Josephson effect in topological superconducting rings coupled to a microwave cavity

Abstract

We theoretically study a one dimensional p-wave superconducting mesoscopic ring interrupted by a weak link and coupled inductively to a microwave cavity. We establish an input-output description for the cavity field in the presence of the ring, and identify the electronic contributions to the cavity response and their dependence on various parameters, such as the magnetic flux, chemical potential, and cavity frequency. We show that the cavity response is 4π periodic as a function of the magnetic flux in the topological region, stemming from the so called fractional Josephson current carried by the Majorana fermions, while it is 2π periodic in the non-topological phase, consistent with the normal Josephson effect. We find a strong dependence of the signal on the cavity frequency, as well as on the parity of the ground state. Our model takes into account fully the interplay between the low-energy Majorana modes and the gaped bulks states, which we show is crucial for visualizing the evolution of the Josephson effect during the transition from the topological to the trivial phase.

Dynamic current susceptibility as a probe of Majorana bound states in nanowire-based Josephson junctions

“…As shown previously [28,[31][32][33][34] (see also the SM [30]), the susceptibility is the sum of three con-…”

“…Here,Î s ≡ −∂H BdG /∂Φ is the current operator in the absence of the perturbation. The time-dependent system is described by the following density matrix evolution [28,[31][32][33][34]:…”

Exact zero modes in twisted Kitaev chains

“…Then, we obtain the condition for the presence of Majorana zero modes in finite chains, as well as explicitly determine the spatial profile of the zero modes. We note that the emergence of Majorana zero modes (protected level crossing) in the fractional Josephson effect has been well-established since the seminal work of Kitaev 2,45,51,[58][59][60] . However, we emphasize that the boundary couplings studied in this paper are more general than those in previous work, as their amplitudes and phases can be arbitrary.…”

“…The usual periodic boundary condition (PBC) and the anti-periodic boundary condition (APBC) are included in the TBC as limiting cases. In practice, such a boundary condition can be realized by magnetic fluxes and Josephson junctions 6,45,[49][50][51][52][53][54][55][56][57][58][59][60][61] . Our work is motivated by the observation that the fermionic parities in the ground states of the Kitaev chain with the PBC and the APBC have opposite signs in the topological phase, hence indicating a level crossing when one continuously changes the parameters so as to connect the PBC with the APBC.…”

Transport in two-dimensional topological materials: recent developments in experiment and theory

“…More recent theories have revisited the fractional JE in the context of MZMs in hybrid structures combining conventional superconductors with topological insulators or semiconductor nanowires [440,448,494,495,496,497,498,499,500,501,502,503,504,505,506,507,455,508,509,510,457,511,512,513,514,515,516,517,518,519,520,521,522,523,524]. Theoretically, the 4π -periodic CPR (69) has been proposed for ferromagnetic weak links in quantum spin-Hall insulator/superconductor structures under assumption of the local fermion-parity conservation [440].…”