Probing Majorana bound states via a $pn$ junction containing a quantum dot

We propose an alternative route to transport experiments for detecting Majorana bound states (MBSs) by combining topological superconductivity with quantum optics in a superconducting $pn$ junction containing a quantum dot (QD). We consider a topological superconductor (TSC) hosting two Majorana bound states at its boundary ($n$ side). Within an effective low-energy model, the MBSs are coherently tunnel-coupled to a spin-split electron level on the QD, which is placed close to one of the MBSs. Holes on the QD are tunnel-coupled to a normal conducting reservoir ($p$ side). Via electron-hole recombination, photons in the optical range are emitted, which have direct information on the MBS properties through the recombined electrons. Using a master equation approach, we calculate the polarization-resolved photon emission intensities (PEIs). In the weak coupling regime between MBSs and QD, we find an analytical expression for the PEI which allows to clearly distinguish the cases of well separated MBSs at zero energy from overlapping MBSs. For separated MBSs, the Majorana spinor polarization is given by the relative widths of the two PEI peaks associated with the two spin states on the QD. For overlapping MBSs, a coupling to the distant (nonlocal) MBS causes a shift of the emission peaks. Additionally, we show that quasiparticle poisoning (QP) influences the PEI drastically and changes its shot noise from super-Poissonian to sub-Poissonian. In the strong coupling regime, more resonances emerge in the PEI due to spin-mixing effects. Finally, we comment on how our proposal could be implemented using a Majorana nanowire.

Figure 1

(a) Exemplary sketch of a $pn$ junction based on semiconducting nanowires in the presence of a magnetic field in $z$ direction with three separated regions. In the topologically nontrivial phase MBSs (yellow stars) emerge at the ends of the $n$ side (blue) which is in proximity to an $s$-wave SC. The $p$ side is coupled to a normal lead and serves as a hole reservoir (red). In the $pn$ junction a QD is formed (gray) which confines electrons and holes and can be tuned via a gate voltage (gate). Due to electron-hole recombination, photons (green arrows) are emitted in wire direction. (b) Sketch of the energy diagram. Shown are the uncoupled levels of the optically active QD, $n$-side (right) and $p$-side (left) leads. Full blue and empty red circles represent electrons and heavy holes, respectively.

Figure 2

Eigenenergies of the coupled QD-MBSs system over QD energy ${\varepsilon }_e$ in different regimes. (a) Separated MBSs. Even and odd states are degenerate and we show the QD states $|n_{d\sigma }\rangle$ away from the avoided crossings. Parameters are $\xi =0, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}$, and $t_2=0$. (b) Overlapping MBSs with finite splitting. Parameters are $\xi =0.5{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.05{\mathrm{\Delta }}_{Z,e}$, and $t_2=0$. (c) Overlapping MBSs with finite coupling to ${\gamma }_2$. Parameters are $\xi =0, 10t_{1\uparrow }=t_{1\downarrow }=0.03{\mathrm{\Delta }}_{Z,e}, i{t}_{2\uparrow }=0.0012{\mathrm{\Delta }}_{Z,e}$, and $i{t}_{2\downarrow }=-0.012{\mathrm{\Delta }}_{Z,e}$. (d) Overlapping MBSs with finite splitting and finite coupling $t_2$. Parameters are $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.03{\mathrm{\Delta }}_{Z,e}, i{t}_{2\uparrow }=0.0009{\mathrm{\Delta }}_{Z,e}$, and $i{t}_{2\downarrow }=-0.009{\mathrm{\Delta }}_{Z,e}$.

Figure 3

Relevant transitions in electronic and hole subsystems. We show the six relevant uncoupled electronic states in (a) and the four hole eigenstates in (b), respectively. Rates included in the master equation are optical transitions for emission of $L/R$ photons with rates $W^L/W^R$ (red/blue), spin relaxation processes with rate ${\mathrm{\Gamma }}_R$ (green), QP with rate ${\mathrm{\Gamma }}_{\mathrm{QP}}$ (yellow), and hole refilling with rate ${\mathrm{\Gamma }}_h$ (violet).

