal {D}\left[ \left| -\right\rangle \left\langle +\right| \right] \rho \nonumber \\&+\,\frac{\Gamma _{\mathrm {so}}}{2}\mathcal {D}\left[ \left| +\right\rangle \left\langle +\right| -\left| -\right\rangle \left\langle -\right| \right] \rho ,\nonumber \end{aligned}$$

(3.124)

where the effective rate

$$\begin{aligned} \Gamma _{\mathrm {so}}=\frac{t_{\mathrm {so}}^{2}}{\varepsilon ^{2}+\Gamma ^{2}}\Gamma \end{aligned}$$

(3.125)

governs decay as well as pure dephasing processes within the triplet subspace. We estimate $\Gamma _{\mathrm {so}}\approx (0.2-0.3)\,\upmu \mathrm {eV}$ which is still fast compared to typical nuclear timescales. In Eq. (3.124) we have also introduced the quantity $\Omega _{\mathrm {so}}=\left( \varepsilon /\Gamma \right) \Gamma _{\mathrm {so}}$. As we are particularly concerned with the nuclear dynamics in the limit where one can eliminate the electronic degrees of freedom, Eq. (3.124) provides an alternative way of accounting for spin-orbital effects: In Eq. (3.124) we encounter a decay term—see the third line in Eq. (3.124)—which pumps the electronic subsystem towards the dark state of the spin-orbit Hamiltonian, namely the state $\left| -\right\rangle$. This state is also dark under the Stark shift and pure dephasing terms in the second and last line of Eq. (3.124), respectively. However, by applying an external magnetic field, the state $\left| -\right\rangle$ dephases due to the induced Zeeman splitting $\omega _{0}$. This becomes apparent when examining the electronic quasisteady state corresponding to the evolution given in Eq. (3.124). In the basis $\left\{ \left| T_{+}\right\rangle ,{\left| T_{-}\right\rangle }\right\}$, it is found to be

$$\begin{aligned} \rho _{\mathrm {ss}}^{\mathrm {el}}=\left( \begin{array}{cc} \frac{1}{2}\left[ 1+\frac{\omega _{0}\Omega _{\mathrm {so}}}{\omega _{0}^{2}+\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}}\right] &{} -\frac{\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}+i\Gamma _{\mathrm {so}}\omega _{0}}{2\left( \omega _{0}^{2}+\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}\right) }\\ -\frac{\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}-i\Gamma _{\mathrm {so}}\omega _{0}}{2\left( \omega _{0}^{2}+\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}\right) } &{} \frac{1}{2}\left[ 1-\frac{\omega _{0}\Omega _{\mathrm {so}}}{\omega _{0}^{2}+\Gamma _{\mathrm {so}}^{2}+\Omega _{\mathrm {so}}^{2}}\right] \end{array}\right) , \end{aligned}$$

(3.126)

which in leading orders of $\omega _{0}^{-1}$ reduces to

$$\begin{aligned} \rho _{\mathrm {ss}}^{\mathrm {el}}\approx \left( \begin{array}{cc} \frac{1}{2}+\frac{\Omega _{\mathrm {so}}}{2\omega _{0}} &{} -i\frac{\Gamma _{\mathrm {so}}}{2\omega _{0}}\\ i\frac{\Gamma _{\mathrm {so}}}{2\omega _{0}} &{} \frac{1}{2}-\frac{\Omega _{\mathrm {so}}}{2\omega _{0}} \end{array}\right) . \end{aligned}$$

(3.127)

Accordingly, for sufficiently large Zeeman splitting $\omega _{0}\gg \Omega _{\mathrm {so}},\Gamma _{\mathrm {so}}$, the electronic subsystem is driven towards the desired equal mixture of blocked triplet states $\left| T_{+}\right\rangle$ and $\left| T_{-}\right\rangle$. Alternatively, the off-diagonal elements of $\left| -\right\rangle \left\langle -\right|$ are damped out in the presence of dephasing processes either mediated intrinsically via cotunneling processes or extrinsically via engineered magnetic noise yielding approximately the equal mixture $\rho _{\mathrm {target}}^{\mathrm {el}}=\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| +\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) /2$ in the quasisteady state.

Fig. 3.19

Electronic quasi-steady-state fidelities in the presence of spin-orbit coupling for the dynamics generated by $\tilde{\mathcal {K}}_{0}$ as a function of the gradient $\Delta$. As expected, in the absence of dephasing [$\Gamma _{\mathrm {deph}}=0$ (black curve)], the system settles into the dark state $\left| -\right\rangle$. For $\Gamma _{\mathrm {deph}}=1\,\upmu \mathrm {eV}$ (blue and red curves), the off-diagonal elements of $\left| -\right\rangle$ are strongly suppressed, leading to a high fidelity $\mathcal {F}_{\mathrm {so}}\gtrsim 0.9$ with the desired mixed state $\rho _{\mathrm {target}}^{\mathrm {el}}$ in the high gradient regime: the blue and red curve refer to $t=20\,\upmu \mathrm {eV}$ and $t=30\,\upmu \mathrm {eV}$, respectively. Other numerical parameters are: $t_{\mathrm {so}}=0.1t$, $\omega _{0}=0$, $\Gamma =25\,\upmu \mathrm {eV}$ and $\varepsilon =30\,\upmu \mathrm {eV}$

Numerical analysis.—To complement the perturbative, analytical study, we carry out a numerical evaluation of the electronic quasisteady state in the presence of spin-orbit coupling. In the two-electron subspace, the corresponding master equation (including spin-orbital effects) under consideration reads

$$\begin{aligned} \dot{\rho }=\tilde{\mathcal {K}}_{0}\rho =-i\left[ H_{\mathrm {el}}+H_{\mathrm {so}},\rho \right] +\mathcal {K}_{\Gamma }\rho +\mathcal {L}_{\mathrm {deph}}\rho . \end{aligned}$$

(3.128)

We evaluate the exact electronic quasisteady state ${{\rho _{\mathrm {ss}}^{\mathrm {el}}}}$ fulfilling $\tilde{\mathcal {K}}_{0}\rho _{\mathrm {ss}}^{\mathrm {el}}=0$. As a figure of merit, we compute the Uhlmann fidelity [115]

$$\begin{aligned} \mathcal {F}_{{{\mathrm {so}}}}=\mathrm {tr}\left[ \left( \sqrt{\rho _{\mathrm {ss}}^{\mathrm {el}}}\rho _{\mathrm {target}}^{\mathrm {el}}\sqrt{\rho _{\mathrm {ss}}^{\mathrm {el}}}\right) ^{1/2}\right] ^{2} \end{aligned}$$

(3.129)

which measures how similar ${{\rho _{\mathrm {ss}}^{\mathrm {el}}}}$ and $\rho _{\mathrm {target}}^{\mathrm {el}}$ are. The results are illustrated in Fig. 3.19: For $\Gamma _{\mathrm {deph}}=0$ the electronic system settles into the pure dark state $\left| -\right\rangle \left\langle -\right|$. However, in the presence of dephasing, the coherences are efficiently damped out. In the low-gradient regime $\rho _{\mathrm {ss}}^{\mathrm {el}}$ has a significant overlap with the triplet $\left| T_{0}\right\rangle$, whereas in the high-gradient regime it is indeed approximately given by the desired mixed target state $\rho _{\mathrm {target}}^{\mathrm {el}}$. Lastly, we have checked that in the high-gradient regime the corresponding asymptotic decay rate can be approximated very well by $\mathrm {ADR}_{\mathrm {el}}\approx -2\Gamma _{\mathrm {so}}$.

3.1.5 3.11.5 Effective Nuclear Master Equation

In this Appendix, we present a detailed derivation of the effective nuclear dynamics presented in Sect. 3.5. We use standard adiabatic elimination techniques to derive an effective simplified description of the dynamics. To do so, we assume that electronic coherences decay quickly on typical nuclear timescales. Conservatively, i.e. not taking into account the detuning of the HF-mediated transitions, this holds for $2\Gamma _{\pm }+\Gamma _{\mathrm {deph}}/4\gg g_{\mathrm {hf}}$, where $g_{\mathrm {hf}}$ quantifies the typical HF interaction strength. Alternatively, one may use a projection-operator based technique [46, 70]; this is done in detail in Appendix 3.11.9 for the high-gradient regime where $\rho _{\mathrm {ss}}^{\mathrm {el}}=\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| +\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) /2$, but a generalization for the electronic quasisteady state in Eq. (3.33) is straightforward.

