Inverse Purcell Suppression of Decoherence in Majorana Qubits via Environmental Engineering

Abstract

We show that the electromagnetic or phononic environment of a topological quantum device can be engineered to actively suppress decoherence. For a Majorana qubit in a superconducting wire, the exponentially small splitting $\epsilon \sim e^{-L/\xi }$ that provides topological protection also makes the qubit vulnerable to low-frequency noise. From a microscopic local interaction, we derive the effective low-energy coupling between the Majorana parity operator and a bosonic field; the coupling strength itself is proportional to $\epsilon$, reflecting the qubit's non-local nature. The resulting pure-dephasing rate scales as ${\Gamma }_{\varphi }\propto {\epsilon }^{2}S(\epsilon )$, with $S(\epsilon )$ the environmental noise power at frequency $\epsilon$. In the experimentally relevant high-temperature regime ($k_{B}T\gtrsim \hslash \epsilon$), this gives ${\Gamma }_{\varphi }\propto \epsilon \rho (\epsilon )T$, where $\rho (\epsilon )$ is the environmental density of states. By engineering an environment with a suppressed low-frequency density of states, ${\rho }_{\text{eng}}(\omega )={\rho }_0(\omega /{\omega }_c{)}^{\alpha }$ for $\omega <{\omega }_c$ ($\alpha >0$), the dephasing rate drops to ${\Gamma }_{\varphi }^{\text{eng}}\propto {\epsilon }^{\alpha +1}\propto e^{-(\alpha +1)L/\xi }$. Thus, longer wires yield exponentially better coherence---a direct synergy with topological protection. This "inverse Purcell" effect, which suppresses the density of states at the qubit frequency, provides a quantitative design principle for environmental engineering. Our work establishes spectral density shaping as a practical method for enhancing coherence in topological quantum devices.

5 pages. Revised version currently under review: Expanded the intro to include dissipative engineering approach, added a microscopic derivation of the effective qubit-environment coupling, showing it scales as $g\epsilon$ (rather than constant g), reflecting non-local nature of the Majorana qubit. Conclusions unchanged