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Proposed experimental probes of non-abelian anyons

Proposed experimental probes of non-abelian anyons. Ady Stern (Weizmann) with: N.R. Cooper, D.E. Feldman, Eytan Grosfeld , Y. Gefen, B.I. Halperin, Roni Ilan , A. Kitaev, K.T. Law , B. Rosenow, S. Simon. Outline:

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Proposed experimental probes of non-abelian anyons

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  1. Proposed experimental probes of non-abelian anyons Ady Stern (Weizmann) with: N.R. Cooper, D.E. Feldman, Eytan Grosfeld, Y. Gefen, B.I. Halperin, Roni Ilan, A. Kitaev, K.T. Law, B. Rosenow, S. Simon

  2. Outline: • Non-abelian anyons in quantum Hall states – what they are, why they are interesting, how they may be useful for topological quantum computation. • How do you identify a non-abelian quantum Hall state when you see one ?

  3. More precise and relaxed presentations: Introductory pedagogical Comprehensive

  4. The quantized Hall effect and unconventional quantum statistics

  5. or a fraction with qodd, The quantum Hall effect • zero longitudinal resistivity - no dissipation, bulk energy gap current flows mostly along the edges of the sample • quantized Hall resistivity B I n is an integer, or q even

  6. Extending the notion of quantum statistics Laughlin quasi-particles Electrons A ground state: Energy gap Adiabatically interchange the position of two excitations

  7. More interestingly, non-abelian statistics(Moore and Read, 91) In a non-abelian quantum Hall state, quasi-particles obey non-abelian statistics, meaning that (for example) with 2N quasi-particles at fixed positions, the ground state is -degenerate. Interchange of quasi-particles shifts between ground states.

  8. ground states position of quasi-particles ….. Permutations between quasi-particles positions unitary transformations in the ground state subspace

  9. 1 2 3 2 1 3 Up to a global phase, the unitary transformation depends only on the topology of the trajectory Topological quantum computation (Kitaev 1997-2003) • Subspace of dimension 2N,separated by an energy gap from the continuum of excited states. • Unitary transformations within this subspace are defined by the topology of braiding trajectories • All local operators do not couple between ground states • – immunity to errors

  10. The goal: • experimentally identifying non-abelian quantum Hall states • The way: the defining characteristics of the most prominent candidate, the n=5/2 Moore-Read state, are • Energy gap. • Ground state degeneracy exponential in the number N of quasi-particles, 2 N/2. • Edge structure – a charged mode and a Majorana fermion mode • Unitary transformation applied within the ground state subspace when quasi-particles are braided.

  11. In this talk: • Proposed experiments to probe ground state degeneracy – thermodynamics • Proposed experiments to probe edge and bulk braiding by electronic transport– • Interferometry, linear and non-linear Coulomb blockade, Noise

  12. Probing the degeneracy of the ground state (Cooper & Stern, 2008 Yang & Halperin, 2008)

  13. Measuring the entropy of quasi-particles in the bulk The density of quasi-particles is Zero temperature entropy is then To isolate the electronic contribution from other contributions:

  14. Leading to (~1.4) (~12pA/mK)

  15. Probing quasi-particle braiding - interferometers

  16. All g’s anti-commute, and g2=1. A localized Majorana operator . • Essential information on the Moore-Read state: • Each quasi-particle carries a single Majorana mode • The application of the Majorana operators takes one ground state to another within the subspace of degenerate ground states When a vortex i encircles a vortex j, the ground state is multiplied by the operator gigj Nayak and Wilczek Ivanov

  17. Interference term Number of q.p.’s in the interference loop, even odd even Brattelli diagram Interferometers: The interference term depends on the number and quantum state of the quasi-particles in the loop.

  18. 1 a Odd number of localized vortices: vortex a around vortex 1 - g1ga The interference term vanishes:

  19. Even number of localized vortices: vortex a around vortex 1 and vortex 2 - g1gag2ga~ g2g1 2 1 a The interference term is multiplied by a phase: Two possible values, mutually shifted by p

  20. Interference in the n=5/2 non-abelian quantum Hall state: The Fabry-Perot interferometer D2 D1 S1

  21. n=5/2 Gate Voltage, VMG (mV) Magnetic Field (or voltage on anti-dot) The number of quasi-particles on the island may be tuned by charging an anti-dot, or more simply, by varying the magnetic field. cell area

  22. Coulomb blockade vs interference (Stern, Halperin 2006, Stern, Rosenow, Ilan, Halperin, 2009 Bonderson, Shtengel, Nayak 2009)

  23. Interferometer (lowest order) Quantum dot For non-interacting electrons – transition from one limit to another via Bohr-Sommerfeld interference of multiply reflected trajectories. Can we think in a Bohr-Sommerfeld way on the transition when anyons, abelian or not, are involved? Yes, we can (BO, 2008) (One) difficulty – several types of quasi-particles may tunnel

  24. Thermodynamics is easier than transport. Calculate the thermally averaged number of electrons on a closed dot. Better still, look at The simplest case, n=1. Energy is determined by the number of electrons Partition function Poisson summation

  25. Sum over windings. • Thermal suppression of high winding number. • An Aharonov-Bohm phase proportional to the winding number. • At high T, only zero and one windings remain • Sum over electron number. • Thermal suppression of high energy configurations

  26. And now for the Moore-Read state • The energy of the dot is made of • A charging energy • An energy of the neutral mode. The spectrum is determined by the number and state of the bulk quasi-particles. The neutral mode partition function χdepends on nqp and their state. Poisson summation is modular invariance (Cappelli et al, 2009)

