Non-Abelian Anyon Interferometry Collaborators:Parsa Bonderson, Microsoft Station Q Alexei Kitaev, Caltech Joost Slingerland, DIAS Kirill Shtengel, UC Riverside

Exchange statistics in (2+1)D Quantum-mechanical amplitude for particles at x 1, x 2, …, x n at time t 0 to return to these coordinates at time t. Feynman: Sum over all trajectories, weighting each one by e iS. Exchange statistics: What are the relative amplitudes for these trajectories?

Exchange statistics in (2+1)D In (3+1)D, T −1 = T, while in (2+1)D T −1 ≠ T ! If T −1 = T then T 2 = 1, and the only types of particles are bosons and fermions.

Exchange statistics in (2+1)D: The Braid Group Another way of saying this: in (3+1)D the particle statistics correspond to representations of the group of permutations. In (2+1)D, we should consider the braid group instead:

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  =  /m for a Fractional Quantum Hall state at a filling factor ν = 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).

Non-Abelian Anyons Exchanging particles 1 and 2: Matrices M 12 and M 23 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 Exchanging particles 2 and 3:

The Quantum Hall Effect Eisenstein, Stormer, Science 248, 1990

“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

Case in Point: A “simple” interferometric experiment Q: What is the expected periodicity in Φ ? A: Φ* = 2π/e* = 3 Φ 0, right? Wrong! Φ* = Φ 0

Goldman, Liu and Zaslavsky, PRB 2005:

Probing Abelian Statistics in FQHE The reason for Φ* = Φ 0 is gauge invariance. At the end, the system is “made”of real electrons, hence adiabatically adding Φ 0 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!

FQHE Quasiparticle interferometer Chamon, Freed, Kivelson, Sondhi, Wen 1997 Fradkin, Nayak, Tsvelik, Wilczek 1998

Measure longitudinal conductivity due to tunneling of edge current quasiholes n

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).

* These are all subject to consistency conditions. Associativity relations for fusion: Braiding rules:

Consistency conditions A. Pentagon equation:

Consistency conditions B. Hexagon equation:

Topological S-matrix

is a parameter that can be experimentally varied. |M ab | < 1 is a smoking gun that indicates non-Abelian braiding.

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):

(Bonesteel, et. al.) Topological Quantum Computation (Kitaev, Preskill, Freedman, Larsen, Wang)

Topological Qubit (Das Sarma, Freedman, Nayak) One (or any odd number) of quasiholes per antidot. Their combined state can be either I or , we can measure the state by doing interferometry: The state can be switched by sending another quasihole through the middle constriction.

Topological Qubit (the Fibonacci kind) The two states can be discriminated by the amplitude of the interference term!

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. Conclusions If we had bacon, we could have bacon and eggs, if we had eggs…

* Normalization factors that make the diagrams isotopy invariant will be left implicit to avoid clutter. State vectors may be expressed diagrammatically in an isotopy invariant formalism: Part Two: Measurements and Decoherence:

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:

Mach-Zehnder Interferometer

Applying an (inverse) F-move to the target system, we have contributions from four diagrams: Remove the loops by using: to get…

… the projection: for the measurement outcome where

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: