Inverse Purcell Suppression of Decoherence in Majorana Qubits via Environmental Engineering

TL;DR

This paper introduces a method to enhance Majorana qubit coherence by engineering the environment to suppress low-frequency noise, leading to exponentially improved decoherence times with longer wires.

Contribution

It proposes an environmental engineering approach that suppresses environmental noise density of states, resulting in a fundamental inverse Purcell effect that improves qubit coherence.

Findings

Engineered environments with suppressed density of states reduce dephasing.

Longer topological wires exhibit exponentially better coherence.

The approach respects fermion parity superselection rules.

Abstract

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 $\epsilon \sim e^{-L/\xi}$ 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 $\Gamma_\phi \propto S(\epsilon)$, where $S(\epsilon)$ is the environmental noise power at the qubit splitting frequency. In the experimentally relevant regime where $k_B T \gtrsim \hbar\epsilon$ (with $T \sim 10-100$ mK), the noise power scales as $S(\epsilon) \propto \rho(\epsilon) k_B T/\hbar\epsilon$, leading to a dephasing rate $\Gamma_\phi \propto \rho(\epsilon) T/\epsilon$. This exposes a fundamental challenge: the dephasing rate diverges as $1/\epsilon$ 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 $\rho_{\text{engineered}}(\epsilon) = \rho_{\text{free}}(\epsilon) (\epsilon/\omega_c)^\alpha$. This creates an ``inverse Purcell effect'' that yields a temperature-independent suppression factor $F_P = (\epsilon/\omega_c)^\alpha$. For $\alpha > 1$, the engineered dephasing rate decreases exponentially with wire length, $\Gamma_{\phi,\text{engineered}} \propto e^{-(\alpha-1)L/\xi}$, 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.