Majorana Fermions and Topological Quantum Information Processing. Liang Jiang Yale University & IIIS. QIP 2013, Beijing

Transcription

1 Majorana Fermions and Topological Quantum Information Processing Liang Jiang Yale University & IIIS QIP 2013, Beijing

2 Conventional Quantum Systems Local degrees of freedom E.g., spins, photons, ions, superconducting devices, Merits: arbitrary unitary operations, distant entanglement, Challenges: vulnerable to various imperfections & decoherences Quantum Systems Topological Quantum Systems Global degrees of freedom E.g., Fractional Quantum Hall Effect, Topological Insulators, Majoranas, Merits: robust against local perturbation/decoherence. Challenges: limited unitary operations, quantum network

3 Majoranas in 1D Outline Kitaev Model & Semiconductor Wires Experimental signatures of Majoranas Duality (Spin v.s. Particle hole) Hybrid platforms Majoranas in 2D (Fu & Kane) Majoranas qubits & SC qubits

4 MAJORANAS IN 1D

5 Majorana Bound States Majoranas: E.g., half of a Dirac Fermion Search for Majorana: c c c c j j 2 1 and j j j 2 j 2 2i Ettore Majorana (1937) P+ip SC (SrRuO) Topological Insulators Read & Green, PRB 2000 Jang, et al., Science 2011 Quantum Wires Cold Atoms Fu & Kane. PRL 2008, Lutchyn, et al., PRL 2010 Oreg, et al., PRL 2010 L.J., Pekker, et al., PRL 2011 Sato, et al., PRL 2009 Zhu, et al., PRL 2010 L J,, Kitagawa, et al., PRL 2011

6 Topological Qubit Four Majoranas encode 1 topological qubit Subspace with odd Dirac fermion { 1 0, 0 1 }. Braiding of Majoranas Non abelian anyons U final AB init / with U AB Comments: N+2 Majoranas encode N topological qubits Braiding Majoranas using quasi 1D T junctions Braiding Majoranas is not universal (for computation) Nayak, et al., RMP 80, 1083 (2008). J. Alicea, et al., Nat. Phys. 7, (2010).

7 Kitaev Quantum Wire Create half of a Dirac Fermion Chain of (spinless) Dirac fermions 1 H cc tcc cc hc.. N1 N j j j j1 j j1 j1 2 j1 Introduce Majoranas:, N 1 H it j1 and, j, B j1, A Consider 0and Δ, Two localized Majoranas at the end of the quantum wire! N,,,,,,,, For, bound Majoranas exist! With exponential tail N Kitaev, arxiv: cond-mat/ (2001).

8 Topological Perspective for Kitaev Quantum Wire 1 H cc tcc cc hc.. N N1 j j j j1 j j1 j1 2 j1 Rewrite in BdG form with Ck ck, ck H 1 2 k C H C k k k H h h h * k k k k, x x k, y y k, zz k k Define unit vector: nˆ h / h k k k nˆ zˆ & nˆ zˆ 0 Key challenges: 1.Spinless fermions 2.P wave pairing The Z topological invarinance ˆ ˆ 0 sgn 1 n n t with h tcos k kz, k i h ih i e sin k kx, ky, k (trivial) (topological)

9 Spin 1/2 Fermions to Create Majoranas Fermions with two internal states ap a, a p, p, Effective energy scale: Δ p H a up a Ba a a a p 2m Kinetic energy 2 p z p H.c. p, p, p, p, Spin orbital coupling: Magnetic field: Define quantity: boundary between MFs appear at the boundary between two phases Δ/B 1 Topological (C<0) Trivial (C>0) μ/b E(p) 0 MF MF MF MF p Effectively Spinless Lutchyn, Sau, Das Sarma, PRL 105, (2010); Oreg, Refael, von Oppen, PRL, 105, (2010) L. J., Kitagawa, Alicea, et al., PRL, 106, (2011); L. J., Pekker, Alicea, et al., PRL, 106, (2011)

10 Braiding of Majoranas in Quasi 1D Systems 1. Move Majoranas along the wire 2. Exchange Majoranas via a T junction U exp / Wire networks for efficient exchange of many Majoranas Alicea, Oreg, Refael, von Oppen, Fisher, Nat. Phys. 7, (2011).

