Passive stabilization of a hole spin qubit using the optical Stark effect

The extrinsic dephasing of a hole spin confined to a self-assembled quantum dot is dominated by charge noise acting on an electric-field-dependent $g$-factor. Here we propose the use of the optical Stark effect to reduce the sensitivity of the effective hole Zeeman energy to fluctuations in the local electric field. Calculations using measured quantum dot parameters are presented, and they demonstrate a factor of 10–100 reduction in the extrinsic dephasing. Compared to active stabilization methods, this technique should benefit from reduced experimental complexity.

Figure 1

Energy-level diagram of heavy-hole trion system in a Voigt geometry. The hole and trion spin states are aligned parallel and antiparallel with the magnetic field along the $x$ direction, and separated by their respective Zeeman energies. The case of a linearly $x$-polarized laser positively detuned from both optical transitions is sketched. For a strongly detuned laser the trion state is not significantly populated, and the different ac Stark shifts on the hole spin states act to reduce the hole Zeeman splitting. For $y$ polarization the cross transitions are activated, and the ac Stark shifts increases the hole Zeeman energy.

Figure 2

(a) Left axis: $E$-field dependence of hole Zeeman energy. $x$ polarization $(B=5\mathrm{T},{\omega }_L-{\omega }_0=-1\mathrm{meV},\mathrm{\Omega }=100\mu \mathrm{eV})$. Right axis: At the turning point TP, the hole Zeeman energy is relatively insensitive to the $E$ field, leading to a drop in the extrinsic dephasing rate due to $E$-field fluctuations by three orders of magnitude. (b) Closeup of the turning point. Red lines indicate a factor of (10, 100, 1000) improvement in extrinsic dephasing time. The top axis indicates the change in gate voltage required to change the electric field, assuming a gate separation of 230 nm [18].

Figure 3

Gray-scale plot of the suppression factor of extrinsic dephasing $S=\frac{{\mathrm{\Gamma }}^{*}}{{\mathrm{\Gamma }}^{*}(\mathrm{\Omega }=0)}$ vs laser detuning and Rabi energy. (a) $x$ polarization; (b) $y$ polarization; (c)–(e) cross sections at fixed Rabi energy $\mathrm{\Omega }$ for $y$ polarization. $B=5$ T. Power fluctuations of $\mathrm{\Delta }{\mathrm{\Omega }}^2=\alpha {\mathrm{\Omega }}^2, \alpha =0.01$ are used.

Figure 4

Gray-scale plot of the suppression factor against detuning and Rabi energy for a magnetic field of $B=1.0\mathrm{T}$.