Preparation of topological modes by Lyapunov control

By Lyapunov control, we present a proposal to drive quasi-particles into a topological mode in quantum systems described by a quadratic Hamiltonian. The merit of this control is the individual manipulations on the boundary sites. We take the Kitaev’s chain as an illustration for Fermi systems and show that an arbitrary excitation mode can be steered into the Majorana zero mode by manipulating the chemical potential of the boundary sites. For Bose systems, taking the noninteracting Su-Schrieffer-Heeger (SSH) model as an example, we illustrate how to drive the system into the edge mode. The sensitivity of the fidelity to perturbations and uncertainties in the control fields and initial modes is also examined. The experimental feasibility of the proposal and the possibility to replace the continuous control field with square wave pulses is finally discussed.

[1]  R. Barnett.  Edge-state instabilities of bosons in a topological band , 2013, 1308.1972.

[2]  Sen Kuang,et al.  Lyapunov control methods of closed quantum systems , 2008, Autom..

[3]  A. R. Akhmerov.  Topological quantum computation away from the ground state using Majorana fermions , 2010 .

[4]  A. Kitaev.  Unpaired Majorana fermions in quantum wires , 2000, cond-mat/0010440.

[5]  Parsa Bonderson.  Measurement-only topological quantum computation via tunable interactions , 2012, 1210.7929.

[6]  S. Sarma,et al.  Tunneling of anyonic Majorana excitations in topological superconductors , 2010, 1006.0452.

[7]  Alexei Kitaev,et al.  Anyons in an exactly solved model and beyond , 2005, cond-mat/0506438.

[8]  E. Bach.  Discrete Logarithms and Factoring , 1984 .

[9]  G. Refael,et al.  Helical liquids and Majorana bound states in quantum wires. , 2010, Physical review letters.

[10]  C. Kane,et al.  Topological Insulators , 2019, Electromagnetic Anisotropy and Bianisotropy.

[11]  L. Fu,et al.  Superconducting proximity effect and majorana fermions at the surface of a topological insulator. , 2007, Physical review letters.

[12]  X. Qi,et al.  Topological insulators and superconductors , 2010, 1008.2026.

[13]  A lattice of double wells for manipulating pairs of cold atoms. , 2006, cond-mat/0602103.

[14]  E. Bakkers,et al.  Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices , 2012, Science.

[15]  S. Sarma,et al.  Realizing a robust practical Majorana chain in a quantum-dot-superconductor linear array , 2011, Nature Communications.

[16]  Chunfeng Wu,et al.  Driving quantum systems into decoherence-free subspaces by Lyapunov control , 2009, 0908.1048.

[17]  N. Read,et al.  Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect , 1999, cond-mat/9906453.

[18]  Lukas Furst,et al.  Superconductivity Of Metals And Alloys , 2016 .

[19]  Xiaoting Wang,et al.  Analysis of Lyapunov Method for Control of Quantum States , 2010, IEEE Transactions on Automatic Control.

[20]  Matthew P. A. Fisher,et al.  Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure , 2013, 1307.4403.

[21]  G. Refael,et al.  Non-Abelian statistics and topological quantum information processing in 1D wire networks , 2010, 1006.4395.

[22]  P. Zoller,et al.  Preparing and probing atomic Majorana fermions and topological order in optical lattices , 2012 .

[23]  Xiaoting Wang,et al.  Entanglement generation between distant atoms by Lyapunov control , 2009, 0906.1830.

[24]  J. Sebby-Strabley.  Lattice of double wells for manipulating pairs of cold atoms (9 pages) , 2006 .

[25]  Jean-Paul Blaizot,et al.  Quantum Theory of Finite Systems , 1985 .

[26]  S. Raghu,et al.  Majorana zero modes in a quantum Ising chain with longer-ranged interactions , 2011, 1110.3072.

[27]  Mazyar Mirrahimi,et al.  Implicit Lyapunov control of finite dimensional Schrödinger equations , 2007, Syst. Control. Lett..

[28]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[29]  Jean-Michel Coron,et al.  Quantum control design by Lyapunov trajectory tracking for dipole and polarizability coupling , 2009 .

[30]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[31]  D. D’Alessandro.  Introduction to Quantum Control and Dynamics , 2007 .

[32]  Yize Jin,et al.  Topological insulators , 2014, Topology in Condensed Matter.

[33]  S. Das Sarma,et al.  Majorana fermions and a topological phase transition in semiconductor-superconductor heterostructures. , 2010, Physical review letters.

[34]  E. Grosfeld,et al.  Unpaired Majorana fermions in a layered topological superconductor , 2011, 1102.0360.

[35]  Y. Oreg,et al.  Zero-bias peaks and splitting in an Al–InAs nanowire topological superconductor as a signature of Majorana fermions , 2012, Nature Physics.

[36]  Zak,et al.  Berry's phase for energy bands in solids. , 1989, Physical review letters.

[37]  Zhao Cheng-da,et al.  Solitons on conducting polymers , 2005 .

[38]  S. Simon,et al.  Non-Abelian Anyons and Topological Quantum Computation , 2007, 0707.1889.

[39]  E. M. Stoudenmire,et al.  Interaction effects in topological superconducting wires supporting Majorana fermions , 2011, 1104.5493.

[40]  Liang Jiang,et al.  Majorana fermions in equilibrium and in driven cold-atom quantum wires. , 2011, Physical review letters.

[41]  Ulrich Hohenester,et al.  Twin-atom beams , 2010, 1012.2348.

[42]  Jason Alicea,et al.  New directions in the pursuit of Majorana fermions in solid state systems , 2012, Reports on progress in physics. Physical Society.

[43]  S Das Sarma,et al.  Generic new platform for topological quantum computation using semiconductor heterostructures. , 2009, Physical review letters.

[44]  M. Freedman,et al.  Measurement-only topological quantum computation. , 2008, Physical review letters.