Different driving protocols and multiple Majorana modes in a Rashba coupled superconducting nanowire

We perform systematic analyses of single and multi-frequency driving protocols on a Rashba nanowire with superconducting correlations induced by proximity effects. The results for the single-mode drive reveal interesting frequency dependencies of the Majorana modes, in the sense that the parameters corresponding to the trivial and topological limits of the undriven (static) case host Majorana zero modes, respectively at low and high frequencies. Further, emergence of long-range interactions are noted that give rise to multiple gap-closing scenarios, which imply occurrence of multiple Majorana modes. On the other hand, the multi-frequency driving protocol, sub-grouped into commensurate and incommensurate ratios of the frequencies, demonstrates intriguing consequences. The commensurate case yields dynamical control over the stability of the edge modes. Moreover, complex driving protocols harm the Majoranas by pushing them into the bulk. Finally, the incommensurate case yields independent Majorana modes occurring at low-symmetry points in the Brillouin zone. While the single and the commensurate multi-frequency driving protocols admit the usage of symmetric time frames for the computation of the topological invariants, the incommensurate case relies on the framework of many-mode Floquet theory, where the topological properties are ascertained via calculating the Berry phase. We present band structure and phase diagrams to substantiate all our results. The robustness and concurrent existence of these unique Majorana modes, even amidst a very dense energy spectrum along with a lack of global time-periodicity, hold promise for applications in quantum computing and the development of Floquet time crystals.