Nuclear-spin polaron formation: Anisotropy effects and quantum phase transition

We study theoretically the formation of the nuclear-spin polaron state in semiconductor nanosystems within the Lindblad equation approach. To this end, we derive a general Lindblad equation for the density operator that complies with the symmetry of the system Hamiltonian and address the nuclear-spin polaron formation for localized charge carriers subject to an arbitrarily anisotropic hyperfine interaction when optically cooling the nuclei. The steady-state solution of the density matrix for an anisotropic central spin model is presented as a function of the electron and nuclear spin bath temperature. Results for the electron-nuclear spin correlator as well as data for the nuclear spin distribution function serve as a measure of spin-entanglement. The features in both of them clearly indicate the formation of the nuclear polaron state at low temperatures where the crossover regime coincides with an enhancement of quantum fluctuations and agrees with the mean-field prediction of the critical temperature line. We can identify two distinct polaron states dependent upon the hyperfine anisotropy, which are separated by a quantum phase transition at the isotropic point. These states are reflected in the temporal spin autocorrelation functions accessible in experiment via spin-noise measurements.

Figure 1

Electron-nuclear spin correlation as a function of the inverse nuclear spin temperature ${\beta }_n$ for various anisotropy factors $\lambda$ of the hyperfine interaction. The inverse electron spin temperature is fixed at ${\beta }_e{\omega }_h=0.5$. The dashed vertical-red lines correspond to the transition temperatures according to the analytical Eq. (30). Mean-field results are added as turquoise dotted lines. Note that from the definition below Eq. (8), $A_0={\omega }_h/\sqrt{N}$.

Figure 2

Fluctuations ${\sigma }_{SJ}^2$ of the electron-nuclear spin correlator, Eq. (31), as a function of the effective inverse nuclear spin temperature ${\beta }_n$ and the effective inverse electron spin temperature ${\beta }_e$ for different values of the hyperfine anisotropy parameter, (a) $\lambda =1$ and (b) $\lambda =2$, and (c) at a fixed electron spin temperature, ${\beta }_e{\omega }_h=0.4$.

Figure 3

Fluctuations ${\sigma }_{SJ}^2$ of the electron-nuclear spin correlator in the isotropic system ($\lambda =1$) as a function of the effective inverse nuclear spin temperature ${\beta }_n$ for a fixed inverse electron spin temperature, (a) ${\beta }_e{\omega }_h=0.4$ and (b) ${\beta }_e{\omega }_h=0.4\sqrt{1000/N}$. For panel (b) we scaled ${\beta }_e$ to follow the shift of the maximum fluctuations to smaller inverse electron temperatures with increasing the nuclear bath size $N$ [see legend in panel (b)].

Figure 4

Distribution function of the nuclear spin quantum numbers $J^z$ (upper panels) and ${J^p}^2$ (lower panels) for three typical values of the anisotropy factor $\lambda$ of the hyperfine interaction with $N=1000$ nuclear spins. The inverse nuclear spin temperature ${\beta }_n$ is displayed on the horizontal axis; the inverse electron spin temperature is fixed at ${\beta }_e{\omega }_h=0.5$.

Figure 5

Distribution function of the nuclear spin quantum numbers $J^z$ (upper panels) and ${J^p}^2$ (lower panels) for various anisotropy factors $\lambda$ of the hyperfine interaction, see legend in panel (c). The inverse nuclear spin temperature is set to ${\beta }_n{\omega }_h=80$ [(a),(c)] or ${\beta }_n{\omega }_h=270$ [(b),(d)]; the inverse electron spin temperature is fixed at ${\beta }_e{\omega }_h=0.5$.

Figure 6

Distribution function of the nuclear spin quantum number $J^z$. (a) Dependence on the anisotropy factor $\lambda$ of the hyperfine interaction for fixed inverse spin temperatures, ${\beta }_n{\omega }_h=1000, {\beta }_e{\omega }_h=0.5$. [(b),(c)] Dependence on the inverse nuclear spin temperature with $\lambda$ adjusted slightly below (b) or above one (c); ${\beta }_e{\omega }_h=0.5$.

Figure 7

Temporal fluctuations of the electron spin components for different values of the hyperfine anisotropy parameter $\lambda$ in (a), (b), and (c). Results for various effective inverse nuclear spin temperatures ${\beta }_n$ are presented respectively whereas ${\beta }_e{\omega }_h=0.5$ is kept constant.

Figure 8

Temporal fluctuations of the nuclear spin $z$ component for (a) the Ising limit, (b) the isotropic system, and (c) the anisotropic system with $\lambda =2$. The inverse electron spin temperature ${\beta }_e$ is fixed; the inverse nuclear spin temperature ${\beta }_n$ is encoded by different colors. (d) Decay time ${\tau }_d$ of the nuclear spin fluctuations for different values of the hyperfine anisotropy parameter $\lambda$. The transition temperature, Eq. (30), is added as a red dashed-vertical line.

Figure 9

Temporal fluctuations of the electron spin components perpendicular to the directions favored by hyperfine interaction for a hyperfine anisotropy parameter (a) $\lambda =0$ and (b) $\lambda =2$. Results for various effective inverse nuclear spin temperatures ${\beta }_n$ are presented respectively whereas ${\beta }_e{\omega }_h=0.5$ is kept constant.