ABSTRACT - DRUM - University of Maryland

states:
ν(ε) =
2ν 0 ε
√
ε2 − ∆ 2 0
Θ(ε − ∆ 0 ). (6.49)
Here ν 0 is the normal-state density of states and ∆ 0 is the bulk superconducting
gap. We consider the limit ∆ 0 T ≫ 1. The long-time asymptotic behavior of the
integral is given by
〈 ˆP 0 (T )〉 ≈ 1 − 8ν −1 2|β|
0|β| sinh
∆
√ 0
4|β|2 + ∆ 2 0
( ) 1
+ O √ . (6.50)
∆0 T
Therefore, the non-adiabatic coupling to the excited continuum causes finite depletion
of the fermion parity in the zero-energy ground state subspace, which can
be regarded as the dissipation of the topological qubit.
The depletion becomes
comparable to 1 if ν 0
( |β|
∆ 0
) 2
∼ 1, rendering the qubit undefined. We notice that our
calculation breaks down for large |β| since then the excited states can not be treated
as being independent. They are coupled through second-order virtual processes via
the zero-energy state, which is weighted by ( |β|
∆ 0
) 2 perturbatively. Thus our results
should be regarded as the leading-order correction in the non-adiabatic perturbation
theory.
6.3 Discussion and Conclusion
In conclusion, we have considered the braiding of non-Abelian anyons as a
dynamical process and calculated the corrections to non-Abelian evolutions due
to non-adiabatic effects.
We discuss several sources of non-adiabaticity: first of
all, tunneling between non-Abelian anyons results in splitting of the degenerate
ground states. The Abelian dynamical phase accumulated in the process of braiding
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