NonAbelian Anyon Interferometry

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Non-Abelian Anyon Interferometry
Kirill Shtengel, UC Riverside
Collaborators Parsa Bonderson, Microsoft
Station Q Alexei Kitaev, Caltech Joost
Slingerland, DIAS
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Exchange statistics in (21)D

• Quantum-mechanical amplitude for particles at
  x1, x2, , xn at time t0 to return to these
  coordinates at time t.
• Feynman Sum over all trajectories, weighting
  each one by eiS.

Exchange statistics What are the
relative amplitudes for these trajectories?
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Exchange statistics in (21)D

• In (31)D, T -1 T, while in (21)D T -1 ? T !
• If T -1 T then T 2 1, and the only types of
  particles are bosons and fermions.

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Exchange statistics in (21)DThe Braid Group
Another way of saying this in (31)D the
particle statistics correspond to representations
of the group of permutations. In (21)D, we
should consider the braid group instead
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Abelian Anyons

• Different elements of the braid group correspond
  to disconnected subspaces of trajectories in
  space-time.
• Possible choice weight them by different overall
  phase factors (Leinaas and Myrrheim, Wilczek).

• These phase factors realise an Abelian
  representation of the braid group.
• E.g q p/m for a Fractional Quantum Hall state
  at a filling factor ? 1/m.
• Topological Order is manifested in the ground
  state degeneracy on higher genus
  manifolds (e.g. a torus) m-fold degenerate
  ground states for FQHE (Haldane, Rezayi 88, Wen
  90).

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Non-Abelian Anyons
Exchanging particles 2 and 3

• Exchanging particles 1 and 2

• Matrices M12 and M23 need not commute, hence
  Non-Abelian Statistics.
• Matrices M form a higher-dimensional
  representation of the braid-group.
• For fixed particle positions, we have a
  non-trivial multi-dimensional Hilbert
• space where we can store information

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The Quantum Hall Effect
Eisenstein, Stormer, Science 248, 1990
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Unusual FQHE states

• Pan et al. PRL 83,1999
• Gap at 5/2 is 0.11 K

Xia et al. PRL 93, 2004 Gap at 5/2 is 0.5K, at
12/5 0.07K
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Case in Point A simple interferometric
experiment

• Q What is the expected periodicity in F?
• A F 2p/e 3F0, right?
• Wrong! F F0

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• Goldman, Liu and Zaslavsky, PRB 2005

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Probing Abelian Statistics in FQHE

• The reason for F F0 is gauge invariance. At
  the end, the system is madeof real electrons,
  hence adiabatically adding F0 should bring the
  system back to its ground state (Byers and Yang,
  1961).

• But this need not be the same ground state
  (Thouless and Gefen, 1991)!
• A transition between these ground states is
  accompanied by tunnelling a quasihole between the
  inner and the outer edge of a ring.
• So, such experiment would pick both charge and
  statistical contributions!

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FQHE Quasiparticle interferometer
Chamon, Freed, Kivelson, Sondhi, Wen
1997 Fradkin, Nayak, Tsvelik, Wilczek 1998
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Measure longitudinal conductivity due to
tunneling of edge current quasiholes
n
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General Anyon Models(unitary braided tensor
categories)Describes two dimensional systems
with an energy gap. Allows for multiple particle
types and variable particle number.
1. A finite set of particle types or anyonic
charges.
2. Fusion rules (specifying how charges can
combine or split).
3. Braiding rules (specifying behavior under
particle exchange).
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Associativity relations for fusion
Braiding rules
These are all subject to consistency conditions.
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Consistency conditions
A. Pentagon equation
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Consistency conditions
B. Hexagon equation
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Topological S-matrix
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is a parameter that can be experimentally varied.
Mab lt 1 is a smoking gun that indicates
non-Abelian braiding.
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Combined anyonic state of the antidot

• Adding non-Abelian anyonic charges for the
  Moore-Read state

Even-Odd effect (Stern Halperin Bonderson,
Kitaev KS, PRL 2006)
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Topological Quantum Computation(Kitaev,
Preskill, Freedman, Larsen, Wang)
(Bonesteel, et. al.)
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Topological Qubit(Das Sarma, Freedman, Nayak)
One (or any odd number) of quasiholes per
antidot. Their combined state can be either I or
y, we can measure the state by doing
interferometry
The state can be switched by sending another
quasihole through the middle constriction.
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Topological Qubit(the Fibonacci kind)
The two states can be discriminated by the
amplitude of the interference term!
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Wish list

• Get experimentalists to figure out how to
  perform these very difficult measurements and,
  hopefully,
• Confirm non-Ableian statistics.
• Find a computationally universal topological
  phase.
• Build a topological quantum computer.

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Part Two Measurements and Decoherence
State vectors may be expressed diagrammatically
in an isotopy invariant formalism
Normalization factors
that make the diagrams isotopy invariant will
be left implicit to avoid clutter.
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Anyonic operators are formed by fusing and
splitting anyons (conserving anyonic charge). For
example a two anyon density matrix is
Traces are taken by closing loops on the
endpoints
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Mach-Zehnder Interferometer
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Applying an (inverse) F-move to the target
system, we have contributions from four diagrams
Remove the loops by using
to get
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the projection
for the measurement outcome where
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Result
Interferometry with many probes gives
binomial distributions in for the
measurement outcomes, and collapses the target
onto fixed states with a definite value of .
Hence, superpositions of charges and
survive only if
and only in fusion channels corresponding to
difference charges with