Dynamics of Majorana tetron qubits under quasiparticle poisoning

Abstract: We study the dissipative dynamics of a Majorana tetron qubit in the presence of extrinsic quasiparticle poisoning due to the coupling to external leads. From the Bloch-Redfield equation describing a finite-size topological superconductor hosting four Majorana zero modes, we recover analytical expressions for the steady state, the parity leakage rate, and the decoherence rate of a Majorana qubit at arbitrary values of the charging energy. The analysis shows that the exponential suppression of the dephasing rate is gradually removed by the energy splitting of the qubit states. These results can be useful to understand time-domain experimental data in Majorana qubit prototypes.

• The paper develops an analytically tractable open-quantum-system theory of Majorana tetron qubit dynamics under quasiparticle poisoning, providing closed-form solutions for decoherence and parity leakage rates.
• The results reveal that exponential suppression of decoherence $\Gamma_{e} \sim \Gamma_{0} e^{-U/T}$ diminishes as qubit energy splitting $\epsilon$ approaches the charging energy $U$.
• The study differentiates between localized Majorana modes and extended Andreev bound states, with cross-correlation slowing down decoherence in the latter case.

Overview

The paper develops an analytically tractable open-quantum-system theory of a Majorana tetron qubit subject to extrinsic quasiparticle poisoning from gapless fermionic leads (2608.18042). The tetron—a floating topological superconducting island hosting four Majorana zero modes $a_1,\dots,a_4$—is modeled by a Hamiltonian combining quadratic overlap couplings $A_{ij}$, a charging-energy term $2U\,a_1a_2a_3a_4$ proportional to total parity, and point-contact tunneling to four thermal reservoirs. Starting from the Born-Markov (Bloch-Redfield) master equation, the authors obtain closed-form expressions for the steady state, parity leakage rate, and qubit decoherence rate at arbitrary ratios of charging energy $U$, level splitting $\epsilon$, tunneling rate $\Gamma_0$, and temperature $T$. The central result is that the exponential suppression of decoherence, $\Gamma_\mathrm{e}\sim \Gamma_0 e^{-U/T}$, is progressively removed as the qubit energy splitting $\epsilon$ grows toward $U$.

Model and energy scales

The isolated tetron spectrum consists of an even-parity doublet and an odd-parity doublet separated by the gap $A_{ij}$0, with intra-doublet splittings set by the canonical form of the antisymmetric matrix $A_{ij}$1 (eigenvalues $A_{ij}$2). The regime of topological protection corresponds to the hierarchy

$A_{ij}$3

where $A_{ij}$4 is the bare tunneling rate and $A_{ij}$5. The Born-Markov approximation requires $A_{ij}$6, i.e. $A_{ij}$7; for a wide-band Lorentzian reservoir with bandwidth $A_{ij}$8, numerical estimates confirm this separation of scales explicitly, with $A_{ij}$9 and variation of $2U\,a_1a_2a_3a_4$0 within a few percent over $2U\,a_1a_2a_3a_4$1 mK.

Because tunneling flips total parity, four odd-to-even Bohr frequencies arise: $2U\,a_1a_2a_3a_4$2 and $2U\,a_1a_2a_3a_4$3. Notably, the transition-frequency differences $2U\,a_1a_2a_3a_4$4 contain only combinations of $2U\,a_1a_2a_3a_4$5—never $2U\,a_1a_2a_3a_4$6—since allowed processes flip parity in opposite directions successively.

Degenerate regime

When all splittings vanish ($2U\,a_1a_2a_3a_4$7), the Bloch-Redfield equation collapses to a GKSL dissipator acting separately on the even and odd parity sectors of the density matrix. The coherences decay at rates

$2U\,a_1a_2a_3a_4$8

In the protected limit $2U\,a_1a_2a_3a_4$9, $U$0 is exponentially suppressed while $U$1 remains of order $U$2. The steady state is a thermal mixture of the two doublets, and total-parity leakage saturates at $U$3 on the fast timescale $U$4. Because the qubit states are degenerate, no distinction exists between relaxation and dephasing: all bilinear parity observables decay at $U$5 conditioned on even initial parity.

For weak but finite overlaps ($U$6), the dissipator is unchanged while coherent precession at $U$7 persists. Since $U$8, coherent oscillations can remain time-domain resolvable even when $U$9 are unresolvable in spectroscopy—an experimentally relevant observation.

Extended zero-energy states. The paper also treats the case where $\epsilon$0 form an extended Andreev bound state coupled to two leads, parametrized by a cross-correlation strength $\epsilon$1. A finite $\epsilon$2 couples the even and odd coherence channels, converting single-exponential decay into bi-exponential decay with rates $\epsilon$3. Strikingly, $\epsilon$4: the cross-correlation slows down decoherence, reaching $\epsilon$5 at $\epsilon$6. This provides a potential dynamical signature distinguishing extended zero-energy Andreev states from well-localized Majoranas, though the authors caution that extended states are also sensitive to local noise sources excluded from the analysis.

Secular regime and crossover

For $\epsilon$7, the secular approximation applies. Working with the symmetric model $\epsilon$8 (relevant to tetrons built from two nanowire segments), the resulting GKSL master equation yields the key formula

$\epsilon$9

which reduces to $\Gamma_0$0 when $\Gamma_0$1. This makes explicit how the splitting softens the exponential protection: once $\Gamma_0$2, suppression disappears even at low temperature. The steady state becomes the Gibbs state including the splitting, with $\Gamma_0$3, and relaxation of longitudinal parities occurs on the same timescale $\Gamma_0$4 as dephasing.

A technically important result is the proof that the universal Lindblad equation (ULE) coincides exactly with the secular dissipator for this model: all interference terms between the two Bohr frequencies $\Gamma_0$5 cancel upon summing over each canonical Majorana pair, using selection rules from total-parity-diagonal density matrices. Consequently, the secular-regime formulas remain valid for arbitrary $\Gamma_0$6, smoothly connecting to the degenerate limit—the central justification for the broad applicability claimed for the decoherence-rate formula.

Limitations and open questions

The analysis rests on several simplifying assumptions stated plainly by the authors: identical tunneling rates across leads, vanishing odd-parity splitting in the secular calculation, wide-band reservoirs with particle-hole symmetry where needed, and neglect of Lamb shifts. Intrinsic quasiparticle poisoning from out-of-equilibrium above-gap excitations is not treated; only extrinsic poisoning via equilibrium leads is covered. The general case with asymmetric couplings and finite odd-sector splittings "is probably amenable only to numerical analysis," which the authors leave open. Whether the predicted bi-exponential decay signatures of extended states survive competition with local noise channels remains unresolved within this framework.

Conclusion

This work supplies quantitative, analytically derived rates governing leakage and decoherence of tetron qubits under lead-induced quasiparticle poisoning, valid across the full crossover from ideal degeneracy to resolved level splittings. The main formula for $\Gamma_0$7 captures both the exponential protection at $\Gamma_0$8 and its erosion as $\Gamma_0$9 approaches $T$0, providing directly interpretable predictions for time-domain experiments on current InAs–Al, InAs–Pb, and Kitaev-chain-based devices.