Simultaneous SU(2) rotations on multiple quantum dot exciton qubits using a single shaped pulse

Recent experimental demonstration of a parallel $(\pi ,2\pi )$ single qubit rotation on excitons in two distant quantum dots [Nano Lett. 13, 4666 (2013)] is extended in numerical simulations to the design of pulses for more general quantum state control, demonstrating the feasibility of full SU(2) rotations of each exciton qubit. Our results show that simultaneous high-fidelity quantum control is achievable within the experimentally accessible parameter space for commercial Fourier-domain pulse shaping systems. The identification of a threshold of distinguishability for the two quantum dots (QDs) for achieving high-fidelity parallel rotations, corresponding to a difference in transition energies of $\sim 0.25\mathrm{meV}$, points to the possibility of controlling more than 10 QDs with a single shaped optical pulse.

Figure 1

Application of OQC to quantum control transferring the excitons in QD1 and QD2 from their respective ground states to the final states in Eq. (8). (a) Bloch sphere dynamics for QD1 (QD2) are indicated by a black solid (red dashed) curve. (b) The spectral amplitude (phase) of the pulse are indicated by a black solid (blue dashed) curve. (c) Temporal intensity of the shaped pulse. (d) The $x,y$, and $z$ components of $\mathbf{\Lambda }$ are indicated by black, blue, and red curves, respectively, with solid curves corresponding to QD1 and the dashed curves corresponding to QD2.

Figure 2

(a) Optimized fidelity as a function of the difference in phase and inversion of the two qubits for the optoelectronic properties used in Fig. 1. (b)–(d) Optimized fidelity of the quantum control process shown in Fig. 1 as a function of (b) the detuning of QD2 from the laser frequency $[\hslash {\mathrm{\Delta }}_{\mathrm{QD}2}=\hslash ({\omega }_{\mathrm{QD}2}-{\omega }_l)]$ and its dipole moment, while holding the properties of QD1 constant $(\hslash {\omega }_{\mathrm{QD}1}=1.00125\mathrm{eV}$ and ${\mu }_{\mathrm{QD}1}=25\mathrm{debye})$, (c) the dipole moment of the two quantum dots while holding the transition frequencies constant $(\hslash {\omega }_{\mathrm{QD}1}=1.00125\mathrm{eV}$ and $\hslash {\omega }_{\mathrm{QD}1}=0.99875\mathrm{eV})$, and (d) the detuning of the QD transitions from the laser frequency while holding the dipole moments constant $({\mu }_{\mathrm{QD}1}=25\mathrm{debye}$ and ${\mu }_{\mathrm{QD}2}=28\mathrm{debye})$.

Figure 3

Same calculation as in Fig. 2 taking the assumption of a higher resolution pulse shaping system, incorporating a 640 pixel SLM.

Figure 4

Application of OQC to the simultaneous manipulation of excitons in three QDs. For this calculation, the final state of the third quantum dot, QD3, with $\hslash {\omega }_{\mathrm{QD}3}=1.00005\mathrm{eV}$ and ${\mu }_{\mathrm{QD}3}=24\mathrm{debye}$, is ${\left|{\psi }_{\mathrm{QD}3}\right.〉}_{\mathrm{f}}=(\left|0\right.〉+e^{-i3\pi /4}\left|1\right.〉)/\sqrt{2}$, while for QD1 and QD2 the final states are given by Eqn. (8). (a) Bloch sphere dynamics for QD1, QD2, and QD3 are indicated by the black solid curve, the red dashed curve, and the blue dot-dashed curve, respectively. The optimal pulse in this case is defined by $q_{\mathrm{opt}}=(0.730\pi ,134\mathrm{fs},-0.726\pi ,2.12\pi )$ and when implemented results in a quantum gate fidelity of $0.985$. (b) The spectral amplitude (phase) of the pulse is indicated by a black solid (blue dashed) curve. (c) Temporal intensity of the shaped pulse. (d) The $x,y$, and $z$ components of $\mathbf{\Lambda }$ are indicated by black, blue, and red curves, respectively, with solid curves corresponding to QD1, dashed curves corresponding to QD2, and dot-dashed curves corresponding to QD3.

Figure 5

Fidelity for the calculation shown in Fig. 1 incorporating excitation-induced dephasing caused by coupling to longitudinal acoustic phonons as a function of the cutoff energy, $\hslash {\omega }_c$, of the exciton-phonon response. The phonon modes of bulk GaAs are assumed, for which $\alpha =0.027{\mathrm{ps}}^2$. The top $x$ axis shows the corresponding width of the norm squared of the carrier wave function. Inset: The calculated phonon response function taking $\hslash {\omega }_c=1.38\mathrm{meV}$, corresponding to $d=5.76\mathrm{nm}$.