Fast Initialization of the Spin State of an Electron in a Quantum Dot in the Voigt Configuration

We consider the initialization of the spin state of a single electron trapped in a self-assembled quantum dot via optical pumping of a trion level. We show that with a magnetic field applied perpendicular to the growth direction of the dot, a near-unity fidelity can be obtained in a time equal to a few times the inverse of the spin-conserving trion relaxation rate. This method is several orders of magnitude faster than with the field aligned parallel, since this configuration must rely on a slow hole spin-flip mechanism. This increase in speed does result in a limit on the maximum obtainable fidelity, but we show that for InAs dots, the error is very small.

Figure 1

The four levels of the electron-trion system in the Voigt basis consists of two Zeeman-split single-electron ground states $|x\pm ⟩$ with spins in the $x$ direction, and two trion levels $|\tau \pm ⟩$ with heavy-hole spins also in the $x$ direction. Arrows indicate allowed optical transitions with $H$, $V$ denoting two orthogonal linear polarizations. State preparation is achieved by resonantly pumping the $V_1$ transition. This populates the trion level $|{\tau }_{+}⟩$, which subsequently relaxes with rate $\Gamma$ back to both ground states, resulting in a partial transfer of population from $|x+⟩$ to $|x-⟩$. This simple picture is complicated by the fact that the same laser also drives the $V_2$ transition, albeit off resonantly. This results in a small pumping of population in the opposite direction and hence a slight decrease in the maximum obtainable purity. We take the relaxation rate $\Gamma =1.2\mu \mathrm{eV}$ and $g$ factors $g_x^e=-0.46$ and $g_x^h=-0.29$. For a field of 1 T, the Zeeman splitting are then $E_B^e=-27\mu \mathrm{eV}$ and $E_B^h=17\mu \mathrm{eV}$.

Figure 2

The characteristic time $T_0$ describing the approach of the system to its asymptotic limit as a function of the Rabi frequency $\Omega$. When $\Omega /\Gamma \gtrsim 1$, $T_0\approx 2/\Gamma$. Inset: The population of the state $|x-⟩$ as a function of time under continuous illumination. The solid black line shows the result of numerical integration of the master equation, whereas the dashed blue line shows the analytic result of Eq. (7). Near-unity fidelity is approached with a characteristic time of $T_0\approx 1\mathrm{ns}$. For both plots, the parameters are the same as in Fig. 1 and for the inset we have further set $\Omega =\Gamma$.

Figure 3

The logarithm of the error $ϵ=1-F$, with $F$ the fidelity, as a function of the Rabi energy $\Omega$. Results are shown for three different fields: 1 T (black, solid), 4 T (blue, dash), and 8 T (green, dash-dot). Other parameters as Fig. 1.