11 Experimental Signatures Zero Bias Peaks Majoranas implies. Δ/B 1 Increasing B field Trivial (C>0) Topological (C<0) 0 1 μ/b 1. V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhoven, Science 336, (2012). 2. M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff and H. Q. Xu, Nano Lett. 12, (2012). 3. A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum and H. Shtrikman, Nat. Phys. 8, (2012).

12 Experimental Signatures Fractional Josephson Effects Conventional Josephson effect i H.c.,,,, H a up a Ba a e a a p p p z p p p p p sin T NT T Majorana Josephson effect sin 2 T NT T Zipper Majorana Josephson effect sin 2 T NT T A. Kitaev, Physics-Uspekhi 44, 131 (2001); Kwon, Sengupta, Yakovenko EPJB 37, 349 (2003); Fu, Kane, PRB 79, (2009); Lutchyn, Sau, Das Sarma PRL 105, (2010); Akhmerov et al, PRL 106, (2011); L. J., Pekker, Alicea, Refael, Oreg, von Oppen, PRL, 106, (2011)

13 Experimental Signatures Fractional Josephson Effects Conventional Josephson effect sin T NT T Majorana Josephson effect sin 2 2 T NT 2 T DC voltage on left-leg. AC voltage on right leg. Current on right leg. Even steps only! evac 12e Ir IM Jn( ) sin Vdctnt 0 n A. Kitaev, Physics-Uspekhi 44, 131 (2001); Kwon, Sengupta, Yakovenko EPJB 37, 349 (2003); Fu, Kane, PRB 79, (2009); Lutchyn, Sau, Das Sarma PRL 105, (2010); Akhmerov et al, PRL 106, (2011); L. J., Pekker, Alicea, Refael, Oreg, von Oppen, PRL, 106, (2011)

14 Experimental Signatures Fractional Josephson Effects B=0 B=1.0T Disappearance of conventional Josephson effects Signatures of unconventional Josephson effects B=1.6T B=2.0T B=2.5T L. P. Rokhinson, X. Liu and J. K. Furdyna, Nat Phys 8 (11), (2012). 2

15 Duality in 1D Quantum Wire p H.c.,,,, H a up a Ba a a a p z p p p p p p Use Nambu spinor basis Ψ,,, H u p B z z z x x p Generalize with,, and. H p cos sin z x y z B cos sin up z z z x y B

16 Duality in 1D Quantum Wires H p cos sin z x y Bz B cos sin vp z z z x y Physical Parameter Δ Charge current Majorana Josephson Effect Dual Parameter Spin current Majorana Spintronics Effect T, NT, T, L. J., Pekker, Alicea, Refael, Oreg, Brataas, von Oppen, arxiv: L. J., Pekker, Alicea, Refael, Oreg, von Oppen, Phys. Rev. Lett. 107, (2011).

17 Duality in 1D Quantum Wires H p 2 z z p vp 2m z B T NT T, 0,0, x y cos sin x y cos sin Δ/B Probe in Quantum Wires 1 Topological (C<0) Trivial (C>0) 0 1 μ/b Charge current j Q r 2e H r Fractional Josephson Effects is 4 periodic in, Fractional Magneto Josephson is 4 periodic in, Shapiro step measurement Rotate the magnetic angle Spin current j S r H r Fractional Phase Driven Spin Current is 4 periodic in, Fractional Spin Josephson is 4 periodic in, Ultrasensitive spin torque Use shift in FMR in nanoparticle L. J., Pekker, Alicea, Refael, Oreg, Brataas, von Oppen, arxiv: & In preparation

18 TOPOLOGICAL & CONVENTIONAL QUANTUM SYSTEMS

19 Hybrid Platforms between topological and conventional systems Goal 1. Coherent coupling 2. Switch on/off coupling without fine tuning (to restore topological protection) Idea 1. Use conventional system to control the evolution/braiding of topological system

20 Another approach to create Majoranas? Topological Insulator Interior: gapped insulator Surface: spin locked conductor Energy TI (bulk) TI (surface) Band gap Momentum Hasan and Kane, RMP 82, 3045 (2010); Qi and Zhang, RMP 83, 1057 (2011). Brüne, Liu, Novik, et al., PRL 106, (2011).