Throughout this Appendix, for convenience we adopt the following notation: $\left| a\right\rangle =\left| T_{+}\right\rangle$, $\left| b\right\rangle =\left| \lambda _{2}\right\rangle$, $\left| c\right\rangle =\left| T_{-}\right\rangle$, $L=L_{2}$, $\mathbb {L}=\mathbb {L}_{2}$ and $\mathcal {D}\left[ c\right] \rho =\mathcal {D}_{c}\rho$. Within this simplified three-level model system, the flip-flop Hamiltonian $H_{\mathrm {ff}}$ reads

$$\begin{aligned} H_{\mathrm {ff}}=\frac{a_{\mathrm {hf}}}{2}\left[ L\left| b\right\rangle \left\langle a\right| +\mathbb {L}\left| b\right\rangle \left\langle c\right| +\mathrm {h.c.}\right] . \end{aligned}$$

(3.130)

For simplicity, we assume $\omega _{0}=0$ and neglect nuclear fluctuations arising from $H_{\mathrm {zz}}$. This approximation is in line with the semiclassical approximation for studying the nuclear polarization dynamics; for more details also see Appendix 3.11.6. Within this reduced scheme, the dynamics are then described by the Master equation

$$\begin{aligned} \dot{\rho }= & {} -i\left[ H_{\mathrm {ff}},\rho \right] -i\varepsilon _{2}\left[ \left| b\right\rangle \left\langle b\right| ,\rho \right] +\frac{\Gamma _{\mathrm {deph}}}{2}\mathcal {D}_{\left| a\right\rangle \left\langle a\right| -\left| c\right\rangle \left\langle c\right| }\rho \nonumber \\&+\,\Gamma _{\pm }\left[ \mathcal {D}_{\left| c\right\rangle \left\langle a\right| }\rho +\mathcal {D}_{\left| a\right\rangle \left\langle c\right| }\rho +\mathcal {D}_{\left| b\right\rangle \left\langle a\right| }\rho +\mathcal {D}_{\left| b\right\rangle \left\langle c\right| }\rho \right] \nonumber \\&+\left( \Gamma _{\pm }+\Gamma _{2}\right) \left[ \mathcal {D}_{\left| a\right\rangle \left\langle b\right| }\rho +\mathcal {D}_{\left| c\right\rangle \left\langle b\right| }\rho \right] . \end{aligned}$$

(3.131)

After adiabatic elimination of the electronic coherences $\rho _{ab}=\left\langle a|\rho |b\right\rangle$, $\rho _{cb}$ and $\rho _{ac}$ we obtain effective equations of motion for the system’s density matrix projected onto the electronic levels $\left| a\right\rangle$, $\left| b\right\rangle$ and $\left| c\right\rangle$ as follows

$$\begin{aligned} \dot{\rho }_{aa}= & {} \Gamma _{\pm }\left( \rho _{cc}-\rho _{aa}\right) +\Gamma _{\pm }\left( \rho _{bb}-\rho _{aa}\right) +\Gamma _{2}\rho _{bb}\end{aligned}$$

(3.132)

$$\begin{aligned}&+\,\gamma \left[ L^{\dagger }\rho _{bb}L-\frac{1}{2}\left\{ L^{\dagger }L,\rho _{aa}\right\} \right] \nonumber \\&+\,i\delta \left[ L^{\dagger }L,\rho _{aa}\right] ,\nonumber \\ \dot{\rho }_{cc}= & {} \Gamma _{\pm }\left( \rho _{aa}-\rho _{cc}\right) +\Gamma _{\pm }\left( \rho _{bb}-\rho _{cc}\right) +\Gamma _{2}\rho _{bb}\\&+\,\gamma \left[ \mathbb {L}^{\dagger }\rho _{bb}\mathbb {L}-\frac{1}{2}\left\{ \mathbb {L}^{\dagger }\mathbb {L},\rho _{cc}\right\} \right] \nonumber \\&+\,i\delta \left[ \mathbb {L}^{\dagger }\mathbb {L},\rho _{cc}\right] ,\nonumber \end{aligned}$$

(3.133)

and

$$\begin{aligned} \dot{\rho }_{bb}= & {} -2\Gamma _{2}\rho _{bb}+\gamma \left[ L\rho _{aa}L^{\dagger }-\frac{1}{2}\left\{ LL^{\dagger },\rho _{bb}\right\} \right] \\&-\,i\delta \left[ LL^{\dagger },\rho _{bb}\right] \nonumber \\&+\,\gamma \left[ \mathbb {L}\rho _{cc}\mathbb {L^{\dagger }}-\frac{1}{2}\left\{ \mathbb {L}\mathbb {L^{\dagger }},\rho _{bb}\right\} \right] -i\delta \left[ \mathbb {L}\mathbb {L^{\dagger }},\rho _{bb}\right] .\nonumber \\&+\,\Gamma _{\pm }\left( \rho _{aa}+\rho _{cc}-2\rho _{bb}\right) .\nonumber \end{aligned}$$

(3.134)

Since this set of equations is entirely expressed in terms of $\rho _{aa}$, $\rho _{bb}$ and $\rho _{cc}$, the full density matrix of the system obeys a simple block structure, given by

$$\begin{aligned} \rho =\rho _{aa}\left| a\right\rangle \left\langle a\right| +\rho _{bb}\left| b\right\rangle \left\langle b\right| +\rho _{cc}\left| c\right\rangle \left\langle c\right| . \end{aligned}$$

(3.135)

Therefore, the electronic decoherence is fast enough to prevent the entanglement between electronic and nuclear degrees of freedom and the total density matrix of the system $\rho$ factorizes into a tensor product for the electronic and nuclear subsystem [33], respectively, that is $\rho =\rho _{\mathrm {el}}\otimes \sigma ,$ where $\sigma =\mathsf {Tr}_{\mathrm {el}}\left[ \rho \right]$ refers to the density matrix of the nuclear subsystem. This ansatz agrees with the projection operator approach where $\mathcal {P}\rho =\sigma \otimes \rho _{\mathrm {el}}$ and readily yields $\rho _{aa}=p_{a}\sigma$, where we have introduced the electronic populations

$$\begin{aligned} p_{a}=\left\langle a|\rho _{\mathrm {el}}|a\right\rangle =\mathsf {Tr}_{\mathrm {n}}\left[ \rho _{aa}\right] , \end{aligned}$$

(3.136)

and accordingly for $p_{b}$ and $p_{c}$; here, $\mathsf {Tr}_{\mathrm {n}}\left[ \dots \right]$ denotes the trace over the nuclear degrees of freedom. With these definitions, Eqs. (3.132), (3.133) and (3.134) can be rewritten as

$$\begin{aligned} \dot{p}_{a}= & {} \, \Gamma _{\pm }\left( p_{c}-p_{a}\right) +\Gamma _{2}p_{b}+\gamma \left[ p_{b}\left\langle LL^{\dagger }\right\rangle -p_{a}\left\langle L^{\dagger }L\right\rangle \right] \nonumber \\&+\,\Gamma _{\pm }\left( p_{b}-p_{a}\right) ,\nonumber \\ \dot{p}_{c}= & {} \,\Gamma _{\pm }\left( p_{a}-p_{c}\right) +\Gamma _{2}p_{b}+\gamma \left[ p_{b}\left\langle \mathbb {L}\mathbb {L}^{\dagger }\right\rangle -p_{c}\left\langle \mathbb {L}^{\dagger }\mathbb {L}\right\rangle \right] \nonumber \\&+\,\Gamma _{\pm }\left( p_{b}-p_{c}\right) ,\nonumber \\ \dot{p}_{b}= & {} -2\Gamma _{2}p_{b}+\Gamma _{\pm }\left( p_{a}+p_{c}-2p_{b}\right) \nonumber \\&+\,\gamma \left[ p_{a}\left\langle L^{\dagger }L\right\rangle -p_{b}\left\langle LL^{\dagger }\right\rangle +p_{c}\left\langle \mathbb {L}^{\dagger }\mathbb {L}\right\rangle -p_{b}\left\langle \mathbb {L}\mathbb {L}^{\dagger }\right\rangle \right] . \end{aligned}$$

