ABSTRACT - DRUM - University of Maryland

Given initial state ˆρ(0) = ˆρ path ⊗ ˆρ s ⊗ ˆρ bath , we can find the off-diagonal component
of the final state ˆρ(t) = Û(t)ˆρ(0)Û † (t), corresponding to the interference, as
]
λ LR = Tr
[ÛL (t)Û † R (t)ˆρ s ⊗ ˆρ bath
[
]
= Tr ρ s Tr L [ˆρ bath,L Û L (t)]Tr R [ˆρ bath,R Û R (t)] .
(5.17)
Now we evaluate Ŵη(t) = Tr η [ˆρ bath,η Û η (t)] (notice Ŵη is still an operator in
the spin Hilbert space). We drop the η index in this calculation. First we switch to
interaction picture and the evolution operator
Û(t) can be represented formally as
Û(t) = T exp{−i ∫ t
0 dt′ Ĥ 1 (t ′ )} where
Ĥ 1 (t) = ∑ k
g k (σ + e i∆t/2 + σ − e −i∆t/2 )(â † k eiω kt + â k e −iω kt ). (5.18)
Following the derivation of the master equation for the density matrix, we can derive
a “master equation” for Ŵ (t) under the Born-Markovian approximation:
dŴ
dt
= −γ(n + 1/2 + σ z /2)Ŵ , (5.19)
where γ = π ∑ k g2 k δ(ω k − ∆), n = γ ∑ −1 k g2 k n kδ(ω k − ∆).
Therefore, the visibility of the interference, proportional to the trace of Ŵ , is
given by
ζ ∝ Tr[Ŵ (t)ρ s] ∝ e −γnt = e −γnL/v . (5.20)
Here L is the length of the inteferometer and v is the average velocity of the fluxon.
We notice that the model we have used is of course a simplification of the real
fluxon. We only focus on the decoherence due to the midgap states and assume
that only one such state is present. In reality, there could be many midgap states
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