Theory of semiconductor quantum-wire-based single- and two-qubit gates

A $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ based two-qubit quantum device that allows the controlled generation and straightforward detection of entanglement by measuring a stationary current-voltage characteristics is proposed. We have developed a two-particle Green’s function method of open systems and calculate the properties of three-dimensional interacting entangled systems nonperturbatively. We present concrete device designs and detailed charge-self-consistent predictions. One of the qubits is an all-electric Mach-Zehnder interferometer that consists of two electrostatically defined quantum wires with coupling windows, whereas the second qubit is an electrostatically defined double quantum dot located in a second two-dimensional electron gas beneath the quantum wires. We find that the entanglement of the device can be controlled externally by tuning the tunneling coupling between the two quantum dots.

Figure 1

(Color online) Schematic view of the proposed quantum transport device. The device is realized by two stacked $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ 2DEGs. The top 2DEG (green) is depleted by external gates to form a double-QWR based Mach-Zehnder interferometer. The bottom 2DEG (blue) contains two electrostatically defined, coupled quantum dots. The larger one is centered beneath and in between the QWRs. There is a bias voltage $V_{\mathrm{SD}}$ applied between the source $\left(S\right)$ and the three drain contacts $(D_1-D_3)$ that leads to currents $(J_1,J_2)$ that are labeled accordingly. There are additional gates acting on the device that are shown in Fig. 2. For sake of clarity, the figure is not drawn to scale.

Figure 2

(a) Vertical cross section of the $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ heterostructure with a single $10\mathrm{nm}$ wide GaAs quantum well that is used for the Mach-Zehnder interferometer. GaAs is shown in black; ${\mathrm{Al}}_{0.37}{\mathrm{Ga}}_{0.63}\mathrm{As}$ is shown in gray. The GaAs cap layer is $5\mathrm{nm}$ thick. A silicon $\delta$ doping layer with a concentration of $2.5\times {10}^{12}{\mathrm{cm}}^{-2}$ is located $25\mathrm{nm}$ below the surface (dashed line). (b) Top view of the Mach-Zehnder interferometer that depicts the side gates (black) with voltage $V_{\mathrm{SG}}\left(\mathrm{S}\right)$ and the mid gates (gray) with voltage $V_{\mathrm{SG}}\left(\mathrm{M}\right)$ that define the two QWRs and the coupling windows between them. On top of the wires, there are two additional phase gates (hatched) at a bias of $\pm V_{\mathrm{SG}}\left(\mathrm{P}\right)$.

Figure 3

(Color online) Top views of the equilibrium electron density (upper panel) and electrostatic potential (lower panel) within the upper 2DEG. The black and the white frames indicate the position of the metal side and mid gates.

Figure 4

(a) Stationary currents $J_1$ (solid), $J_2$ (dashed), and $J_3$ (dotted) through the Mach-Zehnder interferometer as a function of the phase gate voltage $V_{\mathrm{SG}}\left(\mathrm{P}\right)$ for a nominal length of the coupling windows of $72.5\mathrm{nm}$. The source-drain voltage $V_{\mathrm{SD}}$ is set to $50\mu \mathrm{V}$. (b) Same as (a), but for a coupling window length of $67.5\mathrm{nm}$.

Figure 5

(Color online) Charge densities associated with the current-carrying scattering states for a phase gate voltage $V_{\mathrm{SG}}\left(P\right)$ of (a) $0\mathrm{meV}$, (b) $-3.6\mathrm{meV}$, and (c) $-7.5\mathrm{meV}$. In all cases, the source-drain voltage $V_{\mathrm{SD}}$ is set to $50\mu \mathrm{V}$.

Figure 6

Schematic representation of the Mach-Zehnder interferometer. We have indicated the three quantum-operation regions, consisting of the two coupling windows $\widehat{R}$ and one phase gate $\widehat{P}$, and four other segments where the qubit state is well defined (see Table ).

Figure 7

Contour plot of the transmission function $T_1$ between source $S$ and drain $D_1$ as a function of kinetic energy in meV and as a function of the coupling window length $L$ in nm. The total width of the structure equals $W=220\mathrm{nm}$, whereas the width of the inner coupling windows amounts to $W^{\prime }=100\mathrm{nm}$. The darker the color, the smaller the value of $T_1$. The dark lines therefore indicate minima in $T_1$. The arrows point to crossings of resonances where $T_1$ vanishes.

Figure 8

Transmission functions $T_1$ (solid), $T_2$ (dashed), and $T_3$ (dotted), as a function of the kinetic energy of the electron in meV. The length $L$ of the coupling window has been set equal to $82\mathrm{nm}$. The other symbols have been defined in Fig. 1.

Figure 9

Currents $J_1$ and $J_2$ in nA as a function of the phase gate voltage $V_{\mathrm{SG}}\left(\mathrm{P}\right)$ in mV for two different quantum dot tunneling couplings $t$ in $\mu \mathrm{eV}$. The inset is a simplified version of Fig. 1.

Figure 10

Visibility as defined in the text, corresponding to the interference pattern of Fig. 9, as a function of the tunneling coupling in $\mu \mathrm{eV}$.

Figure 11

(Color online) (a) Contour plot of the stationary charge density of the current-carrying states, in units of ${10}^{15}{\mathrm{cm}}^{-3}$. The upper (lower) panel shows the single-particle electron density in the QWRs when the electron in the quantum dot occupies the bonding (antibonding) state. The quantum dot tunneling coupling amounts to $t=10\mu \mathrm{eV}$. (b) Same type of figure, but with the Coulomb interaction between the electron in the wires and the electron in the quantum dot switched off. In case (b), the two possible paths of the scattered electron can interfere with one another. In case (a), however, the electron propagating in the lower wire changes the double-dot state and loses its capability to interfere with the other path, thus reducing the visibility (see text).

Figure 12

Transmission functions $T_1$ and $T_2$ as a function of the phase angle in rad for the analytical model of the entangled Mach-Zehnder double quantum dot device. The full line is for a finite tunneling coupling $t=50$ in units of ${\hslash }^2∕2m{L}^2$, whereas the dashed curve is for vanishing tunneling coupling that corresponds to no entanglement.

Figure 13

Calculated von Neumann entropy, as defined in the text, of the entangled Mach-Zehnder double quantum dot device of Sec. 5C as a function of the tunneling coupling in $\mu \mathrm{eV}$.