Spin-dependent coupling between quantum dots and topological quantum wires

Considering Rashba quantum wires with a proximity-induced superconducting gap as physical realizations of Majorana bound states and quantum dots, we calculate the overlap of the Majorana wave functions with the local wave functions on the dot. We determine the spin-dependent tunneling amplitudes between these two localized states and show that we can tune into a fully spin polarized tunneling regime by changing the distance between dot and Majorana bound state. Upon directly applying this to the tunneling model Hamiltonian, we calculate the effective magnetic field on the quantum dot flanked by two Majorana bound states. The direction of the induced magnetic field on the dot depends on the occupation of the nonlocal fermion formed from the two Majorana end states which can be used as a readout for such a Majorana qubit.

Figure 1

A quantum wire with applied magnetic field along the longitudinal ($x$) axis and spin-orbit vector along the $z$ axis in which the boundary between a topological section (red) and nontopological section (gray) supports a MBS. In the upper panel, a quantum dot (blue) of size $L$ is defined, within the nontopological section, at a distance $\ell$ from the topological section. The second setup (lower panel) is identical to the first with an additional topological section which ends a distance $d-\ell$ from the quantum dot center. The red and blue curves are schematically the probability amplitudes of the MBS and quantum dot wave functions, respectively.

Figure 2

The real and imaginary components of spin-dependent tunneling amplitudes $t_{\uparrow }$ (blue solid) and $t_{\downarrow }$ (red dashed) for a small quantum dot, $k_{\text{SO}}L=0.1$, and ${\kappa }^t/k_{\text{SO}}=10\sqrt{3}$ as a function of distance $\ell$ between the dot and the topological section. Both $t_{\uparrow }$ and $t_{\downarrow }$ oscillate with periodicity of ${\lambda }_{\text{SO}}/2$ but with a relative phase difference of $\pi /2$.

Figure 3

The real and imaginary components of spin-dependent tunneling amplitudes $t_{\uparrow }$ (blue solid) and $t_{\downarrow }$ (red dashed) for a large dot, $k_{\text{SO}}L=10$ and ${\kappa }^t/k_{\text{SO}}=10\sqrt{3}$, as a function of distance $\ell$ between the dot and the topological section. Both $t_{\sigma }$ oscillate with the period ${\lambda }_{\text{SO}}$ and the magnitude of oscillation is smaller as compared with the case of a small dot. Due to the finite size of the dot, we start with a separation $\ell =L/2$ between the center of the dot and the end of the topological section.

Figure 4

Effective magnetic field $B_i^{\pm }$ induced on a small dot situated (upper panel) equidistant to two MBSs $d=2\ell$, (middle panel) slightly asymmetrically to the two MBSs $d=2\ell -0.2l_{\text{SO}}$, and (lower panel) in the case when one fixes the distance between between MBSs $k_{\text{SO}}d=10$, as a function of the distance $k_{\text{SO}}\ell$ between the dot and topological sections for $k_{\text{SO}}L=0.1, {\kappa }^t/k_{\text{SO}}=10\sqrt{3}, {\epsilon }_{\uparrow }=9t_0$, and ${\epsilon }_{\downarrow }=11t_0$. Upper panel: The components $B_x^{\pm }$ oscillate with period ${\lambda }_{\text{SO}}/4$ while the other components are zero. Middle panel: $B_x^{\pm }$ and $B_y^{\pm }$ oscillate with period ${\lambda }_{\text{SO}}/4, B_z^{\pm }=0$. Lower panel: $B_x^{\pm }$ ($B_y^{\pm }$) is a symmetric (antisymmetric) function of $\ell$ around the middle of the nontopological section $k_{\text{SO}}\ell =5$. Both components oscillate with period ${\lambda }_{\text{SO}}/4$.