(3.137)

Similarly, the effective Master equation for the nuclear density matrix $\sigma =\mathsf {Tr}_{\mathrm {el}}\left[ \rho \right]$ is obtained from $\dot{\sigma }=\mathsf {Tr}_{\mathrm {el}}\left[ \dot{\rho }\right] =\dot{\rho }_{aa}+\dot{\rho }_{bb}+\dot{\rho }_{cc}$, leading to

$$\begin{aligned} \dot{\sigma }= & {} \,\gamma \left\{ p_{b}\mathcal {D}_{L^{\dagger }}\left[ {{\sigma }}\right] +p_{b}\mathcal {D}_{\mathbb {L}^{\dagger }}\left[ \sigma \right] +p_{a}\mathcal {D}_{L}\left[ \sigma \right] +p_{c}\mathcal {D}_{\mathbb {L}}\left[ \sigma \right] \right\} \nonumber \\&+\,i\delta \left\{ p_{a}\left[ L^{\dagger }L,\sigma \right] +p_{c}\left[ \mathbb {L}^{\dagger }\mathbb {L},\sigma \right] \right. \nonumber \\&\left. -p_{b}\left[ LL^{\dagger },\sigma \right] -p_{b}\left[ \mathbb {L}\mathbb {L}^{\dagger },\sigma \right] \right\} . \end{aligned}$$

(3.138)

Equation (3.138) along with Eq. (3.137) describe the coupled electron-nuclear dynamics on a coarse-grained timescale that is long compared to electronic coherence timescales. Due to the normalization condition $p_{a}+p_{b}+p_{c}=1$, this set of dynamical equations comprises three coupled equations. Differences in the populations of the levels $\left| a\right\rangle$ and $\left| c\right\rangle$ decay very quickly on timescales relevant for the nuclear evolution; that is,

$$\begin{aligned} \dot{p}_{a}-\dot{p}_{c}= & {} -3\Gamma _{\pm }\left( p_{a}-p_{c}\right) +\gamma \left[ p_{b}\left( \left\langle LL^{\dagger }\right\rangle -\left\langle \mathbb {L}\mathbb {L}^{\dagger }\right\rangle \right) \right. \nonumber \\&\left. -p_{a}\left\langle L^{\dagger }L\right\rangle +p_{c}\left\langle \mathbb {L}^{\dagger }\mathbb {L}\right\rangle \right] \end{aligned}$$

(3.139)

Due to a separation of timescales, as $\Gamma _{\pm }\gg \gamma _{c}=N\gamma \approx 10^{-4}\,\upmu \mathrm {eV}$, in a perturbative treatment the effect of the second term can be neglected and the electronic subsystem approximately settles into $p_{a}=p_{c}$. This leaves us with a single dynamical variable, namely $p_{a}$, entirely describing the electronic subsystem on relevant timescales. Thus, using $p_{c}=p_{a}$ and $p_{b}=1-2p_{a}$, the electronic quasi steady state is uniquely defined by the parameter $p_{a}$ and the nuclear evolution simplifies to

$$\begin{aligned} \dot{\sigma }= & {} \,\gamma \left\{ p_{a}\left[ \mathcal {D}_{L}\left[ \sigma \right] +\mathcal {D}_{\mathbb {L}}\left[ \sigma \right] \right] \right. \\&\left. +\left( 1-2p_{a}\right) \left[ \mathcal {D}_{L^{\dagger }}\left[ \sigma \right] +\mathcal {D}_{\mathbb {L}^{\dagger }}\left[ \sigma \right] \right] \right\} \nonumber \\&+\,i\delta \left\{ p_{a}\left( \left[ L^{\dagger }L,\sigma \right] +\left[ \mathbb {L}^{\dagger }\mathbb {L},\sigma \right] \right) \right. \nonumber \\&\left. -\left( 1-2p_{a}\right) \left( \left[ LL^{\dagger },\sigma \right] +\left[ \mathbb {L}\mathbb {L}^{\dagger },\sigma \right] \right) \right\} ,\nonumber \end{aligned}$$

(3.140)

with $p_{a}$ obeying the dynamical equation

$$\begin{aligned} \dot{p}_{a}= & {} \,\Gamma _{\pm }\left( 1-3p_{a}\right) +\Gamma _{2}\left( 1-2p_{a}\right) \\&-\,\gamma \left[ p_{a}\left\langle L^{\dagger }L\right\rangle +\left( 1-2p_{a}\right) \left\langle LL^{\dagger }\right\rangle \right] . \end{aligned}$$

Neglecting the HF terms in the second line, we recover the projection-operator-based result for the quasisteady state, $p_{a}\approx \left( \Gamma _{\pm }+\Gamma _{2}\right) /\left( 3\Gamma _{\pm }+2\Gamma _{2}\right)$ as stated in Eq. (3.34).

3.1.6 3.11.6 Effective Nuclear Dynamics: Overhauser Fluctuations

In Sect. 3.5 we have disregarded the effect of Overhauser fluctuations, described by $\dot{\rho }=-i\left[ H_{\mathrm {zz}},\rho \right] =-ia_{\mathrm {hf}}\sum _{i}\left[ S_{i}^{z}\delta A_{i}^{z},\rho \right] .$ In the following analysis, this simplification is discussed in greater detail.

First of all, we note that this term cannot induce couplings within the effective electronic three level system, $\left\{ \left| T_{\pm }\right\rangle ,\left| \lambda _{2}\right\rangle \right\}$, since $\left| T_{\pm }\right\rangle$ are eigenstates of $S_{i}^{z}$, that is explicitly $S_{i}^{z}\left| T_{\pm }\right\rangle =\pm \frac{1}{2}\left| T_{\pm }\right\rangle$, which leads to

$$\begin{aligned} \left\langle T_{\pm }|S_{i}^{z}|\lambda _{2}\right\rangle =0. \end{aligned}$$

(3.141)

In other words, different $S_{\mathrm {tot}}^{z}$ subspaces are not coupled by the action of $H_{\mathrm {zz}}$; this is in stark contrast to the flip flop dynamics $H_{\mathrm {ff}}$.

When also accounting for Overhauser fluctuations, the dynamical equations for the coherences read

$$\begin{aligned} \dot{\rho }_{ab}= & {} \left( i\varepsilon _{2}-\tilde{\Gamma }\right) \rho _{ab}-i\left[ L^{\dagger }\rho _{bb}-\rho _{aa}L^{\dagger }\right] \\&-\,ia_{\mathrm {hf}}\sum _{i}\left[ \left\langle S_{i}^{z}\right\rangle _{a}\delta A_{i}^{z}\rho _{ab}-\left\langle S_{i}^{z}\right\rangle _{b}\rho _{ab}\delta A_{i}^{z}\right] ,\nonumber \end{aligned}$$

(3.142)

where $\left\langle S_{i}^{z}\right\rangle _{a}=\left\langle a|S_{i}^{z}|a\right\rangle$; an analog equation holds for $\dot{\rho }_{cb}$. Typically, the second line is small compared to the fast electronic quantities $\varepsilon _{2},\tilde{\Gamma }$ in the first line. Therefore, it will be neglected. In Eqs. (3.132), (3.133) and (3.134), the Overhauser fluctuations lead to the following additional terms

$$\begin{aligned} \dot{\rho }_{aa}= & {} \,\cdots -\frac{i}{2}a_{\mathrm {hf}}\sum _{i}\left[ \delta A_{i}^{z},\rho _{aa}\right] ,\end{aligned}$$

(3.143)

$$\begin{aligned} \dot{\rho }_{cc}= & {} \,\cdots +\frac{i}{2}a_{\mathrm {hf}}\sum _{i}\left[ \delta A_{i}^{z},\rho _{cc}\right] ,\end{aligned}$$

(3.144)

$$\begin{aligned} \dot{\rho }_{bb}= & {} \,\cdots -ia_{\mathrm {hf}}\sum _{i}\left\langle S_{i}^{z}\right\rangle _{b}\left[ \delta A_{i}^{z},\rho _{bb}\right] . \end{aligned}$$

(3.145)

