On the decoherence of Majorana zero modes mediated by gapless fermions

TL;DR

This work addresses decoherence of Majorana zero modes when coupled to a gapless fermionic reservoir and derives a Lindblad master equation in a Majorana basis, revealing non-local jump operators that couple distant modes through the reservoir.Using Born-Markov approximations and Prosen’s third quantization, the authors connect the Liouvillian spectrum to a matrix $Z=oldsymbol{Λ}+ ext{π}oldsymbol{J}$, showing that the spectral gap $oldsymbol{Δ}$ generally decreases with the minimum Majorana separation as a power law in $d=1,2,3$ and that a vanishing gap can yield extra steady states in fine-tuned cases.Gaussian-state analysis yields a tractable covariance-matrix evolution $rac{d}{dt}M=rac{1}{2}[oldsymbol{Λ},M]-rac{ ext{π}}{2}ig"{oldsymbol{J},Mig"}$, illustrating slow parity decay when nonlocal couplings are present and enabling explicit simple-limit solutions.Unbiased numerics with a critical Kitaev-chain reservoir validate the Lindblad predictions qualitatively while uncovering non-Markovian effects like delayed correlation-building and revivals at finite size, informing the regime of validity for the Markov approximation and pointing to experimental observability in future devices.

Abstract

We study the decoherence of a collection of Majorana zero modes weakly coupled to a gapless reservoir of non-interacting fermions. Using the Born-Markov approximation, we derive a Lindblad master equation for the dissipative dynamics of the Majorana zero modes. Due to the long-range coupling between Majorana zero modes mediated by the gapless reservoir, the Lindblad jump operators are non-local linear combinations of the Majorana operators. We show that, as a consequence, the dissipative dynamics can exhibit long relaxation times, i.e. a slow decay of fermion parities. A spectral analysis of the Liouvillian shows that the slow-down is suppressed as a power law of the distance between Majorana zero modes. Finally, we validate the Lindblad equation by comparison with unbiased numerical simulations of the time evolution of the full density matrix. In particular, these illustrate that non-Markovian dynamics establishes non-local correlations at small times.

Figures (8)

• Figure 1: We consider a collection of Majorana zero modes (black dots) coupled by local tunneling (dashed lines) to a gapless reservoir of non-interacting fermions (gray). The reservoir supports trajectories linking different Majorana zero modes, as well as trajectories escaping away.
• Figure 2: Four Majorana zero modes coupled pairwise to two different reservoirs. The red and blue boxes represent two alternative ways to initialize fermion parity, denoted Z-type and X-type in the main text. The decay of fermion parity can be much slower for X-type initial conditions (blue) than for Z-type (red).
• Figure 3: Spectra of the Liouvillian for a free-electron gas reservoir in $d=1,2,3$ dimensions, for ten Majoranas placed randomly in a hypercube of size $L$, smaller in the top row ($k_FL=10)$ and larger in the bottom row ($k_FL=100$). Note that in $d=2$ and $d=3$ panel, the range of the axes shrinks in the bottom row.
• Figure 4: Scaling of the spectral gap with the distance between Majorana zero modes. The plot is generated by computing the spectral gap $\Delta$ and the minimum distance between two Majorana zero modes for one hundred samples with $2N$ Majorana zero modes randomly placed in $[0, L]^d$. We then compute the average gap and average minimum distance over this sample, resulting in a single data point in the scatter plot. We vary $N$ between $2$ and $20$ and $k_FL$ between $1$ and $10^4$ to generate the entire plot.
• Figure 5: Illustration of the model used in the numerical comparison. Majorana zero modes $a_1$ and $a_2$ are coupled via a hopping with strength $w$ to a reservoir given by a chain of Majorana zero modes with uniform nearest neighbor hopping $h$ (a Kitaev chain tuned to its critical point); $a_1$ and $a_2$ are connected to the reservoir $d$ lattice spacings apart. Majorana zero modes $a_3$ and $a_4$ are connected to an identical copy of the same reservoir. Periodic boundary conditions are used for the reservoirs.