Optimal quantum control for conditional rotation of exciton qubits in semiconductor quantum dots

Pulse-shaping protocols for subpicosecond optically controlled quantum gates in semiconductor quantum dots are reported. Our emphasis is the development of shaping schemes for either amplitude or phase control of the pulse that are easily implemented using commercial pulse shapers and femtosecond laser systems. We illustrate the efficacy of our approach through simulations of a controlled-rotation gate in a realistic In(Ga)As quantum dot with electronic structure calculated using eight-band, strain-dependent $\mathit{k}·\mathit{p}$ theory. Our results show that amplitude- and phase-shaping protocols both lead to substantial improvements in fidelity when compared with transform-limited pulses with equivalent gate times. Dephasing was found to have a minimal effect on the gate fidelities due to the ultrafast time scale of the quantum operations.

Figure 1

(a) Energy-level diagram for the exciton and biexciton systems in a QD—the vacuum ground state ($|00\rangle$), two single excitons ($|01\rangle ,|10\rangle$), and a biexciton ($|11\rangle$) with a binding energy of ${\Delta }_b$. The arrows indicate optically allowed, linearly polarized (${\Pi }_x$ or ${\Pi }_y$) transitions. (b) Unitary transformation matrix for the c-rot gate. (c) Truncated pyramid quantum dot structure and (d) ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ composition profile within the dot. The composition is graded from high indium concentration in the shape of an inverted triangle at the center of the dot to low indium concentration at the base.

Figure 2

Exciton transition energy (circles) and biexciton binding energy (triangles) for variations in (a) average indium composition and (b) quantum dot height. In varying the dot height, material was added to the top of the dot [see Fig. 1d], while maintaining the same graded structure, until a complete pyramid was formed ($h=6.5\mathrm{nm}$).

Figure 3

$4f$ pulse shaper consisting of two diffraction gratings, two lenses, and an optical mask shown as a spatial light modulator (SLM). The distance $f$ is the focal length of the lenses. Manipulation of the pulse shape is carried out in the Fourier plane.

Figure 4

Population dynamics and control pulse characteristics for (a) the optimal phase-shaped pulse and (b) the TL pulse. Panels (i) and (ii) show the population dynamics for the initial conditions ${\rho }_{10,10}(t=0)=1$ and ${\rho }_{00,00}(t=0)=1$, respectively. The third panel shows the temporal envelope of the electric-field intensity of the control pulse. The bottom panel shows the amplitude (solid curves) and phase (dashed curves) profiles for the control pulse.

Figure 5

Bloch vector representations of the first qubit for ${\rho }_{00,00}(t=0)=1$ (solid curve) and the second qubit for ${\rho }_{10,10}(t=0)=1$ (dashed curve) for the (a) phase-shaped pulses and (b) TL pulse. Truth table of the gate operation for the (c) optimal phase-shaped pulse and (d) TL pulse.

Figure 6

Population dynamics and control pulse characteristics for (a) the optimal amplitude-shaped pulse and (b) the TL pulse. Panels (i) and (ii) show the population dynamics for the initial conditions ${\rho }_{10,10}(t=0)=1$ and ${\rho }_{00,00}(t=0)=1$, respectively. The third panel shows the temporal envelope of the electric-field intensity of the control pulse. The bottom panel shows the amplitude (solid curves) and phase (dashed curves) profiles for the control pulse.

Figure 7

Bloch vector representations of the first qubit for ${\rho }_{00,00}(t=0)=1$ (solid curve) and the second qubit for ${\rho }_{10,10}(t=0)=1$ (dashed curve) for the (a) amplitude-shaped pulses and (b) TL pulse. Truth table of the gate operation for the (c) optimal amplitude-shaped pulse and (d) TL pulse.

Figure 8

(Top) Fidelity for the optimal phase-shaped pulses (diamonds) and optimal amplitude-shaped pulses (circles) is shown as a function of the biexciton binding energy. The fidelities of the corresponding TL pulses with equivalent gate times are also shown for the phase-shaped pulses ($\times$) and amplitude-shaped pulses ($+$). (Bottom) Gate time of optimal phase-shaped (diamonds) and amplitude-shaped (circles) pulses as a function of binding energy.

Figure 9

Fidelity versus dephasing time $T_2$ for the optimal phase-shaped pulses (diamonds) and optimal amplitude-shaped pulses (circles), and the TL sech pulses with gate time equivalent to that of the phase-shaped pulses ($\times$) and amplitude-shaped pulses ($+$).