Adiabatic optical entanglement between electron spins in separate quantum dots

We present an adiabatic approach to the design of entangling quantum operations with two electron spins localized in separate InAs/GaAs quantum dots via the Coulomb interaction between optically excited localized states. Slowly varying optical pulses minimize the pulse noise and the relaxation of the excited states. An analytical “dressed-state” solution gives a clear physical picture of the entangling process and a numerical solution is used to investigate the error dynamics. For two vertically stacked quantum dots we show that, for a broad range of dot parameters, a two-spin state with concurrence $C>0.85$ can be obtained by four optical pulses with durations $\sim 0.1–1\text{ns}$.

Figure 1

(Color online) (a) Optical scheme to control the entanglement between spins in two InAs/GaAs QDs in the Voigt configuration. Two-dot states are denoted by kets such as $|t+,+⟩$, with $|\pm ⟩$ for the spin states and $|t\pm ⟩$ for the trion states. Arrows indicate the linear polarizations $V_j^{\pm }$ and $H_j^{\pm }$ for the transitions $|\pm ⟩\leftrightarrow |t\pm ⟩$ and $|\pm ⟩\leftrightarrow |t\mp ⟩$ of dot $j=1,2$. (b) Timing of pulses for either dot. $V\left(t\right)$ and $H\left(t\right)$ are envelope functions, for which we use the same shape, rectangular with fronts shaped as ${\text{sin}}^4\left(\pi t/T_f\right)$, for all pulses, and the same amplitudes for both $V$ pulses and for both $H$ pulses. (c) Adiabatic time evolution of the dressed-state energies. Solid lines show the essential energies which drive the operation.

Figure 2

(Color online) Energies of allowed optical transitions versus the optical frequencies (measured in energy units) for $V$ polarization (upper figure) and $H$ polarization (lower figure). The thin solid lines mark the transition energies in zero magnetic field. ${\omega }_{tj}$ is the transition energy between a spin state and a trion state in dot $j$. Their difference between the dots is shown as $\Delta {\omega }_t={\omega }_{t2}-{\omega }_{t1}$. $\Delta$ is the bitrion binding energy, thus making the transition energy between the single and bitrion ${\omega }_{tj}-\Delta$. In a magnetic field, the electron and hole Zeeman splittings ${\omega }_j^e$ and ${\omega }_j^h$ in dot $j$ cause the transition energy splitting $2{\Pi }_j={\omega }_j^e+{\omega }_j^h$ in the $V$ polarization and $2{\Sigma }_j={\omega }_j^e-{\omega }_j^h$ in the $H$ polarization. The Zeeman split transitions used in the quantum operation and off-resonant transitions are denoted by the thick solid lines and thick dashed lines, respectively. The vertical arrows show the central frequencies of the optical pulses and their detuning $\delta$ from the corresponding transitions.

Figure 3

(Color online) Concurrence of the output two-spin state for different bitrion binding energies. Excitation parameters: filled circles $-\delta =-0.1\text{meV}$, $V_0=20\text{meV}$, and $H_0=44\text{meV}$; open squares $-\delta =-0.13\text{meV}$, $V_0=20\text{meV}$, $H_0=44\text{meV}$; filled squares $-\delta =-0.1\text{meV}$, $V_0=10\text{meV}$, and $H_0=44\text{meV}$; open triangles $-\delta =0.12\text{meV}$, $V_0=15\text{meV}$, and $H_0=65\text{meV}$. $V_0$ and $H_0$ denote amplitudes of the $V$- and $H$-polarized fields. Inset: fidelity of the analytical model compared to numerical simulations as a function of $\Delta$.

Figure 4

(Color online) Evolution of a spin state controlled by four optical fields. The Coulomb coupling is $\Delta =0.3\text{meV}$, the detuning is $\delta =-0.1\text{meV}$, the magnetic field is $B=8\text{T}$, and the field amplitudes are $V_0=20\mu \text{eV}$ and $H_0=44\mu \text{eV}$. Durations of the $H$ and $V$ pulses interrelated as $T_H=T_V+2T_f,T_f=250\text{ps}$ is the front duration. (a) Optical pulses are centered at $t=0,T_V=340\text{ps}$. The components not shown in the figure are below ${10}^{-3}$ at the end of the excitation. (b) Spin density matrix as a function of $T_V$.

Figure 5

(Color online) Density matrix of the output two-qubit state prepared from $|+,+⟩$ using an optimized entangling gate. The analytical solution, obtained using Eq. (12), is compared with the numerical simulations. Parameters of the dots and the optical fields are the same as in Fig. 4.