Manipulation of Majorana fermions via single charge control Karsten Flensberg Niels Bohr Institute University of Copenhagen Superconducting hybrids: from conventional to exotic, Villard de Lans, France, Sept 7-10, 2011.
Outline 1D topological superconductor and Majorana bound states Tunnel characteristic of chain of Majorana bound states Spectroscopy with a Coulomb island coupled to Majorana state Non-abelian manipulation via single electron control Transfer between topological and spin qubit systems
Topological superconductor Niels Bohr Institute Sau et al. PRL 104, 040502 (2010) Alicea, PRB 81, 125318 (2010) Oreg, Refael,von Oppen, PRL 105, 177002 (2010) Lutchyn, Sau, das Sarma, PRL 105, 077001 (2010) Semiconductor with strong S0 S-wave superconductor Can now couple to s-wave superconductor
Triplet superconductor by Zeeman & SO Niels Bohr Institute Semiconductor with strong S0 S-wave superconductor Pairing in semiconductor induced by proximity effect: Effective Hamiltonian for the lowest mode µ=0: Spin up/down Majorana modes
Majorana end bound states Niels Bohr Institute Normal lead Topological superconductor V : Spin up in y : Spin down in y Tunneling Hamiltonian: Projection:
Tunneling into disordered chain of Majorana bound states Niels Bohr Institute Flensberg, PRB (RC), 180516 (2010) (Disorder correlation length > Majorana localization length)
General current formula formula PRB, 180516R (2010)
Short chain Law, Lee, Ng, PRL 103, 237001 (2009). Andreev channel: resonant Electrons MBS Holes µ+ µ µ
Long random chain Example with weak link:
What is a weak link? Remember: Width of resonances due to the chain after weak link: Note visible if:
Spectroscopy of Majorana bound states using quantum dot Niels Bohr Institute Leijnse, Flensberg, PRB (RC), in press. arxiv:1012.4650 Tunneling limit Solve QD-TS exactly and use master equation - include relaxation of parity
Eigenstates of QD-TS Even subspace, 4 states: Odd subspace, 4 states: Mixed by tunneling Solve rate equations!
Coulomb blockade diamond Bias voltage Zero bias peak ev MBS Note: Blue=NDR
Finite coupling between Majorana states inelastic cotunnling - relaxation Non-eq. Cotun. 0 ev
Non-abelian manipulations using single charge control By changing the charge on a dot by one electron: Compare to braiding:
Manipulation of the Majorana system by single electron addition/removal Niels Bohr Institute Flensberg, PRL 106, 090503 (2011) Projection to zero mode and one spin direction in dot: M-fermion: Basis states: Total even: Total odd: Protected
More Majorana states and more dots Allows a finite number of operations of the form:
Coupling to two Majorana bound states Full lines: ε=0 Dashed lines: ε=2v 1 M1, M2 and D1: But now even/odd not degenerated (degenerated only if v 1 and v 2 are real)
In degeneracy point: =0 ε Requires: -Constant tunneling amplitudes -Constant flux No dependence on timing
Compare to braiding M1 M2 With Ivanov, PRL 2001 Tunnel braid can mimic real space braiding
General v 1 and v 2 Two-level system Rotation around a line in the x-y plane: Rotation around the z-axis: (Braiding is restricted to π/2 rotations)
Demonstration of non-abelian operations Two fermions: Basis: Initialize: To initialize and read out: detune away from degeneracy point
Coupled spin qubits and Majorana qubits Martin Leijnse, Flensberg, arxiv:1107.5703 Goal: transfer quantum information from spin qubits to topological qubits, without being prone to charge noise
Filling or emptying the dot: entangles Sweep from empty to full dot: Sweep from full to empty dot: :normalization factor Possibilities: - Transferring between spin & topological qubits - Generating long-distance entanglement Read more: arxiv:1107.5703
Summary -Majorana bound states give clear spectroscopic features, both with an without a quantum dot -A Coulomb blockade setup is also sensitive to the parit relaxation -Single charge control allows one-qubit rotations -Non-Abelian manipulation using quantum dots coupled to two Majorana bound states -Spin selective tunneling allows transfer of quantum information between spin and topological qubits, without charge coupling.