21 Another approach to create Majoranas? Topological Insulator 3D Topological Insulator Interior: gapped insulator Surface: spin locked conductor Φ /2 S S wave superconductor Δ.. Similar to p+ip superconductor Support Majoranas at vortices (e.g., Tri Junction of SC islands) Fu & Kane, PRL 100, (2008), Read & Green, PRB 61, (2000).

22 Two Majoranas with Coupling Using two tri junctions connected by a quantum wire Majorana wavefunction controlled by Interact along quantum wire 2 L 2 with, 0,1. 20 / Λ Δ / 2 Fu and Kane, PRL 100, (2008) 0

23 Superposition of Evolutions L Interaction between Two Majoranas 2 2 Overlap along quantum wire Observation ε induces superposition of evolutions (i.e., Ctrl Phase evolution) Highly non linear (good for switch on/off) How to achieve? / Λ Δ / 2 L.J., C. L. Kane and J. Preskill, PRL 106, (2011). Switch on Interaction Switch off Interaction

24 Flux Qubit to achieve Series of Josephson Junctions Josephson (potential) energy, cos Φ 2 with phase constraint and two potential minimum (1/2 1, 1) 0> 1> SC Flux Qubit CW/CCW current Superposition of two values of ε for 1. 4

25 Hybrid System Topological Quantum Wire & Flux Qubit coherently controls the coupling between Majoranas: H topo E E Z for = with 0 0 topo 0 Z for = with 1 1 topo 1 flux flux Coupling for Controlled Phase Gate g H Z Z 4 I flux topo with coupling strength 1 2 g E1E0 E 1E L.J., C. L. Kane and J. Preskill, PRL 106, (2011).

26 Transfer Quantum Information QND Repetitive measurement Topo qubit:. Flux qubit: 0, Φ 2 2 SWAP quantum state between topological qubit to flux qubit Topo qubit:. Flux qubit: H H H H H H L.J., C. L. Kane and J. Preskill, PRL 106, (2011).

27 Other Approaches to Hybrid Platforms Hybrid system of topological and flux qubits 2 2 Measure, probe anyonic statics, connect different topological systems, Various related proposals Φ 2 Semiconductor quantum wire + SC qubit Hassler et al., NJP 12, (2010) Bonderson, Lutchyn, PRL 106, (2011) Pekker, et al., arxiv Topological quantum wire + SC flux/phase qubit Jiang, Kane, Preskill, PRL 106, (2011) Topological quantum networks

28 Summary Kitaev Quantum Wire Majoranas Hybrid Platforms between Topological & Conventional Systems,,,,,,,, Semiconductor Wire & Braiding 2 2 Φ 2 Duality in 1D Wire (Spin & Particle-hole) T, NT T,

29 Conventional Quantum Systems Local degrees of freedom E.g., spins, photons, ions, superconducting devices, Merits: arbitrary unitary operations, distant entanglement, Challenges: vulnerable to various imperfections & decoherences Quantum Systems Topological Quantum Systems Global degrees of freedom E.g., Fractional Quantum Hall Effect, Topological Insulators, Majoranas, Merits: robust against local perturbation/decoherence. Challenges: limited unitary operations, quantum network New hybrid systems combine merits from both systems

30 Acknowledgement Mikhail Lukin (Harvard) Eugene Demler (Harvard) Takuya Kitagawa (Harvard) Peter Zoller (Innsbruck) Anton Akhemrov (Leiden) Yuval Oreg (Weizmann) Felix von Oppen (Berlin) Ignacio Cirac (MPQ) John Preskill (Caltech) Gil Refael (Caltech) Jason Alicea (UCI/Caltech) David Pekker (Caltech) Alexey Gorshkov (Caltech) Charlie Kane (UPenn) Sherman Fairchild Foundation, NSF-IQI, NSF-CUA