Enhancement of Electron Spin Coherence by Optical Preparation of Nuclear Spins

We study a large ensemble of nuclear spins interacting with a single electron spin in a quantum dot under optical excitation and photon detection. At the two-photon resonance between the two electron-spin states, the detection of light scattering from the intermediate exciton state acts as a weak quantum measurement of the effective magnetic (Overhauser) field due to the nuclear spins. In a coherent population trapping state without light scattering, the nuclear state is projected into an eigenstate of the Overhauser field operator, and electron decoherence due to nuclear spins is suppressed: We show that this limit can be approached by adapting the driving frequencies when a photon is detected. We use a Lindblad equation to describe the driven system under photon emission and detection. Numerically, we find an increase of the electron coherence time from 5 to 500 ns after a preparation time of $10\mu \mathrm{s}$.

Figure 1

Three-level system. State 1 (2) is a spin-up (-down) conduction-band ($E_C$) electron, with splitting $g{\mu }_B{B}_{\mathrm{tot}}+\delta h_z$, where $\delta h_z$ is the $z$ component of the nuclear (Overhauser) field fluctuations. State 3 is a trion with $J_{z^{\prime }}=3/2$. Two laser fields with frequencies ${\omega }_p$ and ${\omega }_c$ are applied near the 13 and 23 resonances with detunings ${\Delta }_{1,2}$. For a ${\sigma }_{+}$ circularly polarized excitation (along $z^{\prime }$), both transitions are allowed for $\theta \ne 0$ and transitions to the $J_{z^{\prime }}=-3/2$ states are forbidden. Inset: Structural axis $z^{\prime }$, leading to a splitting in $E_V$ and spin quantization axis $z\parallel {\mathbf{B}}_{\mathrm{tot}}$ in $E_C$ where $\cos \theta =z·z^{\prime }<1$.

Figure 2

Conditional evolution of the nuclear spin distribution $\nu (\delta h_z^k)={\nu }_{kk}$. (a) During the first period $t_1$ without photon emission, the initial Gaussian distribution (solid line) develops a peak at the two-photon resonance (dashed line). (b) Change of $\nu (\delta h_z)$ after emission at $t_1$ (solid line), until before emission time $t_2$ of the second photon (dashed line). The two-photon resonance $\delta$ is shifted to the position of the left maximum (adaptive technique). The depleted region around $\delta h_z^k=0$ develops at $t_1$. (c) Analogous situation between $t_{11}$ and $t_{12}$. (d) $\nu (\delta h_z)$ is obtained after a total time of $10\mu \mathrm{s}$. Inset: Magnification of peak in (d). The width of $\nu (\delta h_z)$ is reduced by a factor of $\approx 100$ compared to the initial width in (a). The parameters are ${\Omega }_c={\Omega }_p=0.2{\mathrm{ns}}^{-1}$, $\Delta =0$, ${\Gamma }_{X\uparrow }={\Gamma }_{X\downarrow }=1{\mathrm{ns}}^{-1}$, and ${\gamma }_{\downarrow }={\gamma }_X=0.001{\mathrm{ns}}^{-1}$.

Figure 3

Electron coherence function $|⟨S_{+}(t)⟩|/|⟨S_{+}(0)⟩|$ vs electronic precession time $t$ calculated from $\nu (\delta h_z)$ in Fig. 2 after emission of the $n$th photon ($n=1,6,\dots ,26$). The initial decay is approximately Gaussian.

Figure 4

Characteristic time $t_0$ of the initial Gaussian decay of $|⟨S_{+}(t)⟩|/|⟨S_{+}(0)⟩|$ in Fig. 3 as a function of the optical preparation time $t$, averaged over 50 numerical runs (error bars indicate the standard deviation).