Exact master equation and decoherence dynamics of Majorana zero modes under the gate-induced charge fluctuation

TL;DR

This paper derives an exact master equation to analyze the decoherence of Majorana zero modes in a topological superconductor under charge fluctuations, revealing conditions for their protection or decay due to environment effects.

Contribution

It introduces a novel exact master equation approach for Majorana zero modes under charge fluctuations, capturing non-Markovian dynamics and environmental influences.

Findings

Zero-energy localized bound state protects Majorana modes at zero temperature.

Finite temperature environment prevents bound state formation, leading to inevitable decoherence.

The formalism describes detailed non-Markovian decoherence dynamics.

Abstract

In this paper, we derive the exact master equation to investigate the decoherence dynamics of Majorana zero modes in the Kitaev model, a 1D $p\,$-wave spinless topological superconducting chain (TSC), that is disturbed by charge fluctuations through gate controls. The exact master equation is derived by extending Feynman-Vernon influence functional approach to fermionic open systems involving pairing excitations. We obtain the exact master equations for the zero-energy bogoliubon in the TSC, and then transfer it into the master equation for Majorana zero modes. Within this exact master equation formalism, we can describe in detail the non-Markovian decoherence dynamics of zero-energy bogolibons as well as the Majorana zero modes under local perturbations. We find that at zero temperature, there is a zero-energy localized bound state which is not the original zero-energy bogoliubon or the original Majorana zero mode but a localized bound state of Majorana zero mode after the charge fluctuation is taken into account. It is this zero-energy localized bound state that protects Majorana zero modes from decoherence, as a long-time non-Markovain memory effect. However, for the environment at finite temperature, the zero-energy localized bound state cannot be formed when Majorana zero modes are locally perturbed, and decoherence is inevitable.