Figure 4

Photon emission for separated MBSs ($\xi =t_2=0$). Transition schemes for ${\varepsilon }_e=-{\mathrm{\Delta }}_{Z,e}$ (a) and ${\varepsilon }_e=+{\mathrm{\Delta }}_{Z,e}$ (b). Here, $W^L$ and $W^R$ are recombination rates giving $L$ and $R$ photons, respectively, whereas ${\mathrm{\Gamma }}_R$ is the spin relaxation rate. The total PEI $I_P$ is shown for ${\mathrm{\Gamma }}_R=0$ in (c) and for ${\mathrm{\Gamma }}_R=10W_{\max }$ in (d). (e) Frequency-dependent PEI $i_P$ over the QD energy ${\varepsilon }_e$ and the photon energy $\hslash \omega$ for same parameters as in (c), where $\mathrm{\Delta }{E}_h^{L/R}=U_h+{\varepsilon }_{h\Downarrow /\Uparrow }$. Shown are only transitions from the doubly occupied hole state. Gray lines correspond to the excitation spectrum of the coupled QD-MBSs system. Further parameters are $10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}, {\mathrm{\Gamma }}_h=100W_{\text{max}}$, and ${\mathrm{\Gamma }}_{\mathrm{QP}}=0$.

Figure 5

Photon emission for overlapping MBSs. In (a) and (b), we show the total PEI $I_P$ and the PEI $I_{\sigma }$ calculated from the spinless model for $t_2=0$ and for including a nonlocal coupling $t_2\ne 0$, respectively. In (c) and (d), we show the frequency-dependent PEI $i_P$ over the QD energy ${\varepsilon }_e$ and the photon energy $\hslash \omega$ for the same parameter regimes as in (a) and (b), respectively, where $\mathrm{\Delta }{E}_h^{L/R}=U_h+{\varepsilon }_{h\Downarrow /\Uparrow }$. Shown are only transitions from the doubly occupied hole state. Gray lines correspond to the excitation spectrum of the coupled QD-MBSs-system. In (e) and (f), we show the total PEI $I_P$ where we add a QP with ${\mathrm{\Gamma }}_{\mathrm{QP}}=W_{\max }$ again for $t_2=0$ and $t_2\ne 0$, respectively. In the case for $t_2=0$, we used $\xi =0.04{\mathrm{\Delta }}_{Z,e}$ and $10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}$. In the case for $t_2\ne 0$, we used $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.03{\mathrm{\Delta }}_{Z,e}, i{t}_{2\uparrow }=0.0009{\mathrm{\Delta }}_{Z,e}$, and $i{t}_{2\downarrow }=-0.009{\mathrm{\Delta }}_{Z,e}$. Other parameters are ${\mathrm{\Gamma }}_h=100W_{\text{max}}$ and ${\mathrm{\Gamma }}_R=0$.

Figure 6

(a) Emission cycles involving the uncoupled QD-MBSs states $|n_{d\sigma },n_c\rangle$ for ${\varepsilon }_{e\sigma }=-\xi$. Here, even states are maximally hybridized due to resonant tunneling, whereas odd states are not well hybridized, so PEI is weak and the system stays mostly in the unoccupied QD state $|0,1\rangle$. A finite QP leads to a large PEI, since the weakly hybridized odd states are now connected to the even states via ${\mathrm{\Gamma }}_{\mathrm{QP}}$. (b) Fano factor $F$ for overlapping MBSs. We show $F$ over the QD energy ${\varepsilon }_e$. Without QP the system is in the super-Poissonian regime with $F\ge 1$ (green). A finite QP leads to sub-Poissonian noise with $F\le 1$ (orange). These effects are strongest at ${\varepsilon }_{e\sigma }=\pm \xi$ where odd, respectively, even states hybridize. We used $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}{t}_2=0, {\mathrm{\Gamma }}_h=100W_{\max }, {\mathrm{\Gamma }}_R=0, {\mathrm{\Gamma }}_{\mathrm{QP}}=0$ (green), and ${\mathrm{\Gamma }}_{\mathrm{QP}}=W_{\max }$ (orange) [cf. PEIs in Figs. 5 and 5].

