# How to read out a Majorana qubit in a topological insulator

```How to read out a Majorana qubit in a
topological insulator
Anton Akhmerov
with Johan Nilsson and Carlo Beenakker
PRL 102, 216404 (2009)
(Related work: Fu&Kane PRL 102, 216403 (2009) )
Lyon mini-school on topological insulators, December 11 2009
InstituutLorentz
Majorana fermions & edge state interferometry
1. Majorana bound states form qubit
a = γ1 + iγ2 , a† = γ1 − iγ2
(Kitaev):
Majorana fermions & edge state interferometry
Das Sarma et al., Stern & Halperin, Bonderson et al.
1. Majorana bound states form qubit
a = γ1 + iγ2 , a† = γ1 − iγ2
(Kitaev):
2. Majorana mode at the edges ψ(ε) = ψ † (−ε)
Majorana fermions & edge state interferometry
Das Sarma et al., Stern & Halperin, Bonderson et al.
1. Majorana bound states form qubit
a = γ1 + iγ2 , a† = γ1 − iγ2
(Kitaev):
2. Majorana mode at the edges ψ(ε) = ψ † (−ε)
3. Can measure qubit in Fabry-Perot interferometer
due to non-abelian braiding statitics
Majorana fermions & edge state interferometry
Das Sarma et al., Stern & Halperin, Bonderson et al.
Main question:
How to do the same with Majorana qubit in TI?
Topological insulators
Has time reversal and electron-hole symmetries
v σp − EF
∆
H=
∆∗
EF − v σp
But no edge modes
Topological insulators
Edge Majorana mode (Fu&Kane)
v σp + mσz − EF
H=
∆∗
∆
EF − v σp + mσz
Neutral unlike 5/2 QHE ⇒ no charge current
Topological insulators
Edge Majorana mode
v σp − mσz − EF
H=
∆∗
∆
EF − v σp − mσz
Direction controlled by magnetization (Can this help?)
Electron to Majorana converter
Electron to Majorana converter
1. No backscattering
Electron to Majorana converter
1. No backscattering
2. Electron-hole symmetry completely fixes coupling
Electron to Majorana converter
1. No backscattering
2. Electron-hole symmetry completely fixes coupling
3. c → ψ1 + iψ2 , c † → ψ1 − iψ2
Electron to Majorana converter
1.
2.
3.
4.
No backscattering
Electron-hole symmetry completely fixes coupling
c → ψ1 + iψ2 , c † → ψ1 − iψ2
Charge is not conserved (goes into superconductor)
±1 → 0 + 0
Setup 1: Mach-Zehnder
cin → ψ1 + iψ2 → cout
G =0
Setup 1: Mach-Zehnder
†
cin → ψ1 + iψ2 → ψ1 − iψ2 → cout
G =2
e2
h
Setup 1: Mach-Zehnder
Measures parity of the number of vortices
Setup 2: Fabry-Perot
Setup 2: Fabry-Perot
e2
(tγ1 + (−1)nf tγ2 )2
h
Allows for the Majorana qubit readout
(if coherent phase slips may occur)
I ∼V
Setup 2: Fabry-Perot
e2
(tγ1 + (−1)nf tγ2 )2
h
Measures the fermion number parity
I ∼V
Conclusions
1. Deterministic conversion of an electron into a pair of
Majorana fermions
2. Maximal current via two neutral modes
3. Vortex parity measurement
4. Readout of a topological qubit. (Potentially)
Conclusions
Thank you all.
The end.
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