Majorana Fermions in Wire Networks as non-Abelian Anyons

59 Pages·2014·0.88 MB·English

IQOQI
AUSTRIAN ACADEMY OF SCIENCES
Majorana Fermions in Wire Networks
as non-Abelian Anyons
M. Baranov
Institute for Quantum Optics and Quantum Information, OAW, Innsbruck
International School of Physics "Enrico Fermi"
Quantum Matter at Ultralow Temperatures
Varenna 7 – 15 July 2014
Outline:
Exchange and statistics
Majorana fermions as non-Abelian anyons
“Making” Majorana fermions (Kitaev wire)
Majorana fermions in atomic wires
- “Making” Kitaev wire with cold atoms/molecules
- Braiding protocol
Demonstration of non-Abelian statistics
Using Majorana fermions for QC
- Deutsch-Jozsa algorithm
Conclusion
Innsbruck Majorana Team:
Ying Hu Christina Kraus Catherine Laflamme
Marcello Dalmonte Sebastian Diehl Andreas Läuchli Peter Zoller
Exchange and statistics
Particle exchange and statistics
Behavior of the state (wave function)
Statistics under the exchange of two
identical (quasi)particles
Exchange of two (quasi)particles
1
2
     
Ψ(r , r ,) → Ψ(r , r ,) = ? Ψ(r , r ,)
1 2 2 1 1 2
Properties of many-body wave functions:
 
 
ψ(R , R ,; r , r ,)
1 2 1 2

R
positions of quasiparticles
i

r positions of particles
j

r
has to be single-valued with respect to particle coordinates
j

R
but not necessarily with respect to quasiparticle coordinates
i
Exchange as adiabatic dynamical evolution
General statements:
1. Adiabatic theorem: States in a (possibly degenerate) energy subspace
separated from others by a gap remain in the subspace when the system
is changed adiabatically without closing the gap.
2. Change under adiabatic transport (holonomy) = combination of Berry’s
phase/matrix and transformation of instantaneous energy eigenstate (explicit
holonomy)
3. Holonomy is invariant, but Berry’s phase/matrix and eigenstate transformation
depend on choice of gauge (and can be shifted from one to the other)
Exchange as adiabatic dynamical evolution
For a unique ground state Ψ (single-valued)
2 1
separated by a gap from excited states
1
t ψ → eiϕψ , ϕ= − ∫ dt E(t) +α

e
m 
ti
dynamical phase
d
1 2
Berry phase α= i ∫ dt ψ| ψ =α (path) +ϑ
g
dt
1
2
α (path)
geometrical phase
g
ϑ
statistical angle - of interest!
ψ → eiϑψ
Exchange statistics

Constraint on the exchange R
relative position
12
σˆ 2
σˆ
exchange double exchange
 
R R
12,f 12,f
2 1 1 2

t

e
m
ti
1  2 1  2
R R
12,i 12,i


0 R
0
R 12,i
12,i
 
R R
12,f 12,f
 
R = −R
12,f 12,i  
R = R
12,f 12,i
σˆ 2
Is an identity?
3D case: σˆ 2 = 1 (identity!)
C
The contour C can be deformed
  
0
R to a point R = const (= R )
12,i
12 12,i

R (i.e., to the case when nothing happens)
12,f

without crossing the origin R = 0
12
0

R
12,i
Conclusion: in 3D only bosons (ϑ = 0 ) or fermions ( ϑ = π )
ψ → ± ψ