First, this leaves the electronic populations $p_{a}=\mathsf {Tr}_{\mathrm {n}}\left[ \rho _{aa}\right]$ untouched; $H_{\mathrm {zz}}$ does not induce any couplings between them. Second, the dynamical equation for the nuclear density matrix $\sigma =\mathsf {Tr}_{\mathrm {el}}\left[ \rho \right]$ is modified as

$$\begin{aligned} \dot{\sigma }= & {} \,\cdots -ia_{\mathrm {hf}}\sum _{i}\left[ \frac{1}{2}\left( p_{a}-p_{c}\right) +p_{b}\left\langle S_{i}^{z}\right\rangle _{b}\right] \left[ \delta A_{i}^{z},\sigma \right] ,\nonumber \\\approx & {} \,\cdots -i\left( 1-2p_{a}\right) a_{\mathrm {hf}}\sum _{i}\left\langle S_{i}^{z}\right\rangle _{b}\left[ \delta A_{i}^{z},\sigma \right] . \end{aligned}$$

(3.146)

In the second step, we have used again that differences in $p_{a}$ and $p_{c}$ are quickly damped to zero with a rate of $3\Gamma _{\pm }$. Now, let us examine the effect of Eq. (3.146) for different important regimes: In the high gradient regime, where $p_{b}$ is fully depleted, it does not give any contribution since the electronic quasi steady state does not have any magnetization $\left[ \left\langle S_{i}^{z}\right\rangle _{b}=\left\langle S_{i}^{z}\right\rangle _{\mathrm {ss}}=0\right]$ and $p_{a}=1/2$. In the low gradient regime, $\left| b\right\rangle$ approaches the triplet $\left| T_{0}\right\rangle$ and again (since $\left\langle S_{i}^{z}\right\rangle _{b}=0$) this term vanishes. Finally, the intermediate regime has been studied within a semiclassical approximation (see Sect. 3.6): Note that Eq. (3.146), however, leaves the dynamical equation for the nuclear polarizations $I_{i}^{z}$ unchanged, since they commute with $H_{\mathrm {zz}}$.

3.1.7 3.11.7 Ideal Nuclear Steady State

In this Appendix we analytically construct the ideal (pure) nuclear steady-state $\left| \xi _\mathrm {ss}\right\rangle$, fulfilling $L_{2}\left| \xi _\mathrm {ss}\right\rangle =\mathbb {L}_{2}\left| \xi _\mathrm {ss}\right\rangle =0$, in the limit of identical dots $(a_{1j}=a_{2j}\forall j=1,\dots ,N_{1}\equiv N_{2}=N)$ for uniform HF-coupling where $a_{ij}=N/N_{i}$. In this limit, the non-local nuclear jump operators simplify to

$$\begin{aligned} L_{2}= & {} \, \nu I_{1}^{+}+\mu I_{2}^{+},\end{aligned}$$

(3.147)

$$\begin{aligned} \mathbb {L}_{2}= & {} \, \mu I_{1}^{-}+\nu I_{2}^{-}. \end{aligned}$$

(3.148)

Here, to simplify the notation, we have replaced $\mu _{2}$ and $\nu _{2}$ by $\mu$ and $\nu$, respectively. The common proportionality factor is irrelevant for this analysis and therefore has been dropped. The collective nuclear spin operators $I_{1,2}^{\alpha }$ form a spin algebra and the so-called Dicke states $\left| J_{1},k_{1}\right\rangle \otimes \left| J_{2},k_{2}\right\rangle \equiv \left| J_{1},k_{1};J_{2},k_{2}\right\rangle$, where the total spin quantum numbers $J_{i}$ are conserved and the spin projection quantum number $k_{i}=0,1,\dots ,2J_{i}$, allow for an efficient description. Here, we restrict ourselves to the symmetric case where $J_{1}=J_{2}=J$; analytic and numerical evidence for small $J_{i}\approx 3$ shows, that for $J_{1}\ne J_{2}$ one obtains a mixed nuclear steady state. The total spin quantum numbers $J_{i}=J$ are conserved and we set $\left| J,k_{1};J,k_{2}\right\rangle =\left| k_{1},k_{2}\right\rangle$. Using standard angular momentum relations, one obtains

$$\begin{aligned} L_{2}\left| k_{1},k_{2}\right\rangle= & {} \,\nu j_{k_{1}}\left| k_{1}+1,k_{2}\right\rangle +\mu j_{k_{2}}\left| k_{1},k_{2}+1\right\rangle ,\end{aligned}$$

(3.149)

$$\begin{aligned} \mathbb {L}_{2}\left| k_{1},k_{2}\right\rangle= & {} \, \mu g_{k_{1}}\left| k_{1}-1,k_{2}\right\rangle +\nu g_{k_{2}}\left| k_{1},k_{2}-1\right\rangle . \end{aligned}$$

(3.150)

Here, we have introduced the matrix elements

$$\begin{aligned} j_{k}= & {} \sqrt{J\left( J+1\right) -\left( k-J\right) \left( k-J+1\right) },\end{aligned}$$

(3.151)

$$\begin{aligned} g_{k}= & {} \sqrt{J\left( J+1\right) -\left( k-J\right) \left( k-J-1\right) }. \end{aligned}$$

(3.152)

Note that $j_{2J}=0$ and $g_{0}=0$. Moreover, the matrix elements obey the symmetry

$$\begin{aligned} j_{k}= & {} \, j_{2J-k-1},\end{aligned}$$

(3.153)

$$\begin{aligned} g_{k+1}= & {} \,g_{2J-k}. \end{aligned}$$

(3.154)

Now, we show that $\left| \xi _{\mathrm {ss}}\right\rangle$ fulfills $L_{2}\left| \xi _{\mathrm {ss}}\right\rangle =\mathbb {L}_{2}\left| \xi _{\mathrm {ss}}\right\rangle =0$. First, using the relations above, we have

$$\begin{aligned} L_{2}\left| \xi _{\mathrm {ss}}\right\rangle= & {} \sum _{k=0}^{2J}\xi ^{k}\left[ \nu j_{k}\left| k+1,2J-k\right\rangle \right. \\&\left. +\mu j_{2J-k}\left| k,2J-k+1\right\rangle \right] \\= & {} \sum _{k=0}^{2J-1}\xi ^{k}\left[ \nu j_{k}\left| k+1,2J-k\right\rangle \right. \\&\left. +\xi \mu j_{2J-k-1}\left| k+1,2J-k\right\rangle \right] \\= & {} \sum _{k=0}^{2J-1}\xi ^{k}\nu \underset{=0}{\underbrace{\left[ j_{k}-j_{2J-k-1}\right] }}\left| k+1,2J-k\right\rangle . \end{aligned}$$

In the second step, since $j_{2J}=0$, we have redefined the summation index as $k\rightarrow k+1$. Along the same lines, one obtains

$$\begin{aligned} \mathbb {L}_{2}\left| \xi _{\mathrm {ss}}\right\rangle= & {} \sum _{k=0}^{2J}\xi ^{k}\left[ \mu g_{k}\left| k-1,2J-k\right\rangle \right. \\&\left. +\nu g_{2J-k}\left| k,2J-k-1\right\rangle \right] \\= & {} \sum _{k=0}^{2J-1}\xi ^{k}\left[ \xi \mu g_{k+1}\left| k,2J-k-1\right\rangle \right. \\&\left. +\nu g_{2J-k}\left| k,2J-k-1\right\rangle \right] \\= & {} \sum _{k=0}^{2J-1}\xi ^{k}\nu \underset{=0}{\underbrace{\left[ g_{2J-k}-g_{k+1}\right] }}\left| k,2J-k-1\right\rangle . \end{aligned}$$

This completes the proof. For illustration, the dark state $\left| \xi _{\mathrm {ss}}\right\rangle$ is sketched in Fig. 3.20. In particular, the bistable polarization character inherent to $\left| \xi _{\mathrm {ss}}\right\rangle$ is emphasized, as (in contrast to the bosonic case) the modulus of the parameter $\xi$ is not confined to $\left| \xi \right| <1$.

