Optically controlled logic gates for two spin qubits in vertically coupled quantum dots

We describe an interaction mechanism between electron spins in a vertically stacked double quantum dot that can be used for controlled two-qubit operations. This interaction is mediated by excitons confined within, and delocalized over, the double dot. We show that gates equivalent to the $\sqrt{\mathrm{SWAP}}$ gate can be obtained in times much less than the exciton relaxation time and that the negative effects of hole mixing and spontaneous emission do not seriously affect these results.

Figure 1

(Color online) (a) Sketch of the two vertically stacked quantum dots. Growth direction is the $z$ direction, and the two dots are positioned symmetrically at $z=\pm d$. The dots have heights $2a$ and the width of the barrier between them is $b\equiv 2(d-a)$. The lower $\left(A\right)$ and upper $\left(B\right)$ dots have Darwin radii of $L_A$ and $L_B$, respectively. (b) Two electronic and two hole levels in each dot take part in gate operation. The lowest electron levels in each dot (labeled 1 and 2) are the two-qubit levels and an electron permanently resides in each of them. Laser illumination is tuned such that it creates an exciton in the excited levels of dot $A$ only (levels 3). (c) Due to a tunnel coupling ${\tau }_{34}$ between the dots and a resonance condition met through the tuning of the gate voltage $V_g$, the excitonic electron can tunnel back and forth. The exchange interaction between this electron and the qubits gives rise to the optically induced interaction between the qubits that can be used to perform a quantum gate.

Figure 2

Logarithm of ${\tau }_{34}$, the tunneling amplitude between excited states in different dots as a function of the interdot distance $d$. The solid black line shows the approximate result of Eq. (7) and the black points show a numerical evaluation of full single-particle matrix element. The dashed line and gray boxes show the analogous quantities for heavy-hole tunneling. The dot height is $2a=2\mathrm{nm}$ and the radial confinement energies are $\hslash {\omega }_A^e=35\mathrm{meV}$ and $\hslash {\omega }_B^e=30\mathrm{meV}$ for all carriers. The vertical confinement depth is $V_0^e=680\mathrm{meV}$ for electrons and $V_0^h=100\mathrm{meV}$ for holes. We took the electron mass to be $0.04m_0$ in InAs and $0.067m_0$ in the surrounding GaAs matrix. Heavy holes have masses $m_{hz}=0.34m_0$ and $m_{h\perp }=0.04m_0$ for both InAs and GaAs.

Figure 3

(Color online) The absolute value of the optimal C-PHASE angle ${\Phi }_{\mathrm{opt}}$ as a function of the pulse duration scaled by the qubit isolation time. Results are shown for three different tunnel strengths ${\tau }_{34}=0.05\mathrm{meV}$ (blue squares), ${\tau }_{34}=0.1\mathrm{meV}$ (green triangles), and ${\tau }_{34}=0.2\mathrm{meV}$ (red diamonds). As is evident, a phase of $\Phi =\pi$ can be obtained at ${\tau }_{34}=0.05\mathrm{meV}$ with $T∕T_Q\gtrsim 0.064$. In real time units, this equates to $T\approx 84\mathrm{ps}$. This value of $\Phi$ can also be achieved with higher tunneling amplitudes, but this requires longer pulses.