Fingerprints of Majorana fermions in high-harmonic spectroscopy

Majorana fermions [1], due to the equivalent character of their particle and antiparticle, have found applications in various scientific areas like solid state physics, nuclear and particle physics. Till date, the presence of Majorana fermions in solid state systems is under vigorous debate. HHG in solids became a method of choice to probe different aspects of solids such as examining the dynamics of the defects in solids [2,3] and probe localisation-delocalisation phase transition [4] in condensed matter systems. In this work, we provide an alternative method to see the signatures of the Majorana fermions without ambiguity using ultrashort laser pulses and high-harmonic spectroscopy. We show that the nonlinear optical response of a system is able to sense the presence of Majorana fermions. The one-dimensional superconducting chain originally proposed by A. Kitaev that hosts Majorana edge modes in its topological phase is considered as the model system. We show non-resonant light fields probe topological–trivial superconducting phase transition in a system with edges, i.e., open boundary conditions. The sensitivity of high-harmonic spectroscopy to the superconducting phase transition to trace the signature of Majorana edge modes as their population dynamics are different than other modes in the bulk. Moreover, the high-harmonic spectroscopy becomes insensitive to the phase transition with similar harmonic profiles in both phases for systems with periodic boundary conditions when Majorana-zero-modes (MZMs) are absent. Moreover, we show that the harmonic spectra obtained from Bogoliubov-de Gennes (BdG) form of the Hamiltonian and from the many-body Kitaev superconducting Hamiltonian are same.