Figure 5

Effective magnetic field $B_i^{\pm }$ induced on a large dot (upper panel) situated equidistant to two MBSs $d=2\ell$, (middle panel) slightly asymmetrically to the two MBSs $d=2\ell -0.2l_{\text{SO}}$, and (lower panel) in the case when one fixes the distance between between MBSs $k_{\text{SO}}d=20$, as a function of the distance $k_{\text{SO}}\ell$ between the dot and topological sections for $k_{\text{SO}}L=10, {\kappa }^t/k_{\text{SO}}=10\sqrt{3}$, and ${\epsilon }_{\uparrow }={\epsilon }_{\downarrow }=t_0/10$. Upper panel: The components $B_x^{\pm }$ oscillate with period ${\lambda }_{\text{SO}}/2$ while the other components are zero. Middle panel: $B_x^{\pm }$ and $B_y^{\pm }$ oscillate with period ${\lambda }_{\text{SO}}/2, B_z^{\pm }=0$. Lower panel: $B_x^{\pm }$ ($B_y^{\pm }$) is symmetric (antisymmetric) function of $\ell$ around the middle of the nontopological section $k_{\text{SO}}\ell =10$.

Figure 6

System setup of the $N$-site tight-binding model. In the upper panel, the topological section (red), defined from site $N_r$ to the end of the chain, is realized due to proximity-induced superconductivity. The nontopological section (gray) is driven to the topologically trivial phase by depleting the wire. A quantum dot (blue) of size $L_d$ defined by gates is located at $N_d<N_r$. In the lower panel, the setup is the same with the addition of a second topological section, realized in the same way, from the beginning of the chain to $N_l<N_d$. In both setups, the magnetic field, inducing a Zeeman splitting ${\mathrm{\Delta }}_Z$, is applied along the $x$ axis and the spin-orbit vector, with magnitude $\tilde{\alpha }$, is along the $z$ axis.

Figure 7

Local density of states (LDOS) of the wire with one topological section [Fig. 6]. The lowest quantum dot level (blue) is centered at $N_d=50$ and of the size $L_d=31$ while the MBS wave functions (red) are peaked roughly at the beginning ($N_r=200$) and at the end ($N=350$) of the topological section. Here ${\mathrm{\Delta }}_s=0.06, {\mathrm{\Delta }}_Z=0.12, {\mu }_n=2.27, {\mu }_t=2$, and ${\mu }_d=2.245$. Inset: Energy spectrum indicates two MBS levels (red) at zero energy, two quantum dot levels (blue) with energies at ${\tilde{E}}_{\uparrow }\approx {\tilde{E}}_{\downarrow }$, and the next lowest energy levels of the dot (black).

Figure 8

Tunneling amplitudes $|{\tilde{t}}_{\uparrow }|$ and $|{\tilde{t}}_{\downarrow }|$ between the quantum dot and MBS [Fig. 6] as a function of the distance between the position of the dot and the end of the topological section of the wire. All parameters are the same as in Fig. 7.

Figure 9

Local density of states of a chain with two identical topological sections [Fig. 6] between $N=1$ and $N_l=200$ and between $N_r=400$ and the end of the chain ($N=600$). The quantum dot wave function (blue) is centered around the dot position $N_d=300$ and there are four MBSs at the interfaces of the topological and nontopological sections. Inset: Energy spectrum indicates four zero energy states corresponding to the MBSs (red), the lowest energy dot levels (blue), and second lowest dot levels (black). The system parameters are the same as in Fig. 7.

Figure 10

The spin components of the lowest quantum dot level along the $y$ and $x$ axes, respectively, as a function of the distance between the dot and end of the left and right topological sections, which are kept equidistant to the dot up to one lattice constant, $N_d-N_l-1=N_r-N_d-1$. Both components oscillate with period ${\lambda }_{\text{SO}}/2$, and depend exponentially on the distance between the dot and the MBSs. The spin projection on the $y$ axis goes to zero when the dot is far from MBSs while the $x$ component saturates at the value determined by the external magnetic field (black solid line around $\approx 0.271$). We note that the component ${\tilde{S}}_z$ is always zero due to the symmetry of the problem.

Figure 11

The spin component $S_x$ of the lowest positive energy level of the dot (blue solid line) as a function of the position within the dot when the magnetic field is weak (${\mathrm{\Delta }}_Z=0.04$, left panel) and strong (${\mathrm{\Delta }}_Z=0.12$, right panel). The black dashed line stands for the symmetric axis of the blue curve corresponding to the average spin projection $S_x$ on the dot. The system parameters are the same as in Fig. 9.