Non-Abelian Anyon Interferometry Collaborators:Parsa Bonderson, Microsoft Station Q Alexei Kitaev,...
-
date post
19-Dec-2015 -
Category
Documents
-
view
214 -
download
0
Transcript of Non-Abelian Anyon Interferometry Collaborators:Parsa Bonderson, Microsoft Station Q Alexei Kitaev,...
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