Quantum limit for nuclear spin polarization in semiconductor quantum dots

A recent experiment [E. A. Chekhovich et al., Phys. Rev. Lett. 104, 066804 (2010)] has demonstrated that high nuclear spin polarization can be achieved in self-assembled quantum dots by exploiting an optically forbidden transition between a heavy hole and a trion state. However, a fully polarized state is not achieved as expected from a classical rate equation. Here, we theoretically investigate this problem with the help of a quantum master equation and we demonstrate that a fully polarized state cannot be achieved due to formation of a nuclear dark state. Moreover, we show that the maximal degree of polarization depends on structural properties of the quantum dot.

Figure 1

(a) Level scheme for electronic and nuclear states. The hyperfine sublevels are denoted with their total nuclear spin quantum numbers $j$ and $m$. The trion state $|{\mathrm{t}}_{\downarrow }\rangle$ with angular momentum (along $z$) $M=-\frac{1}{2}$ is pumped by the laser light with Rabi frequency $\Omega$ from the heavy-hole state $|{\mathrm{h}}_{\downarrow }\rangle$ with $M=-\frac{3}{2}$. The hyperfine interaction couples the trion state $|{\mathrm{t}}_{\downarrow }\rangle$ and $|{\mathrm{t}}_{\uparrow }\rangle$ ($M=\frac{1}{2}$) and changes the $m$ quantum number of the nuclear system. The trion states can relax by spontaneous emission with the rate ${\Gamma }_{\mathrm{sp}}$. (b) Reduced level scheme with mechanisms for nuclear polarization in the trion–heavy-hole system by spin-forbidden relaxation with rate ${\Gamma }_{\chi {\chi }^{\prime }}$ from $|{\mathrm{t}}_{\downarrow }\rangle$, or by spin-forbidden optical transitions between $|{\mathrm{h}}_{\downarrow }\rangle$ and $|{\mathrm{t}}_{\uparrow }\rangle$.

Figure 2

Nuclear spin polarization calculated for the case of homogeneous hyperfine coupling constant. (a) Saturation of the polarization for $N=30$ by pumping the spin-forbidden transition at a laser detuning $\Delta =-{\omega }_Z^{\mathrm{e}}=-220\mu \mathrm{eV}$ and the allowed transition at $\Delta =0$. (b) Same as (a), but with $N=100$. (c) The degree of nuclear polarization for $N=30$ and 100 after a pumping time $t_{\mathrm{pump}}=0.2\mathrm{ms}$ as function of laser detuning $\Delta$. The maximal degree of polarization observed for $\Delta =-{\omega }_Z^{\mathrm{e}}$ reduces for increasing number of spins. The vertical lines at $\Delta =-{\omega }_Z^{\mathrm{e}}$ and $\Delta =0$ are visual guides to emphasize laser dragging effects and the change of optical resonance. Inset: magnification around $\Delta =0$ showing a difference in the polarization degree between $N=30$ and 100.

Figure 3

Distribution $p_j\left(j\right)$ of the total angular momentum $j$. The larger the system gets, the smaller the value of $p_j$ becomes for the most likely $j\sim \sqrt{N/2}$ and the faster it converges to 0.

Figure 4

Nuclear polarization calculated for inhomogeneous hyperfine interaction. Due to the limitation of the computing power at our disposal, we have simulated two groups of nuclear spins with coupling strengths $A_1^{\mathrm{e}}={10}^7$ Hz and $A_2^{\mathrm{e}}={10}^5$ Hz: the saturation of the polarization for the laser detuning at the forbidden ($\Delta =-{\omega }_Z^{\mathrm{e}}$) and allowed ($\Delta =0$) transitions for (a) $N_1=4,N_2=2$ and (b) $N_1=6,N_2=4$. (c) Nuclear polarization for different laser detunings and for different number of spins divided into two groups for $t_{\mathrm{pump}}=0.08\mathrm{s}$. The vertical lines at $\Delta =-{\omega }_Z^{\mathrm{e}}$ and 0 are visual guides to emphasize the shift of the optical resonance.

Figure 5

Saturation of nuclear polarization calculated for nuclear spins divided into three groups and inhomogeneously coupled to the electronic spin. The coupling strengths are $A_1^{\mathrm{e}}={10}^8$Hz, $A_2^{\mathrm{e}}={10}^7$Hz, and $A_3^{\mathrm{e}}={10}^6$Hz. (a) Polarization at $\Delta =-{\omega }_Z^{\mathrm{e}}$ and (b) at $\Delta =0$.

Figure 6

(a) Hyperfine coupling constants $A_k$ and (b) number of nuclear spins $N_k$ as a function of the distance to the maximum of the wave function. The distance is given in units of the lattice constant $a$. We have plotted in red the continuous distribution of hyperfine coupling constants $A\left(\mathit{r}\right)=A_{0}exp[-{(r/r_0)}^2]$. (c) Same as (a) for two sublattices $s=\mathrm{A},\mathrm{B}$. We denote the hyperfine coupling constant by $A_k^s$. (d) Number of nuclear spins $N_k^s$ in each sublattice.

Figure 7

(a) Hyperfine coupling constants $A_k$ and (b) number of nuclear spins $N_k$ as a function of the distance to the maximum of the wave function, which is located at a lattice site. The distance is measured in units of the lattice constant $a$. The continuous distribution of hyperfine coupling constants $A\left(\mathit{r}\right)=A_{0}exp[-{(r/r_0)}^2]$ is shown in red. (c) Same as (a) for a system made of two nuclear species divided into two sublattices ($s=\mathrm{A},\mathrm{B}$). (d) Number of nuclear spins $N_k^s$ for $s=\mathrm{A},\mathrm{B}$.