Inverse Purcell Suppression of Decoherence in Majorana Qubits via Environmental Engineering

We propose a novel approach for optimizing topological quantum devices: instead of merely isolating qubits from environmental noise, we engineer the environment to actively suppress decoherence. For a Majorana qubit in a topological superconducting wire, the exponentially small energy splitting $\epsilon \sim e^{-L\mathrm{/}\xi }$ provides protection against local perturbations but renders it highly susceptible to pure dephasing from low-frequency environmental noise. We show that coupling via a parity-conserving operator ($i{\gamma }_L{\gamma }_R$) to a bosonic environment yields a dephasing rate ${\mathrm{\Gamma }}_{\phi }\propto S(\epsilon )$, where $S(\epsilon )$ is the environmental noise power at the qubit splitting frequency. In the experimentally relevant regime where $k_{B}T\gtrsim \mathrm{\hslash }\epsilon$ (with $T\sim 10-100$ mK), the noise power scales as $S(\epsilon )\propto \rho (\epsilon )k_{B}T\mathrm{/}\mathrm{\hslash }\epsilon$, leading to a dephasing rate ${\mathrm{\Gamma }}_{\phi }\propto \rho (\epsilon )T\mathrm{/}\epsilon$. This exposes a fundamental challenge: the dephasing rate diverges as $1\mathrm{/}\epsilon$ for a standard environment, e.g., a 1D system with linear dispersion where $\rho (\epsilon )$ is constant. We overcome this by designing environments with a suppressed density of states following ${\rho }_{\text{engineered}}(\epsilon )={\rho }_{\text{free}}(\epsilon )(\epsilon \mathrm{/}{\omega }_c{)}^{\alpha }$. This creates an ``inverse Purcell effect’’ that yields a temperature-independent suppression factor $F_P=(\epsilon \mathrm{/}{\omega }_c{)}^{\alpha }$. For $\alpha >1$, the engineered dephasing rate decreases exponentially with wire length, ${\mathrm{\Gamma }}_{\phi ,\text{engineered}}\propto e^{-(\alpha -1)L\mathrm{/}\xi }$, meaning longer wires provide better coherence protection. This provides a quantitative design principle where environmental engineering transforms detrimental noise into a tool for coherence stabilization, while respecting fermion parity superselection rules.