Fig. 3.20

Sketch of the ideal nuclear dark state for uniform HF coupling $\left| \xi _{\mathrm {ss}}\right\rangle$. The Dicke states are labeled according to their spin projection $k_{i}=0,1,\dots 2J$. Since $k_{1}=k$ is strongly correlated with $k_{2}=2J-k$, the two Dicke ladders are arranged in opposite order. The bistability inherent to $\left| \xi _{\mathrm {ss}}\right\rangle$ is schematized as well: The size of the spheres refers to $\left| \left\langle k_{1},k_{2}|\xi _{\mathrm {ss}}\right\rangle \right| ^{2}$ for $\left| \xi \right| <1$ (red) and $\left| \xi \right| >1$ (blue), respectively. As indicated by the arrows for individual nuclear spins, $\left| \xi \right| <1$ $\left( \left| \xi \right| >1\right)$ corresponds to a nuclear OH gradient $\Delta _{\mathrm {OH}}>0$ $\left( \Delta _{\mathrm {OH}}>0\right)$, respectively

3.1.8 3.11.8 Numerical Results for DNP

In this Appendix the analytical findings of the semiclassical model are corroborated by exact numerical simulations for small sets of nuclear spins. This treatment complements our analytical DNP analysis in several aspects: First, we do not restrict ourselves to the effective three level system $\left\{ \left| T_{\pm }\right\rangle ,\left| \lambda _{2}\right\rangle \right\}$. Second, the electronic degrees of freedom are not eliminated adiabatically from the dynamics. Lastly, this approach does not involve the semiclassical decorrelation approximation stated in Eq. (3.46).

Technical details.—We consider the idealized case of homogeneous hyperfine coupling for which an exact numerical treatment is feasible even for a relatively large number of coupled nuclei as the system evolves within the totally symmetric low-dimensional subspace $\left\{ \left| J,m\right\rangle ,m=-J,\dots ,J\right\}$, referred to as Dicke ladder. We restrict ourselves to the fully symmetric subspace where $J_{i}=N_{i}/2\approx 3$. Moreover, to mimic the separation of timescales in experiments where $N\approx 10^{6}$, the HF coupling is scaled down appropriately to the constant value $g_{\mathrm {hf}}\approx 0.1\,\upmu \mathrm {eV}$; also compare the numerical results presented in Fig. 3.7.

Fig. 3.21

Exact time evolution for $N=8$ and $N=12$ (red dashed curves) nuclear spins, four and six in each quantum dot, respectively. Depending on the initial value of the gradient, the nuclear system either runs into the trivial, unpolarized state or into the highly polarized one, if the initial gradient exceeds the critical value; the blue dotted, black dash-dotted and all other refer to $\Delta _{\mathrm {ext}}=-5\,\upmu \mathrm {eV}$, $\Delta _{\mathrm {ext}}=0$ and $\Delta _{\mathrm {ext}}=5\,\upmu \mathrm {eV}$, respectively. For $\omega _{\mathrm {ext}}\ne 0$, also a homogeneous OH field $\omega _{\mathrm {OH}}$ builds up which partially compensates $\omega _{\mathrm {ext}}$: here, $\omega _{\mathrm {ext}}=0.1\,\upmu \mathrm {eV}$ (magenta dash-dotted) and $\omega _{\mathrm {ext}}=-0.1\,\upmu \mathrm {eV}$ (cyan dash-dotted). Other numerical parameters: $t=10\,\upmu \mathrm {eV}$, $\varepsilon =30\,\upmu \mathrm {eV}$, $\Gamma =25\,\upmu \mathrm {eV}$, $\Gamma _{\pm }=\Gamma _{\mathrm {deph}}=0.1\,\upmu \mathrm {eV}$

Our first numerical approach is based on simulations of the time evolution. Starting out from nuclear states with different initial Overhauser gradient $\Delta _{\mathrm {OH}}\left( t=0\right)$, we make the following observations, depicted in Fig. 3.21: First of all, the tri-stability of the Overhauser gradient with respect to the initial nuclear polarization is confirmed. If the initial gradient $\Delta _{\mathrm {OH}}\left( t=0\right) +\Delta _{\mathrm {ext}}$ exceeds a certain threshold value, the nuclear system runs into the highly-polarized steady state, otherwise it gets stuck in the trivial, zero-polarization solution. There are two symmetric high-polarization solutions that depend on the sign of $\Delta _{\mathrm {OH}}\left( t=0\right) +\Delta _{\mathrm {ext}}$; also note that the Overhauser gradient $\Delta _{\mathrm {OH}}$ may flip the sign as determined by the total initial gradient $\Delta _{\mathrm {OH}}\left( t=0\right) +\Delta _{\mathrm {ext}}$. Second, in the absence of an external Zeeman splitting $\omega _{\mathrm {ext}}$, a potential initial homogeneous Overhauser polarizations $\bar{\omega }_{\mathrm {OH}}$ is damped to zero in the steady state. For finite $\omega _{\mathrm {ext}}\ne 0$, a homogeneous Overhauser polarization $\bar{\omega }_{\mathrm {OH}}$ builds up which partially compensates $\omega _{\mathrm {ext}}$. Lastly, the high-polarization solutions $\Delta _{\mathrm {OH}}^{\mathrm {ss}}\approx 2\,\upmu \mathrm {eV}$ are far away from full polarization. This is an artifact of the small system sizes $J_{i}\approx 3$: As we deal with very short Dicke ladders, even the ideal, nuclear two-mode squeezedlike target state $\left| \xi \right\rangle _{\mathrm {ss}}$ given in Eq. (3.8) does not feature a very high polarization. Pictorially, it leaks with a non-vanishing factor $\sim \!\xi ^{m}$ into the low-polarization Dicke states. This argument is supported by the fact that (for the same set of parameters) we observe tendency towards higher polarization for an increasing number of nuclei N (which features a larger Dicke ladder) and confirmed by our second numerical approach to be discussed below.

Fig. 3.22

Instability towards nuclear self-polarization: Exact numerical results for small system sizes $J_{i}=N_{i}/2$. The exact steady state of the coupled electron-nuclear dynamics is computed as a function of the gradient $\Delta$. The circles (squares) refer to the polarization in the left (right) dot, respectively. a For $\Delta >\left| \Delta _{\mathrm {OH}}^{\mathrm {crt}}\right|$, we find $\Delta _{\mathrm {OH}}>0$, whereas for $\Delta <-\left| \Delta _{\mathrm {OH}}^{\mathrm {crt}}\right|$ we get $\Delta _{\mathrm {OH}}<0$, i.e., outside of the small-gradient regime [see inset (c)] the nuclear system is seen to be unstable towards the buildup of a OH gradient with opposite polarizations in the two dots. The nuclear polarization depends on the system size $J_{i}$ and the parameter $\left| \xi \right|$; compare inset (b). c The critical value of $\Delta _{\mathrm {OH}}^{\mathrm {crt}}\approx 3\,\upmu \mathrm {eV}$ agrees with the semiclassical estimate; it becomes smaller for smaller values of $\Gamma _{\pm }$. Numerical parameters in $\,\upmu \mathrm {eV}$: $\varepsilon =30$, $\Gamma =10$, $\Gamma _{\pm }=\Gamma _{\mathrm {deph}}=0.3$, $\omega _{\mathrm {ext}}=0$ and $t=10$ except for the cyan curve where $t=20$ and $\Gamma _{\pm }=\Gamma _{\mathrm {deph}}=0.6$ for the orange curve in (c)

Our second numerical approach is based on exact diagonalization: As we tune the parameter $\Delta$, we compute the steady state for the full electronic-nuclear system directly giving the corresponding steady-state nuclear polarizations $\left\langle I_{i}^{z}\right\rangle _{\mathrm {ss}}$. We see a clear instability towards the buildup of an Overhauser gradient $\Delta _{\mathrm {OH}}^{\mathrm {ss}}$ (Fig. 3.22): Inside the small-gradient region $\left( \left| \Delta \right| <\left| \Delta _{\mathrm {OH}}^{\mathrm {crt}}\right| \right)$ we observe negative feedback $\mathrm {sgn}\left( \Delta _{\mathrm {OH}}^{\mathrm {ss}}\right) =-\mathrm {sgn}\left( \Delta \right)$, whereas outside of it $\left( \left| \Delta \right| >\left| \Delta _{\mathrm {OH}}^{\mathrm {crt}}\right| \right)$ the nuclear system experiences positive feedback $\mathrm {sgn}\left( \Delta _{\mathrm {OH}}^{\mathrm {ss}}\right) =\mathrm {sgn}\left( \Delta \right)$. The latter leads to the build-up of large OH gradients, in agreement with our semiclassical analysis.