Figure 7

Photon emission in the spin-mixing regime. [(a) and (b)] The total PEI $I_P$ shows two additional peaks at ${\varepsilon }_e=-{\mathrm{\Delta }}_{Z,e}\pm \xi$ due to a spin-mixing effect which do not appear in the spinless model PEI $I_{\sigma }$. We used $\xi =0.5{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.05{\mathrm{\Delta }}_{Z,e}$, and respectively $t_2=0$ in (a) and $i{t}_{2\uparrow }=0.01{\mathrm{\Delta }}_{Z,e}, i{t}_{2\downarrow }=-0.01{\mathrm{\Delta }}_{Z,e}$ in (b). Further parameters are ${\mathrm{\Gamma }}_h=100W_{\text{max}}$ and ${\mathrm{\Gamma }}_{\mathrm{QP}}={\mathrm{\Gamma }}_R=0$. (c) In the emission cycle for ${\varepsilon }_e=-{\mathrm{\Delta }}_{Z,e}-\xi$, we show the strength of hybridization (black/gray arrows) between the states $|n_{d\sigma },n_c\rangle$ and optical transitions with emission of $L$ (red arrows) or $R$ photons (blue arrows). In the spin-mixing regime, the most likely emission cycle is denoted by red/blue solid arrows, whereas the dotted paths are rather unlikely.

Figure 8

Relevant recombination rates over QD energy ${\varepsilon }_e$ for different MBSs regimes at the resonance with $\downarrow$ electrons (${\varepsilon }_e={\mathrm{\Delta }}_{Z,e}$). (a) Separated MBSs. Even and odd states are degenerate, which leads to the same rates for each parity. Parameters are $\xi =0, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}$, and $t_2=0$. (b) Overlapping MBSs with finite splitting. Parameters are $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}$, and $t_2=0$. (c) Overlapping MBSs with finite coupling to ${\gamma }_2$. Parameters are $\xi =0, 10t_{1\uparrow }=t_{1\downarrow }=0.03{\mathrm{\Delta }}_{Z,e}, i{t}_{2\uparrow }=0.0012{\mathrm{\Delta }}_{Z,e}$, and $i{t}_{2\downarrow }=-0.012{\mathrm{\Delta }}_{Z,e}$. (d) Overlapping MBSs with finite splitting and finite coupling $t_2$. Parameters are $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.03{\mathrm{\Delta }}_{Z,e}, i{t}_{2\uparrow }=0.0009{\mathrm{\Delta }}_{Z,e}$, and $i{t}_{2\downarrow }=-0.009{\mathrm{\Delta }}_{Z,e}$.

Figure 9

Transitions between holes on the QD and the coupled reservoirs. Holes tunnel onto the QD from the hole reservoir via ${\mathrm{\Gamma }}_h$ and recombine with electrons to photons via photon emission with rates $W^P$. The calculated current $I$ flows from the left to the right.

Figure 10

Correlation function $g^{\left(2\right)}\left(\tau \right)$. We show $g^{\left(2\right)}\left(\tau \right)$ over $\tau$ for the QD energy ${\varepsilon }_e={\mathrm{\Delta }}_{Z,e}-\xi$ for the full model (solid lines) and $g_{\mathrm{spinless}}^{\left(2\right)}\left(\tau \right)$ over $\tau$ according to Eq. (C8) for the spinless model (dashed lines). The inset shows longer times $\tau$ where $g^{\left(2\right)}\left(\tau \right)\to 1$. For the full model, we used $\xi =0.04{\mathrm{\Delta }}_{Z,e}, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}{t}_2=0, {\mathrm{\Gamma }}_h=100W_{\max }, {\mathrm{\Gamma }}_R=0, {\mathrm{\Gamma }}_{\mathrm{QP}}=0$ (green), and ${\mathrm{\Gamma }}_{\mathrm{QP}}=W_{\max }$ (orange). For the spinless model, we used $\xi =0.04{\mathrm{\Delta }}_{Z,e}, t_{1\sigma }=0.025{\mathrm{\Delta }}_{Z,e}, {\mathrm{\Gamma }}_{\mathrm{QP}}=0$ (green), and ${\mathrm{\Gamma }}_{\mathrm{QP}}=W_{\max }$ (orange).

Figure 11

Fano factor and correlation function $g^{\left(2\right)}\left(\tau \right)$ for slow holes. We show the Fano factor $F$ over the QD energy ${\varepsilon }_e$ in (a) and $g^{\left(2\right)}\left(\tau \right)$ over $\tau$ for ${\varepsilon }_e={\mathrm{\Delta }}_{Z,e}$ in (b). For both plots we used $\xi =0, 10t_{1\uparrow }=t_{1\downarrow }=0.025{\mathrm{\Delta }}_{Z,e}, t_2=0, {\mathrm{\Gamma }}_h=0.001W_{\max }, {\mathrm{\Gamma }}_R=0$, and ${\mathrm{\Gamma }}_{\mathrm{QP}}=0$.