Parity qubits and poor man's Majorana bound states in double quantum dots

We study a double quantum dot connected via a common superconducting lead and show that this system can be tuned to host one Majorana bound state (MBS) on each dot. We call them “poor man's Majorana bound states” since they are not topologically protected, but otherwise share the properties of MBS formed in topological superconductors. We describe the conditions for the existence of the two spatially separated MBS, which include breaking of spin degeneracy in the two dots, with the spins polarized in different directions. Therefore, we propose to use a magnetic field configuration where the field directions on the two dots form an angle. By control of this angle the cross Andreev reflection and the tunnel amplitudes can be tuned to be approximately equal, which is a requirement for the formation of the MBS. We show that the fermionic state encoded in the two Majoranas constitutes a parity qubit, which is nonlocal and can only be measured by probing both dots simultaneously. Using a many-particle basis for the MBS, we discuss the role of interactions and show that interactions between electrons on different dots always shift the condition for degeneracy. We also show how the MBS can be probed by transport measurements and discuss how the combination of several such double dot systems allows for entanglement of parity qubits and measurement of their dephasing times.

Figure 1

Sketch of setup. Two quantum dots are coupled to each other via a superconductor, which mediates both normal tunneling between the dots and cross Andreev reflection. Each dot has only one level close to the chemical potential of the superconductor (energies ${\varepsilon }_1$ and ${\varepsilon }_2$), which are controlled by the gate voltages $V_g^1$ and $V_g^2$. The dots are fully spin-polarized, but in different directions because of the inhomogeneous magnetic field (${\mathbf{B}}_1$ at dot 1 and ${\mathbf{B}}_2$ at dot 2). The normal metal tunnel probe, coupled to dot 1, is used in Sec. 3 to detect the MBS.

Figure 2

Calculated differential conductance, $G=dI/dV$, at zero temperature. Upper panel: $G$ plotted on a color scale as a function of bias voltage and one more parameter, while all other parameters are fixed at their sweet spot values, with $t=\Delta =8\Gamma$. Lower panel: $G$ as a function of bias voltage along three different horizontal cuts in the corresponding upper panel conductance map (the position being marked there with the same color and line style as the corresponding curve in the lower panel). (a) ${\varepsilon }_1$ is varied, cuts in lower panel are at ${\varepsilon }_1=0$ (green solid curve), ${\varepsilon }_1=4\Gamma$ (magenta dashed curve), and ${\varepsilon }_1=8\Gamma$ (brown fine-dashed curve). (b) ${\varepsilon }_2$ is varied, cuts in lower panel are at ${\varepsilon }_2=0$ (green solid curve), ${\varepsilon }_2=4\Gamma$ (magenta dashed curve), and ${\varepsilon }_2=8\Gamma$ (brown fine-dashed curve). (c) $\varepsilon ={\varepsilon }_1={\varepsilon }_2$ is varied, cuts in lower panel are at $\varepsilon =0$ (green solid curve), $\varepsilon =4\Gamma$ (magenta dashed curve), and $\varepsilon =8\Gamma$ (brown fine-dashed curve). (d) $t$ is varied, cuts in lower panel are at $t=\Delta =8\Gamma$ (green solid curve), $t=4\Gamma$ (magenta dashed curve), and $t=\Gamma /2$ (brown fine-dashed curve).

Figure 3

(a) Sketch of setup with two double dots, tunnel coupled with amplitudes $t_{AC}$ and $t_{BC}$ to an additional dot ($C$). Dot $C$ has a single orbital with energy ${\varepsilon }_C$, which is controlled by the gate voltage $V_g^C$. (b) The eigenenergies of the even total parity sector of Eq. (24) for $t_{AC}=t_{BC}$, plotted as a function of ${\varepsilon }_C$.