3.1.9 3.11.9 Effective Nuclear Master Equation in High-Gradient Regime

This Appendix provides background material for the derivation of the effective nuclear master equation as stated in Eq. (3.55) using projection-operator techniques [46, 70]. We start with

$$\begin{aligned} \mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {V}\mathcal {P}\rho \right] =\mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {ff}}\mathcal {P}\rho \right] +\mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {zz}}\mathcal {P}\rho \right] \end{aligned}$$

(3.155)

The first term is readily found to be

$$\begin{aligned} \mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {ff}}\mathcal {P}\rho \right] =-i\frac{a_{\mathrm {hf}}}{2}\sum _{i,\alpha =\pm }\left\langle S_{i}^{\alpha }\right\rangle _{\mathrm {ss}}\left[ A_{i}^{\bar{\alpha }},\sigma \right] , \end{aligned}$$

(3.156)

where $\left\langle \cdot \right\rangle _{\mathrm {ss}}=\mathsf {Tr}_{\mathrm {el}}\left[ \cdot \rho _{\mathrm {ss}}^{\mathrm {el}}\right]$ denotes the steady-state expectation value. An analog calculation yields

$$\begin{aligned} \mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {zz}}\mathcal {P}\rho \right] =-ia_{\mathrm {hf}}\sum _{i}\left\langle S_{i}^{z}\right\rangle _{\mathrm {ss}}\left[ \delta A_{i}^{z},\sigma \right] . \end{aligned}$$

(3.157)

Using that $\left\langle S_{i}^{\alpha }\right\rangle _{\mathrm {ss}}=0$ and $\left\langle S_{i}^{z}\right\rangle _{\mathrm {ss}}=0$ [the Knight shift seen by the nuclear spins is zero since the electronic quasi steady-state carries no net magnetization], the first two Hamiltonian terms vanish.

The second-order term of interest

$$\begin{aligned} \mathcal {K}\sigma =\mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {V}\mathcal {Q}\left( -\mathcal {L}_{0}^{-1}\right) \mathcal {Q}\mathcal {V}\mathcal {P}\rho \right] \end{aligned}$$

(3.158)

can be decomposed as $\mathcal {K}\sigma =\mathcal {K}_{\mathrm {ff}}\sigma +\mathcal {K}_{\mathrm {zz}}\sigma$, where

$$\begin{aligned} \mathcal {K}_{\mathrm {ff}}\sigma= & {} \,\mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {ff}}\mathcal {Q}\left( -\mathcal {L}_{0}^{-1}\right) \mathcal {Q}\mathcal {L}_{\mathrm {ff}}\mathcal {P}\rho \right] ,\end{aligned}$$

(3.159)

$$\begin{aligned} \mathcal {K}_{\mathrm {zz}}\sigma= & {} \,\mathsf {Tr}_{\mathrm {el}}\left[ \mathcal {P}\mathcal {L}_{\mathrm {zz}}\mathcal {Q}\left( -\mathcal {L}_{0}^{-1}\right) \mathcal {Q}\mathcal {L}_{\mathrm {zz}}\mathcal {P}\rho \right] . \end{aligned}$$

(3.160)

All other second order terms containing combinations of the superoperators $\mathcal {L}_{\mathrm {ff}}$ and $\mathcal {L}_{\mathrm {zz}}$ can be shown to vanish. In the following, we will evaluate the two terms separately.

Hyperfine flip-flop dynamics.—First, we will evaluate $\mathcal {K}_{\mathrm {ff}}$ which can be rewritten as

(3.161)

Here, we used the Laplace transform $-\mathcal {L}_{0}^{-1}=\int _{0}^{\infty }d\tau e^{\mathcal {L}_{0}\tau }$ and the property $e^{\mathcal {L}_{0}\tau }\mathcal {P}=\mathcal {P}e^{\mathcal {L}_{0}\tau }=\mathcal {P}$ [70]. The first term labeled as is given by

(3.162)

Then, using the relations

$$\begin{aligned} \mathcal {L}_{0}\left[ \left| \lambda _{k}\right\rangle \left\langle T_{\pm }\right| \right]= & {} -i\left( \delta _{k}^{\pm }-i\tilde{\Gamma }_{k}\right) \left| \lambda _{k}\right\rangle \left\langle T_{\pm }\right| ,\end{aligned}$$

(3.163)

$$\begin{aligned} \mathcal {L}_{0}\left[ \left| T_{\pm }\right\rangle \left\langle \lambda _{k}\right| \right]= & {} +i\left( \delta _{k}^{\pm }+i\tilde{\Gamma }_{k}\right) \left| T_{\pm }\right\rangle \left\langle \lambda _{k}\right| , \end{aligned}$$

(3.164)

where (to shorten the notation) $\delta _{k}^{+}=\Delta _{k}$ and $\delta _{k}^{-}=\delta _{k}$, respectively, we find

$$\begin{aligned} e^{\mathcal {L}_{0}\tau }\left( H_{\mathrm {ff}}\sigma \rho _{\mathrm {ss}}^{\mathrm {el}}\right)= & {} \frac{a_{\mathrm {hf}}}{4}\sum _{k}\left[ e^{-i\left( \delta _{k}^{+}-i\tilde{\Gamma }_{k}\right) \tau }\left| \lambda _{k}\right\rangle \left\langle T_{+}\right| L_{k}\sigma \right. \nonumber \\&\left. +\,e^{-i\left( \delta _{k}^{-}-i\tilde{\Gamma }_{k}\right) \tau }\left| \lambda _{k}\right\rangle \left\langle T_{-}\right| \mathbb {L}_{k}\sigma \right] , \end{aligned}$$

(3.165)

and along the same lines

$$\begin{aligned} e^{\mathcal {L}_{0}\tau }\left( \sigma \rho _{\mathrm {ss}}^{\mathrm {el}}H_{\mathrm {ff}}\right)= & {} \frac{a_{\mathrm {hf}}}{4}\sum _{k}\left[ e^{+i\left( \delta _{k}^{+}+i\tilde{\Gamma }_{k}\right) \tau }\left| T_{+}\right\rangle \left\langle \lambda _{k}\right| \sigma L_{k}^{\dagger }\right. \nonumber \\&\left. +\,e^{+i\left( \delta _{k}^{-}+i\tilde{\Gamma }_{k}\right) \tau }\left| T_{-}\right\rangle \left\langle \lambda _{k}\right| \sigma \mathbb {L}_{k}^{\dagger }\right] . \end{aligned}$$

(3.166)

Plugging Eqs. (3.165) and (3.166) into Eq. (3.162), tracing out the electronic degrees of freedom, performing the integration in $\tau$ and separating real and imaginary parts of the complex eigenvalues leads to

(3.167)

(3.168)

This corresponds to the flip-flop mediated terms given in Eq. (3.55) in the main text. The second term labeled as can be computed along the lines: due to the additional appearance of the projector $\mathcal {P}$, it contains factors of $\left\langle S_{i}^{\alpha }\right\rangle _{\mathrm {ss}}$ and is therefore found to be zero.

