High-fidelity all-optical control of quantum dot spins: Detailed study of the adiabatic approach

Confined electron spins are preferred candidates for embodying quantum information in the solid state. A popular idea is the use of optical excitation to achieve the “best of both worlds,” i.e., marrying the long spin decoherence times with rapid gating. Here, we study an all-optical adiabatic approach to generating single qubit phase gates. We find that such a gate can be extremely robust against the combined effect of all principal sources of decoherence, with an achievable fidelity of 0.999 even at finite temperature. Crucially, this performance can be obtained with only a small time cost: the adiabatic gate duration is within about an order of magnitude of a simple dynamic implementation. An experimental verification of these predictions is immediately feasible with only modest resources.

Figure 1

(Color online) Pauli blocking effect: Heavy holes occupy the lowest energy valence band states with spin $m_z=\pm 3∕2$, whereas light holes have spin $m_z=\pm 1∕2$ at a slightly higher energy. Therefore, ${\sigma }^{+}$ light cannot excite electron hole pairs if the qubit is in the state ∣0⟩ with $m_z=-1∕2$. On the other hand, if the qubit is in the ∣1⟩ configuration, the excitation is possible, leading to a three particle trion $\mid X⟩$ state.

Figure 2

(Color online) Energy of the dressed states, shown as a function of the detuning $\Delta$ in arbitrary units. The ∣−⟩ state tends to ∣1⟩, and ∣+⟩ to $\mid X⟩$, for a large positive detuning. At resonance, $\Delta =0$, the dressed states are an equal superposition of the bare basis states. The decoupled ∣0⟩ state is also shown, its energy remaining constant. Adiabatic following: population in the ∣0⟩ and ∣1⟩ states follows the respective instantaneous eigenstates if the mixing angle $\theta$ changes sufficiently slowly. ∣0⟩ is unaffected by the laser pulse, but ∣1⟩ goes into ∣−⟩ and back. Due to the energy shift of the ∣−⟩ state, a dynamical phase accumulates relative to ∣0⟩.

Figure 3

(Color online) Radiative decay from manifold $\mathcal{M}\left(N\right)$ to $\mathcal{M}(N-1)$. Left: The uncoupled basis. The energy between the two states in each manifold is the detuning $\Delta$, and the solid arrows correspond to absorption and stimulated emission processes, whereas the wavy arrows denote spontaneous emission. Right: Allowed spontaneous emission transitions between the dressed states. The energetic splitting in each manifold is the effective Rabi frequency ${\Omega }^{\prime }=\sqrt{{\Delta }^2+{\Omega }^2}$ and the spacing between adjacent manifolds is the laser frequency ${\omega }_l$. The left emission process is at frequency ${\omega }_l+{\Omega }^{\prime }$, the two center lines emit at the frequency of the laser, and the right at ${\omega }_l-{\Omega }^{\prime }$; this is the Mollow triplet well known in quantum optics (Ref. 17).

Figure 4

(Color online) Any significant population of the excitonic state makes decay events inevitable. The expected number of decays (for ${\Gamma }_0=0.01{\mathrm{ps}}^{-1}$) during an adiabatic $R_Z\left(\pi \right)$ gate is shown as a function of $\Delta$. We depict the analytical prediction (red) along with full numerical solutions, for which ${\Omega }_0$ is given in meV. The inset shows the corresponding pulse durations $\tau$ in ps.

Figure 5

(Color online) Phonons affect dynamic and adiabatic quantum operations differently. To characterize decoherence, we present the purity of the system’s density matrix during an $R_z\left(\pi \right)$ operation for deformation potential coupling at $T=5\mathrm{K}$. The dynamic gate uses $\Omega =0.1\mathrm{meV}$ and time is shown in units of $2\pi ∕\Omega$. Adiabatic gates are performed with ${\Omega }_0=1\mathrm{meV}$ and the units of $\Delta$ are meV. We have scaled the time axis by dividing by $6\tau$ and adding 0.5 (thus mapping the actual integration interval from $-3\tau$ to $3\tau$ to the scale of this figure). The inset shows the deformation potential spectral density of GaAs [Eq. (16)].

Figure 6

(Color online) Comparison of the dynamic versus the adiabatic gate for an $R_Z\left(\pi \right)$ operation by showing the overall gate fidelity. Left: Fidelity of the dynamic operation as a function of the coupling strength $\Omega$. Right: Fidelity of the adiabatic operation for a fixed ${\Omega }_0=1\mathrm{meV}$ as a function of the detuning.