Steady-State Entanglement in the Nuclear Spin Dynamics of a Double Quantum Dot

We propose a scheme for the deterministic generation of steady-state entanglement between the two nuclear spin ensembles in an electrically defined double quantum dot. Because of quantum interference in the collective coupling to the electronic degrees of freedom, the nuclear system is actively driven into a two-mode squeezedlike target state. The entanglement buildup is accompanied by a self-polarization of the nuclear spins towards large Overhauser field gradients. Moreover, the feedback between the electronic and nuclear dynamics leads to multistability and criticality in the steady-state solutions.

Figure 1

(a) Schematic illustration of nuclear entanglement generation via electron transport. Whenever the Pauli blockade is lifted via the HF interaction with the nuclear spins, a nuclear flip can occur in either of the two dots. The local nature of the HF interaction is masked by the nonlocal character of the electronic level $|{\lambda }_2⟩$. (b) Spectrum of $H_{\mathrm{el}}$ for $\Delta =40\mu \mathrm{eV}$ and $t=30\mu \mathrm{eV}$. The three eigenstates $|{\lambda }_k⟩$ are displayed in red. The triplets $|T_{\pm }⟩$ are degenerate for ${\omega }_0=0$. In this setting, lifting of the spin blockade due to HF interaction is predominantly mediated by the nonlocal jump operators required for two-mode squeezing, namely, $L_2$ and ${\mathbb{L}}_2$. The ellipse refers to a potential operational area of our scheme. (c) The resulting effective three-level system $\{|T_{\pm }⟩,|{\lambda }_2⟩\}$ including coherent HF coupling and the relevant dissipative processes: $|{\lambda }_2⟩$ decays according to its overlap with $|S_{02}⟩$ with an effective decay rate ${\Gamma }_2=|⟨{\lambda }_2|S_{02}⟩{|}^2\Gamma$ [43]. Within this three-level subspace, purely electronic Pauli-blockade lifting mechanisms like cotunneling or spin-orbital effects result in effective dephasing and dissipative mixing rates, labeled as ${\gamma }_{\mathrm{deph}}$ and ${\gamma }_{\pm }$, respectively.

Figure 2

Steady-state entanglement between the two nuclear spin ensembles quantified via (a) the EPR uncertainty ${\Delta }_{\mathrm{EPR}}$ and (b) fidelity $\mathcal{F}$ of the nuclear steady state with the two-mode squeezed target state. The black solid curve refers to the idealized setting where the undesired HF coupling to $|{\lambda }_{1,3}⟩$ has been ignored and where $J_1=J_2=p{J}_{\max }$, $p=0.8$, and $N_1=N_2=2J_{\max }={10}^6$, corresponding to ${\Delta }_{\mathrm{OH}}=40\mu \mathrm{eV}$. The blue-dashed line then also takes into account coupling to $|{\lambda }_{1,3}⟩$ while the red-dashed curve, in addition, accounts for an asymmetric dot size: $N_2=0.8N_1=8\times {10}^5$. The amount of entanglement decreases for a smaller nuclear polarization: $p=0.7$ (green dashed curve). Classical uncertainty (symbols) in the total spin $J_i$ quantum numbers leads to less entanglement, but does not destroy it. Here, we have set the range of the distribution to ${\Delta }_{J_i}=50\sqrt{N_i}$. Other numerical parameters: ${\omega }_0=0$, $\Gamma =25\mu \mathrm{eV}$, $ϵ=30\mu \mathrm{eV}$, and ${\gamma }_{\pm }+{\gamma }_{\mathrm{deph}}/2=1\mu \mathrm{eV}$.

Figure 3

Semiclassical solution to the nuclear polarization dynamics. (a) Instantaneous nuclear polarization rate ${\dot{\Delta }}_{I^z}$ as a function of the gradient $\Delta$ for $t=20\mu \mathrm{eV}$ (blue dashed), $t=30\mu \mathrm{eV}$ (red solid), and $t=50\mu \mathrm{eV}$ (black dash-dotted). FPs are found at ${\dot{\Delta }}_{I^z}=0$. The ovals mark stable high-gradient steady-state solutions. The background coloring refers to the sign of ${\dot{\Delta }}_{I^z}$ (for $t=30\mu \mathrm{eV}$) which determines the stable FP the nuclear system is attracted to (see arrows). (b) Zoom-in of (a) into the low-gradient regime: The unpolarized FP lies at $\Delta =0$, whereas critical, instable points ${\Delta }_{\mathrm{OH}}^{\mathrm{crt}}$ (marked by stars) can be identified with ${\dot{\Delta }}_{I^z}=0$ and $d{\dot{\Delta }}_{I^z}/d\Delta >0$. (c) Stable high-polarization FPs ${\Delta }_{\mathrm{OH}}^{\mathrm{ss}}$ (see ovals) as a function of $t$; for $t\approx 10\mu \mathrm{eV}$ we obtain a nuclear polarization of $\sim 90\%$. Other numerical parameters: $\Gamma =25\mu \mathrm{eV}$, $ϵ=30\mu \mathrm{eV}$, ${\gamma }_{\pm }=0.3\mu \mathrm{eV}$, and ${\gamma }_{\mathrm{deph}}=0.5\mu \mathrm{eV}$.