Overhauser fluctuations.—In the next step, we investigate the second-order effect of Overhauser fluctuations with respect to the effective QME for the nuclear dynamics. Our analysis starts out from the second-order expression $\mathcal {K}_{\mathrm {zz}}$ which, as above, can be rewritten as

(3.169)

First, we evaluate the terms labeled by and separately. We find

(3.170)

where we used the Quantum Regression theorem yielding the electronic auto-correlation functions

$$\begin{aligned} \left\langle S_{i}^{z}\left( \tau \right) S_{j}^{z}\right\rangle _{\mathrm {ss}}= & {} \, \mathsf {Tr}_{\mathrm {el}}\left[ S_{i}^{z}e^{\mathcal {L}_{0}\tau }\left( S_{j}^{z}\rho _{\mathrm {ss}}^{\mathrm {el}}\right) \right] ,\end{aligned}$$

(3.171)

$$\begin{aligned} \left\langle S_{j}^{z}S_{i}^{z}\left( \tau \right) \right\rangle _{\mathrm {ss}}= & {} \,\mathsf {Tr}_{\mathrm {el}}\left[ S_{i}^{z}e^{\mathcal {L}_{0}\tau }\left( \rho _{\mathrm {ss}}^{\mathrm {el}}S_{j}^{z}\right) \right] . \end{aligned}$$

(3.172)

In a similar fashion, the term labeled by is found to be

(3.173)

Putting together the results for and , we obtain

$$\begin{aligned} \mathcal {K}_{\mathrm {zz}}\sigma= & {} \sum _{i,j}\Pi _{ij}\left[ \delta A_{j}^{z}\sigma \delta A_{i}^{z}-\delta A_{i}^{z}\delta A_{j}^{z}\sigma \right] \nonumber \\&+\Upsilon _{ij}\left[ \delta A_{j}^{z}\sigma \delta A_{i}^{z}-\sigma \delta A_{i}^{z}\delta A_{j}^{z}\right] , \end{aligned}$$

(3.174)

which can be rewritten as

$$\begin{aligned} \mathcal {K}_{\mathrm {zz}}\sigma= & {} \sum _{i,j}\left( \Pi _{ij}+\Upsilon _{ij}\right) \left[ \delta A_{j}^{z}\sigma \delta A_{i}^{z}-\frac{1}{2}\left\{ \delta A_{i}^{z}\delta A_{j}^{z},\sigma \right\} \right] \nonumber \\&-\frac{i}{2}\left[ \frac{1}{i}\left( \Pi _{ij}-\Upsilon _{ij}\right) \delta A_{i}^{z}\delta A_{j}^{z},\sigma \right] . \end{aligned}$$

(3.175)

Here, we have introduced the integrated electronic auto-correlation functions [70]

$$\begin{aligned} \Pi _{ij}= & {} a_{\mathrm {hf}}^{2}\int _{0}^{\infty }d\tau \left( \left\langle S_{i}^{z}\left( \tau \right) S_{j}^{z}\right\rangle _{\mathrm {ss}}-\left\langle S_{i}^{z}\right\rangle _{\mathrm {ss}}\left\langle S_{j}^{z}\right\rangle _{\mathrm {ss}}\right) ,\\ \Upsilon _{ij}= & {} a_{\mathrm {hf}}^{2}\int _{0}^{\infty }d\tau \left( \left\langle S_{i}^{z}S_{j}^{z}\left( \tau \right) \right\rangle _{\mathrm {ss}}-\left\langle S_{i}^{z}\right\rangle _{\mathrm {ss}}\left\langle S_{j}^{z}\right\rangle _{\mathrm {ss}}\right) . \end{aligned}$$

For an explicit calculation, we use the relation

$$\begin{aligned} S_{j}^{z}\rho _{\mathrm {ss}}^{\mathrm {el}}= & {} \rho _{\mathrm {ss}}^{\mathrm {el}}S_{j}^{z}=\frac{1}{4}\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| -\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) , \end{aligned}$$

(3.176)

and the fact that $\left| T_{+}\right\rangle \left\langle T_{+}\right| -\left| T_{-}\right\rangle \left\langle T_{-}\right|$ is an eigenvector of $\mathcal {L}_{0}$ with eigenvalue $-5\Gamma _{\pm },$ which readily yield $\Pi _{ij}=\Upsilon _{ij}=\gamma _{\mathrm {zz}}/2$. From this, we immediately obtain the corresponding term appearing in the effective nuclear dynamics as

$$\begin{aligned} \mathcal {K}_{\mathrm {zz}}\sigma =\gamma _{\mathrm {zz}}\sum _{i,j}\left[ \delta A_{j}^{z}\sigma \delta A_{i}^{z}-\frac{1}{2}\left\{ \delta A_{i}^{z}\delta A_{j}^{z},\sigma \right\} \right] . \end{aligned}$$

(3.177)

3.1.10 3.11.10 Diagonalization of Nuclear Dissipator

The flip-flop mediated terms $\mathcal {K}_{\mathrm {ff}}$ in Eq. (3.55) can be recast into the following form

$$\begin{aligned} \dot{\sigma }=\sum _{i,j}\frac{\gamma _{ij}}{2}\left[ A_{i}\sigma A_{j}^{\dagger }-\frac{1}{2}\left\{ A_{j}^{\dagger }A_{i},\sigma \right\} \right] +i\frac{\Delta _{ij}}{2}\left[ A_{j}^{\dagger }A_{i},\sigma \right] , \end{aligned}$$

(3.178)

where we have introduced the vector $\mathbf {A}$ containing the local nuclear jump operators as $\mathbf {A}=\left( A_{1}^{+},A_{2}^{+},A_{2}^{-},A_{1}^{-}\right)$. The matrices $\gamma$ and $\Delta$ obey a simple block-structure according to

$$\begin{aligned} \gamma= & {} \gamma ^{+}\oplus \gamma ^{-},\end{aligned}$$

(3.179)

$$\begin{aligned} \Delta= & {} \Delta ^{+}\oplus \Delta ^{-}, \end{aligned}$$

(3.180)

where the 2-by-2 block entries are given by

$$\begin{aligned} \gamma ^{\pm }=\left( \begin{array}{cc} \gamma _{11}^{\pm } &{} \gamma _{12}^{\pm }\\ \gamma _{21}^{\pm } &{} \gamma _{22}^{\pm } \end{array}\right) =\left( \begin{array}{cc} \sum _{k}\gamma _{k}^{\pm }\nu _{k}^{2} &{} \sum _{k}\gamma _{k}^{\pm }\mu _{k}\nu _{k}\\ \sum _{k}\gamma _{k}^{\pm }\mu _{k}\nu _{k} &{} \sum _{k}\gamma _{k}^{\pm }\mu _{k}^{2} \end{array}\right) , \end{aligned}$$

(3.181)

and similarly

$$\begin{aligned} \Delta ^{\pm }= & {} \left( \begin{array}{cc} \Delta _{11}^{\pm } &{} \Delta _{12}^{\pm }\\ \Delta _{21}^{\pm } &{} \Delta _{22}^{\pm } \end{array}\right) \\= & {} \left( \begin{array}{cc} \sum _{k}\Delta _{k}^{\pm }\nu _{k}^{2} &{} \sum _{k}\Delta _{k}^{\pm }\mu _{k}\nu _{k}\\ \sum _{k}\Delta _{k}^{\pm }\mu _{k}\nu _{k} &{} \sum _{k}\Delta _{k}^{\pm }\mu _{k}^{2} \end{array}\right) .\nonumber \end{aligned}$$

(3.182)

The nuclear dissipator can be brought into diagonal form

$$\begin{aligned} \tilde{\gamma }=U^{\dagger }\gamma U=\mathrm {diag}\left( \tilde{\gamma }_{1}^{+},\tilde{\gamma }_{2}^{+},\tilde{\gamma }_{1}^{-},\tilde{\gamma }_{2}^{-}\right) , \end{aligned}$$

(3.183)

where

$$\begin{aligned} \tilde{\gamma }_{1}^{\pm }= & {} \frac{1}{2}\left[ \gamma _{11}^{\pm }+\gamma _{22}^{\pm }+\sqrt{\left( \gamma _{11}^{\pm }-\gamma _{22}^{\pm }\right) ^{2}+4\left( \gamma _{12}^{\pm }\right) ^{2}}\right] ,\end{aligned}$$

(3.184)

$$\begin{aligned} \tilde{\gamma }_{2}^{\pm }= & {} \frac{1}{2}\left[ \gamma _{11}^{\pm }+\gamma _{22}^{\pm }-\sqrt{\left( \gamma _{11}^{\pm }-\gamma _{22}^{\pm }\right) ^{2}+4\left( \gamma _{12}^{\pm }\right) ^{2}}\right] , \end{aligned}$$

(3.185)

and $U=U^{+}\oplus U^{-}$ with

$$\begin{aligned} U^{\pm }=\left( \begin{array}{cc} \cos \left( \theta _{\pm }/2\right) &{} -\sin \left( \theta _{\pm }/2\right) \\ \sin \left( \theta _{\pm }/2\right) &{} \cos \left( \theta _{\pm }/2\right) \end{array}\right) . \end{aligned}$$

(3.186)

