Non-Abelian Anyon Interferometry

Collaborators: Parsa Bonderson, Microsoft Station QAlexei Kitaev, CaltechJoost Slingerland, DIAS

Kirill Shtengel, UC Riverside

Exchange statistics in (2+1)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 relativeamplitudes for thesetrajectories?

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).

.2 angle lstatistica the,2 ,for E.g.

)1 units (in the

22 phase Bohm-Aharonov The

π/n/neΦeq

c

qΦqΦqΦ

'82) (Wilczek, composites flux - charge

:AnyonsAbelian of model)(toy Example

Φq

Non-Abelian Anyons

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

Exchanging particles 2 and 3:

The Quantum Hall Effect

Eisenstein, Stormer, Science 248, 1990

h

e 2

q

p

“Unusual” FQHE states

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

Xia et al. PRL 93, 2004Gap at 5/2 is 0.5K, at 12/5: 0.07K

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

Measure longitudinal conductivity due to tunneling of edge current quasiholes

n

xx

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).

vacuum."" to ingcorrespondtype particle a is There I

.

Ia

aa

give to withfusemay whichtype particle unique the

is thatconjugate" charge" a hastype particle Each

:vertices trivalent allowed of set a gives this

ically,Diagrammat .into andfusing for channels

of number the isinteger the where

:as specified are rules fusion The

cba

N cab

cNbac

cab

f

efabcdF ][

b a b a

abcR

* These are all subject to consistency conditions.

Associativity relations for fusion:

Braiding rules:

Consistency conditions

A. Pentagon equation:

Consistency conditions

B. Hexagon equation:

Abelian.-non is braiding the iff

and braiding Abelianto scorrespond

1

1

ab

ab

M

M

IbIa

IIabab SS

SSM

Topological S-matrix

xx calculate to Need

:setup alexperiment the to Back

abU 1 abU 2

abi Re 2

ab

2

2211 ababxx UtUt ababxx UUtttt 2

112

*1

2

2

2

1 Re2 ababxx Mtttt cos2 21

2

2

2

1

abiabababab eMRM 2

b

a

ababxx Mtttt cos2 21

2

2

2

1

12 /arg tt is a parameter that can be experimentally varied.

|Mab| < 1 is a smoking gun that indicates non-Abelian braiding.

evenfor

oddfor:chargeanyonic carry quasiholes

:chargeanyonic carry electrons

:chargeanyonic carry quasiholes

:Monodromy

:rules Fusion

:types particle Ising

chargeelectric to due factor Abelianfamiliar a is U(1)

IsingU(1)MR be to believed is

or

, ,

, ,

nIne

nnen

e

e

M

II

IIIII

I

),4/(

),4/(

),(

),4/(

111

101

111

,

,,

2/5

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):

),5/(

),(

),5/(

1

11

,

5/12

22

51

2

.38)(

, ,

FibU(1)PfU(1)RR

or:chargeanyonic carry quasiholes

:chargeanyonic carry electrons

:chargeanyonic carry quasiholes

ratio golden the iswhere

:Monodromy

:rules Fusion

:types particle Fib

be to believed is

computer.) quantum universal a simulate can anyons Fibonacci :(Note

33

Inen

Ie

e

M

IIIII

I

forandforwhere

:even

:odd

For

10

4/cos12

:2/5

21

2

2

2

1

2

2

2

1

NIN

nttttn

ttnN

xx

xx

38.

10

5/4cos2

5/12

2

221

2

2

2

1

r

:Note

foandcharge Fibonacci forwhere

For

:2006) PRB Stone & Chung

2006; PRL dSlingerlan & KS ,(Bonderson

NIN

nttttN

xx

0 1

eterInterferom

0c 1c

a b a b

(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)

)()1(05/4cos2 221

2

2

2

1 r fo , INnttttN

xx

The two states can be discriminated by the amplitude of the interference term!

orbemay antidots two of state combined the :qubit I5/12

Wish listGet 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:

4/1/ bac ddd

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-ZehnderInterferometer

122

**

jj

jj

jjj

rt

tr

rtT

Applying an (inverse) F-move to the target system, we have contributions from four diagrams:

Remove the loops by using:

0ba

abab SS

SSM

00

00

to get…

… the projection:

for the measurement outcome where

,s

Result:

'baab MM

1ebM

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:

a 'a

e

bsp

sp