Non-Abelian fermion parity interferometry of Majorana bound states in a Fermi sea

We study the quantum dynamics of Majorana and regular fermion bound states coupled to a quasi-one-dimensional metallic lead. The dynamics following the quench in the coupling to the lead exhibits a series of dynamical revivals as the bound state propagates in the lead and reflects from the boundaries. We show that the nature of revivals for a single Majorana bound state depends uniquely on the presence of a resonant level in the lead. When two spatially separated Majorana modes are coupled to the lead, the revivals depend only on the phase difference between their host superconductors. Remarkably, the quench in this case effectively performs a fermion-parity interferometry between Majorana bound states, revealing their unique non-Abelian braiding. Using both analytical and numerical techniques, we find the pattern of fermion parity transfers following the quench, study its evolution in the presence of disorder and interactions, and thus, ascertain the fate of Majorana bound states in a Fermi sea.

Figure 1

(Top) Schematic of the system; one or both tunnel couplings, ${\lambda }_{\text{I}/\text{II}}\left(t\right)$, are switched on at $t=0$. (Bottom) Coupling of Majorana bound states to the lead, shown in its Majorana basis as two independent Majorana chains. The resonant lead with odd number of sites [(a) and (b)] and the off-resonant lead with even number of sites [(c) and (d)] couple to Majorana bound states ${\gamma }_{\text{I}}$ and ${\gamma }_{\text{II}}$ in superconductors with phase difference $\mathrm{\Delta }\varphi =0$ [(a) and (c)] and $\mathrm{\Delta }\varphi =\pi$ [(b) and (d)].

Figure 2

Fidelity after the quench $\lambda =0.5w$ at $t=0$ couples a lead to a single SSH chain with $m=0.8w$ (a), one Kitaev (b), and two Kitaev chains with gap $\mathrm{\Delta }=0.3w, \mathrm{\Delta }\varphi =\pi$, (c) and $\mathrm{\Delta }\varphi =0$ (d). The disorder strength $W=0.2w$ and the shaded areas show the standard deviation in the disorder average.

Figure 3

The parity $P(r,t)$ to the left of a cut between sites $r$ and $r+1$ in the lead after a quench $\lambda =w$ at $t=0$ couples the lead to two Kitaev chains with gap $\mathrm{\Delta }=0.5w$ and $\mathrm{\Delta }\varphi =0$ [(a) and (b)], $\mathrm{\Delta }\varphi =\pi$ (c), and to two SSH chains with $m=0.8w$ [(d) and (e)]. The lead interaction strength $U=0.5w$ in (b) and (e). The insets show magnified regions for comparison.