Here, we have defined $\theta _{\pm }$ via the relation $\tan \left( \theta _{\pm }\right) =2\gamma _{12}^{\pm }/\left( \gamma _{11}^{\pm }-\gamma _{22}^{\pm }\right)$, $0\le \theta _{\pm }<\pi$. Introducing a new set of operators $\tilde{\mathbf {A}}=\left( \tilde{A}_{1},\tilde{A}_{2},\tilde{B}_{1},\tilde{B}_{2}\right)$ according to

$$\begin{aligned} \tilde{\mathbf {A}}_{k}=\sum _{j}U_{jk}A_{j}, \end{aligned}$$

(3.187)

that is explicitly

$$\begin{aligned} \tilde{A}_{1}= & {} \cos \left( \theta _{+}/2\right) A_{1}^{+}+\sin \left( \theta _{+}/2\right) A_{2}^{+},\end{aligned}$$

(3.188)

$$\begin{aligned} \tilde{A}_{2}= & {} -\sin \left( \theta _{+}/2\right) A_{1}^{+}+\cos \left( \theta _{+}/2\right) A_{2}^{+},\end{aligned}$$

(3.189)

$$\begin{aligned} \tilde{B}_{1}= & {} \sin \left( \theta _{-}/2\right) A_{1}^{-}+\cos \left( \theta _{-}/2\right) A_{2}^{-},\end{aligned}$$

(3.190)

$$\begin{aligned} \tilde{B}_{2}= & {} \cos \left( \theta _{-}/2\right) A_{1}^{-}-\sin \left( \theta _{-}/2\right) A_{2}^{-}, \end{aligned}$$

(3.191)

the effective nuclear flip-flop mediated dynamics simplifies to

$$\begin{aligned} \dot{\sigma }= & {} \sum _{l}\frac{\tilde{\gamma }_{l}}{2}\left[ \tilde{\mathbf {A}}_{l}\sigma \tilde{\mathbf {A}}_{l}^{\dagger }-\frac{1}{2}\left\{ \tilde{\mathbf {A}}_{l}^{\dagger }\tilde{\mathbf {A}}_{l},\sigma \right\} \right] \nonumber \\&+\,i\sum _{k,l}\frac{\tilde{\Delta }_{kl}}{2}\left[ \tilde{\mathbf {A}}_{l}^{\dagger }\tilde{\mathbf {A}}_{k},\sigma \right] , \end{aligned}$$

(3.192)

where the matrix $\tilde{\Delta }_{kl}=\sum _{ij}U_{ki}^{\dagger }\Delta _{ij}U_{jl}$ associated with second-order Stark shifts is in general not diagonal. This gives rise to the Stark term mediated criticality in the nuclear spin dynamics.

In general, the matrices $\gamma ^{\pm }$ have $\mathrm {rank}\left( \gamma ^{\pm }\right) =2$, yielding four non-zero decay rates $\tilde{\gamma }_{1,2}^{\pm }$ and four linear independent Lindblad operators $\tilde{\mathbf {A}}_{l}$; therefore, in general, no pure, nuclear dark state $\left| \Psi _{\mathrm {dark}}\right\rangle$ fulfilling $\tilde{\mathbf {A}}_{l}\left| \Psi _{\mathrm {dark}}\right\rangle =0$ $\forall l$ exists. In contrast, when keeping only the supposedly dominant coupling to the electronic eigenstate $\left| \lambda _{2}\right\rangle$, they simplify to

$$\begin{aligned} \gamma _{\mathrm {ideal}}^{\pm }=\gamma _{2}^{\pm }\left( \begin{array}{cc} \nu _{2}^{2} &{} \mu _{2}\nu _{2}\\ \mu _{2}\nu _{2} &{} \mu _{2}^{2} \end{array}\right) , \end{aligned}$$

(3.193)

which fulfills $\mathrm {rank}\left( \gamma _{\mathrm {ideal}}^{\pm }\right) =1$. Still, also in the non-ideal setting, for realistic experimental parameters we observe a clear hierarchy in the eigenvalues, namely $\tilde{\gamma }_{2}^{\pm }/\tilde{\gamma }_{1}^{\pm }\,\lesssim \,0.1$.

Fig. 3.23

a Knight shift $\omega _{\mathrm {hf}}$ due to nonzero populations $p_{k}$ in the electronic quasisteady state for $t=20\,\upmu \mathrm {eV}$ (solid) and $t=30\,\upmu \mathrm {eV}$ (dashed). b In the high gradient regime, for $\Gamma \gg \Gamma _{\pm }$ the levels $\left| \lambda _{k}\right\rangle$ get depleted efficiently, such that $p_{k}<1\,\%\ll p$. Other numerical parameters: $\varepsilon =30\,\upmu \mathrm {eV}$ and $x_{\pm }=10^{-3}$

3.1.11 3.11.11 Nonidealities in Electronic Quasisteady State

In Sect. 3.7 we have analyzed the nuclear spin dynamics in the submanifold of the electronic quasisteady state $\rho _{\mathrm {ss}}^{\mathrm {el}}=\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| +\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) /2$. In this Appendix we consider (small) deviations from this ideal electronic quasisteady state due to populations of the levels $\left| \lambda _{k}\right\rangle \left( k=1,2,3\right)$, labeled as $p_{k}$. Since all coherences are damped out on electronic timescales, the generalized electronic quasisteady state under consideration is

$$\begin{aligned} \rho _{\mathrm {ss}}^{\mathrm {el}}=p\left( \left| T_{+}\right\rangle \left\langle T_{+}\right| +\left| T_{-}\right\rangle \left\langle T_{-}\right| \right) +\sum _{k}p_{k}\left| \lambda _{k}\right\rangle \left\langle \lambda _{k}\right| . \end{aligned}$$

(3.194)

Using detailed balance, $p_{k}$ can be calculated via the equations $p_{k}\left( \kappa _{k}^{2}+x_{\pm }\right) =px_{\pm }$, where $x_{\pm }=\Gamma _{\pm }/\Gamma$ and $p=\left( 1-\sum p_{k}\right) /2$ gives the population in $\left| T_{\pm }\right\rangle$, respectively. The electronic levels $\left| \lambda _{k}\right\rangle$ get depleted efficiently for $\Gamma _{k}\gg \Gamma _{\pm }$: In contrast to the low-gradient regime where $p_{2}\approx 1/3$, in the high-gradient regime, we obtain $p_{k}<1\,\%\ll p$ such that the electronic system settles to a quasisteady state very close to the ideal limit where $p=1/2$; compare Fig. 3.23. In describing the effective nuclear dynamics, nonzero populations $p_{k}$ lead to additional terms which are second order in $\varepsilon$, but strongly suppressed further as $p_{k}\ll 1$.

Knight shift.—For nonzero populations $p_{k}$, the Knight shift seen by the nuclear spins does not vanish, leading to the following (undesired) additional term for the effective nuclear spin dynamics

$$\begin{aligned} \dot{\sigma }=-i\omega _{\mathrm {hf}}\left[ \delta A_{1}^{z}-\delta A_{2}^{z},\sigma \right] , \end{aligned}$$

(3.195)

where

$$\begin{aligned} \omega _{\mathrm {hf}}=\frac{a_{\mathrm {hf}}}{2}\sum _{k}p_{k}\left( \mu _{k}^{2}-\nu _{k}^{2}\right) . \end{aligned}$$

(3.196)

with $a_{\mathrm {hf}}\approx 10^{-4}\,\upmu \mathrm {eV}$. As shown in Fig. 3.23, however, $\omega _{\mathrm {hf}}\approx 10^{-7}\,\upmu {{\mathrm {eV}}}$ is further suppressed by approximately three orders of magnitude; in particular, $\omega _{\mathrm {hf}}$ is small compared to the dissipative gap of the nuclear dynamics $\mathrm {ADR}\approx 2\times 10^{-5}\,\upmu \mathrm {eV}$ and can thus be neglected.

Hyperfine flip-flop dynamics.—Moreover, nonzero populations $p_{k}$ lead to additional Lindblad terms of the form $\dot{\sigma }=\cdots +p_{k}\gamma _{k}^{+}\mathcal {D}[L_{k}^{\dagger }]\sigma$. They contain terms which are incommensurate with the ideal two-mode squeezedlike target state. Since $p_{k}\ll p$, however, they are strongly suppressed compared to the ones absorbed into $\mathcal {L}_{\mathrm {nid}}$ and thus do not lead to any significant changes in our analysis.