ABSTRACT - DRUM - University of Maryland

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J v SC J s ,Q Figure 5.1: Mach-Zehnder interferometer proposed in Ref. [122] for topological qubit detection. Due to the Aharonov-Casher effect, the vortex current is sensitive to the charge enclosed. Long Josephson junction between two topological superconductors carries allows for Josephson vortices (fluxons) that carry Majorana zero-energy modes. 5.2 Interferometry in the Presence of Midgap States We now discuss the effect of the midgap states in interferometry experiments designed for the qubit read-out [115, 116, 117, 118] There is a number of recent proposals for interferometry experiments in topological superconductors [119, 120, 121]. Here we use an example of the Mach-Zehnder interferometer proposed by Grosfeld and Stern [122] based on Aharonov-Casher (AC) effect. In this proposal, a Josephson vortex (fluxon) is driven by supercurrent J s to circumvent a superconducting island with charge Q and flux Φ, see Fig. 5.1. The fluxon appearing at the interface 84
etween two topological p x + ip y superconductors (represented by the shaded region in Fig. 5.1) carries a zero-energy Majorana modes, and behaves as a non-Abelian anyon. Therefore, the vortex current around the central superconductor is sensitive to the topological content of the enclosed superfluid. (We refer the reader to Ref. [122] for more details.) Indeed, the vortex current is proportional to the total tunneling amplitude: J v ∝ |(t L Û L + t L Û R )|Ψ 0 〉| 2 = |t L | 2 + |t R | 2 + 2Re{t ∗ Lt R 〈Ψ 0 |Û −1 L ÛR|Ψ 0 〉} (5.9) = |t L | 2 + |t R | 2 + 2Re{t ∗ Lt R e iϕ AC 〈Ψ 0 | ˆM|Ψ 0 〉}. Here |Ψ 0 〉 is the initial state of the system and ÛL and ÛR are the unitary evolution operators for the fluxon taking the two respective paths. ϕ AC is the Aharonov- Casher phase accumulated by the fluxon: ϕ AC = πQ/e. Here Q is the total charge enclosed by the trajectory of the fluxon, including the offset charge Q ext set by external gate and the fermion parity n p of the low-energy fermionic states: Q = Q ext + en p . (5.10) ˆM encodes the transformation solely due to the braiding statistics of the non-Abelian fluxon around n non-Abelian vortices. If the superconducting island contains no vortices, then ˆM = 1 and the interference term is solely determined by the AC phase. The magnitude of the vortex current shows an oscillation: ( πQ J v = J v0 [1 + ζ cos e )] . (5.11) Here ζ is the visibility of the interference. 85
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ABSTRACT Title of dissertation: MAJ
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MAJORANA QUBITS IN NON-ABELIAN TOPO
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Acknowledgments First and foremost,
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Table of Contents List of Figures L
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List of Figures 1.1 Braiding of Maj
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Chapter 1 Introduction The subject
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closed throughout the path. If one
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its original configuration, we requ
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mathematics. It is easy to check th
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Here τ x is the Pauli matrix actin
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y constructing the ground state pro
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here and leaves its proof to Append
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the zero mode (see Chapter 3 for de
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In the case of Ising anyons, 2N vor
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find a realistic material in nature
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states near the Fermi circle. We fi
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pairing: ( ) p H = ψ † 2 2m −
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dimensional non-chiral Majorana mod
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is formed out of two fermions, does
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gapless and should not be neglected
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zero modes. We study a one-dimensio
Page 43 and 44: of possible paired states which is
Page 45 and 46: where d ≡ (d 1 , d 2 , d 3 ) = (R
Page 47 and 48: If the order parameter is proportio
Page 49 and 50: 0.015 T/t 0.01 Normal p x + ip y 0.
Page 51 and 52: trivial p x - and p y -states lead
Page 53 and 54: with the gap operator being ˆ∆ =
Page 55 and 56: 0.2 T/t 0.1 Normal f p x + ip y µ
Page 57 and 58: 2.3 Discussion and Conclusions In c
Page 59 and 60: Chapter 3 Majorana Bound States in
Page 61 and 62: satisfied for m = 0. The singlevalu
Page 63 and 64: We summarize this section by provid
Page 65 and 66: Here γ a and Γ a are 4 × 4 Dirac
Page 67 and 68: with λ = µξ/v. It has the follow
Page 69 and 70: where n = (R∆, −I∆) field des
Page 71 and 72: Chapter 4 Topological Degeneracy Sp
Page 73 and 74: ound states are usually splitted an
Page 75 and 76: sponding quasiparticle operator can
Page 77 and 78: Of particular importance is the sig
Page 79 and 80: [61]. Although analytical expressio
Page 81 and 82: eyond perturbation theory and the r
Page 83 and 84: eaks down in this regime and one sh
Page 85 and 86: 4.3 Conclusions In this chapter, we
Page 87 and 88: the fermion bath, and consequently
Page 89 and 90: tion for the reduced density matrix
Page 91 and 92: like phonons. However, such process
Page 93: We also notice that in this formula
Page 97 and 98: neglect the AC phase. The Hamiltoni
Page 99 and 100: which in principle lead to a strong
Page 101 and 102: We conclude that the geometric phas
Page 103 and 104: The first-order term vanishes due t
Page 105 and 106: apply to any realistic system where
Page 107 and 108: ound states are typically close to
Page 109 and 110: Chapter 6 Non-adiabaticity in the B
Page 111 and 112: minor modifications in the details,
Page 113 and 114: {α(T )} to {α(0)}: |α(T )〉 =
Page 115 and 116: educed to 2 N−1 by fermion parity
Page 117 and 118: Berry connection of this state then
Page 119 and 120: wavefunction is u n (r), v n (r). W
Page 121 and 122: asis. The most general form of BdG
Page 123 and 124: 6.2.1 Effect of Tunneling Splitting
Page 125 and 126: Because E + ∼ ∆ 0 e −R/ξ , |
Page 127 and 128: tunneling amplitudes between them a
Page 129 and 130: ˆΓ i = iˆγ i ˆξiˆη i , i =
Page 131 and 132: considerations, we should have β 1
Page 133 and 134: states: ν(ε) = 2ν 0 ε √ ε2
Page 135 and 136: necessary as a matter of principle
Page 137 and 138: Chapter 7 Majorana Zero Modes Beyon
Page 139 and 140: We then demonstrate that in a finit
Page 141 and 142: The standard Abelian bosonization r
Page 143 and 144: and define the chiral fields by ˜
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of (7.2) has a product of the Klein
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7.3 Stability of the Degeneracy We
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quence. The splitting is then propo
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where α = 1 2 (K + K−1 − 2). A
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effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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which allows further simplification
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List of Publications Publications t
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Bibliography [1] K. von Klitzing, G
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[46] S. Das Sarma, C. Nayak, and S.
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[92] A. W. W. Ludwig, D. Poilblanc,
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[133] M. V. Berry, Proc. R. Soc. Lo
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