  27. The components of the vector correspond to the different possible states of the bulk quasi-particles, one state for an odd nqp (“s”), and two states for an even nqp(“1” and “ψ”). A different thermal suppression factor for each component. The modular S matrix. Sabencodes the outcome of a quasi-particle of type a going around one of type b

  28. Low T High T

  29. Probing excited states at the edge – non linear transport in the Coulomb blockade regime (Ilan, Rosenow, Stern, 2010)

  30. A nu=5/2 quantum Hall system n=2 Goldman’s group, 80’s

  31. Non-linear transport in the Coulomb blockade regime: dI/dV at finite voltage – a resonance for each many-body state that may be excited by the tunneling event. dI/dV Vsd

  32. Energy spectrum of the neutral mode on the edge Single fermion: For an odd number of q.p.’s En=0,1,2,3,…. For an even number of q.p.’s En= ½, 3/2, 5/2, … Many fermions: For an odd number of q.p.’s Integers only For an even number of q.p.’s Both integers and half integers (except 1!) The number of peaks in the differential conductance varies with the number of quasi-particles on the edge.

  33. Even number Odd number Current-voltage characteristics (Ilan, Rosenow, AS 2010) Source-drain voltage Magnetic field

  34. Interference in the n=5/2 non-abelian quantum Hall state: Mach-Zehnder interferometer

  35. The Mach-Zehnder interferometer: (Feldman, Gefen, Kitaev, Law, Stern, PRB2007) D1 S D2

  36. D1 S D2 D2 D1 S1 Compare: M-Z F-P Main difference: the interior edge is/is not part of interference loop For the M-Z geometry every tunnelling quasi-particle advances the system along the Brattelli diagram (Feldman, Gefen, Law PRB2006)

  37. G4 G2 G3/2 G1/2 G1 G4/2 G2/2 G3 Interference term Number of q.p.’s in the interference loop • The system propagates along the diagram, with transition rates assigned to each bond. • The rates have an interference term that • depends on the flux • depends on the bond (with periodicity of four)

  38. If all rates are equal, current flows in “bunches” of one quasi-particle each – Fano factor of 1/4. The other extreme – some of the bonds are “broken” Charge flows in “bursts” of many quasi-particles. The maximum expectation value is around 12 quasi-particles per burst – Fano factor of about three.

  39. Interference magnitude depends on the parity of the number of quasi-particles Phase depends on the eigenvalue of Summary: Temperature dependence of the chemical potential and the magnetization reflect the ground state entropy Coulomb blockade I-V characteristics may measure the spectrum of the edge Majorana mode Fano factor changing between 1/4 and about three – a signature of non-abelian statistics in Mach-Zehnder interferometers Mach-Zehnder:

  40. even odd even Das-Sarma-Freedman-Nayak qubit For a Fabry-Perot interferometer, the state of the bulk determines the interference term. D2 D1 S1 Interference term Number of q.p.’s in the interference loop, The interference phases are mutually shifted by p.

  41. even odd even D2 D1 S1 Interference term Number of q.p.’s in the interference loop, The sum of two interference phases, mutually shifted by p. The area period goes down by a factor of two.

  42. Gate Voltage, VMG (mV) Magnetic Field (or voltage on anti-dot) Ideally, The magnetic field Quasi-particles number The gate voltage Area cell area

  43. Are we getting there? (Willett et al. 2008)

  44. From electrons at n=5/2 to non-abelian quasi-particles: Read and Green (2000) Step I: A half filled Landau level on top of two filled Landau levels Step II: the Chern-Simons transformation from:electrons at a half filled Landau level to: spin polarized composite fermions at zero (average) magnetic field GM87 R89 ZHK89 LF90 HLR93 KZ93

  45. (c) B 20 (b) CF B B1/2 = 2ns0 B Electrons in a magnetic fieldB e- H y = E y Composite particles in a magnetic field Mean field (Hartree) approximation

  46. Step III: fermions at zero magnetic field pair into Cooper pairs Spin polarization requires pairing of odd angular momentum a p-wave super-conductor of composite fermions Step IV: introducing quasi-particles into the super-conductor - shifting the filling factor away from 5/2 The super-conductor is subject to a magnetic field and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles.

  47. Step III: fermions at zero magnetic field pair into Cooper pairs Spin polarization requires pairing of odd angular momentum a p-wave super-conductor of composite fermions Step IV: introducing quasi-particles into the super-conductor - shifting the filling factor away from 5/2 The super-conductor is subject to a magnetic field and thus accommodates vortices. The vortices, which are charged, are the non-abelian quasi-particles. For a single vortex – there is a zero energy mode at the vortex’ core Kopnin, Salomaa (1991), Volovik (1999)

  48. A zero energy solution is a spinor g(r) is a localized function in the vortex core All g’s anti-commute, and g2=1. A localized Majorana operator . A subspace of degenerate ground states, with the g’s operating in that subspace. In particular, when a vortex i encircles a vortex j, the ground state is multiplied by the operator gigj Nayak and Wilczek (1996) Ivanov (2001)

  49. Effective charge span the range from 1/4 to about three. The dependence of the effective charge on flux is a consequence of unconventional statistics. Charge larger than one is due to the Brattelli diagram having more than one “floor”, which is due to the non-abelian statistics In summary, flux dependence of the effective charge in a Mach-Zehnder interferometer may demonstrate non-abelian statistics at n=5/2

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