Interplay and optimization of decoherence mechanisms in the optical control of spin quantum bits implemented on a semiconductor quantum dot

We study the influence of the environment on an optically induced rotation of a single electron spin in a charged semiconductor quantum dot. We analyze the decoherence mechanisms resulting from the dynamical lattice response to the charge evolution induced in a trion-based optical spin control scheme. Moreover, we study the effect of the finite trion lifetime and of the imperfections of the unitary evolution such as off-resonant excitations and the nonadiabaticity of the driving. We calculate the total error of the operation on a spin-based qubit in an $\mathrm{In}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot system and discuss possible optimization against the different contributions. We indicate the parameters which allow for coherent control of the spin with a single qubit gate error as low as ${10}^{-4}$.

Figure 1

(a) Schematic plot of the QD with the direction of incidence of the ${\sigma }_{+}$-polarized laser beam and the orientation of the magnetic field. (b) Three-level system in a single doped QD.

Figure 2

(a) The tipping angle and its derivative as a function of time for a detuning $\hslash \Delta =1\mathrm{meV}$ and a pulse duration ${\tau }_p=5\mathrm{ps}$. (b) The adjusted amplitude of the control pulse for the $\pi ∕2$ rotation about the $z$ axis as a function of detuning for different fixed pulse durations.

Figure 3

The error due to the unitary corrections as a function of detuning for two values of the Zeeman splitting ${\Delta }_{\mathrm{B}}$ and three values of the pulse duration: ${\tau }_{\mathrm{p}}=5\mathrm{ps}$ (solid lines), ${\tau }_{\mathrm{p}}=10\mathrm{ps}$ (dashed lines), and ${\tau }_{\mathrm{p}}=30\mathrm{ps}$ (dotted lines).

Figure 4

Spectral density of the phonon reservoir at different temperatures $T$.

Figure 5

(a) and (b) Spectral function $s_1^{\mathrm{ph}}\left(\omega \right)$. (c) and (d) Spectral function $s_2^{\mathrm{ph}}\left(\omega \right)$. In (b) and (d) the line styles denote pulse durations as defined in (d).

Figure 6

(a) The dependence of the error due to pure dephasing for growing detuning for ${\tau }_{\mathrm{p}}=10\mathrm{ps}$. (b) The pure dephasing error as a function of pulse duration at $T=5\mathrm{K}$ and for five values of detuning: $\hslash \Delta =0.066\mathrm{meV}$ (solid line), $\hslash \Delta =0.66\mathrm{meV}$ (longer dashed line), $\hslash \Delta =1.32\mathrm{meV}$ (shorter dashed line), $\hslash \Delta =2\mathrm{meV}$ (dotted line), and $\hslash \Delta =2.64\mathrm{meV}$ (dashed-dotted line). (c) and (d) As in (a) and (b) but for the error due to phonon-assisted trion generation. (e) and (f) As in (a) and (b) but for the total error resulting from the interaction of the carriers with the phonon reservoir. In (a), (c), and (e), the line styles denote temperatures as defined in (a). In (b), (d), and (f), the line styles denote detunings as defined in (b).

Figure 7

The phonon-induced error for negative detunings (a) at different temperatures for growing detuning and (b) for growing pulse duration.

Figure 8

The dependence of the radiative error for growing (a) detuning and (b) pulse duration.

Figure 9

Interplay of the error contributions as a function of (a) and (c) detuning and (b) and (d) pulse duration at $T=5\mathrm{K}$. Different lines represent various error sources, as defined in (b).

Figure 10

The total error for growing (a) detuning and (b) pulse duration at $T=5\mathrm{K}$.

Figure 11

The radiative error calculated by means of the Lindblad (solid lines) and perturbative (dashed lines) methods for growing detuning.