Detecting and using Majorana fermions in superconductors

Detecting and using Majorana fermions in superconductors Anton Akhmerov with Carlo Beenakker, Jan Dahlhaus, Fabian Hassler, and Michael Wimmer New J. Phys. 13, 053016 (2011) and arxiv:1105.0315 Superconductor hybrids: from conventional to exotic, Villard de Lans, 8 September 2011 Instituut- Lorentz

Outline Detecting Majorana fermions Using Majorana fermions

Outline Detecting Majorana fermions 1. Scattering theory Using Majorana fermions

Outline Detecting Majorana fermions 1. Scattering theory 2. Fractional quantization of Andreev conductance Using Majorana fermions

Outline Detecting Majorana fermions 1. Scattering theory 2. Fractional quantization of Andreev conductance Using Majorana fermions 1. Quantum computation with Majoranas by manipulation of topological charge

Majorana fermions: Basic properties 1. Majorana fermion: protected zero energy solution to a Bogoliubov-de Gennes equation: ( ) H T HT 1 ψ = 0 2. No charge, no spin, cannot be removed by local perturbations (=topological protection) 3. Requires broken time-reversal and spin-rotation symmetries 4. Predicted to occur in various superconducting systems

Majorana fermion in tunneling Tunneling: G = 2e2 (Law&Lee&Ng) h with Majorana and G = 0 without.

Majorana fermion in tunneling Tunneling: G = 2e2 (Law&Lee&Ng) h with Majorana and G = 0 without. What about trying to detect the topological superconductor, not the Majorana fermions?

Reflection from a topological superconductor Reflection matrix r has Current conservation: rr = 1 det r = 1 Particle-hole symmetry: ( ) ( ) ree r r = he ree r = he Im det r = 0 r eh r hh r he r ee

Reflection from a topological superconductor det r = 1 det(r 1) = 0 bound state at zero energy.

Topological invariant Q = sign det r

Topological invariant Q = sign det r Phase transition is accompanied by a single fully transmitted mode.

Topological invariant Q = sign det r Phase transition is accompanied by a single fully transmitted mode. Does Q alter Andreev transport?

Andreev reflection II: quantum point contact

Andreev reflection II: quantum point contact Topologically trivial superconductor: G = n 4e 2 /h 4 = 2 (for Andreev reflection) 2 (for Béri degeneracy)

Andreev reflection II: quantum point contact Topologically trivial superconductor: G = n 4e 2 /h 4 = 2 (for Andreev reflection) 2 (for Béri degeneracy) Topologically nontrivial one mode stays non-degenerate: G = (n + 1/2) 4e 2 /h.

QPC robustness Disorder: Quantization of lowest 2 modes robust against disorder.

QPC robustness Finite voltage/temperature: Transparent QPC more robust against temperature.

Conclusions part I 1. Topological invariant of an open system is Q = sign det r 2. Quantum point contact with Majorana fermion has conductance (n + 1/2)g 0 instead of ng 0 3. Conductance quantization is more robust if transmission is large

Majoranas fermions: basic properties II 1. Two Majoranas hide a single quasiparticle excitation c = γ 1 + iγ 2, c = γ 1 iγ 2 2. If γ 1 and γ 2 separated, c is at zero energy and forms a protected qubit (Kitaev, 2000) charge of c is hidden by the superconductor 3. Exchanging Majoranas brings qubits into superposition

Majorana for quantum computation Ingredients needed for a much better quantum computer (Bravyi, Kitaev): 1. Braiding 2. Measure two Majoranas: (1 2c c) = 2iγ 1 γ 2 3. Phase gate: exp(iαc c) 4. Measure four Majoranas (hardest): (1 2c 1 c 1)(1 2c 2 c 2) = 4γ 1 γ 2 γ 3 γ 4

Coupling Majoranas Three options: 1. Bring Majorana close, make wave functions overlap (Fu&Kane, Fu, Flensberg, Jiang&Kane&Preskill) 2. Put Majorana on a small capacitor, use Coulomb energy (Hassler&AA&Hou&Beenakker) 3. Do both (Sau&Tewari&Das Sarma, AA&Nilsson&Beenakker, Grosfeld&Seradjeh&Vishveshwara, Clarke&Shtengel)

Coupling Majoranas Three options: 1. Bring Majorana close, make wave functions overlap (Fu&Kane, Fu, Flensberg, Jiang&Kane&Preskill) No excitations above the ground state are allowed Easy to switch on and off 2. Put Majorana on a small capacitor, use Coulomb energy (Hassler&AA&Hou&Beenakker) Required to measure four Majoranas (single particle Hamiltonian not enough) Protected by fermion parity as long as there are no delocalized quasiparticles 3. Do both (Sau&Tewari&Das Sarma, AA&Nilsson&Beenakker, Grosfeld&Seradjeh&Vishveshwara, Clarke&Shtengel)

Fermion parity meter: Cooper pair box 1. Measure qubit splitting to readout the Majorana qubit(s)

Fermion parity meter: Cooper pair box 1. Measure qubit splitting to readout the Majorana qubit(s) 2. Control voltage to apply phase gate

Fermion parity meter: Cooper pair box 1. Measure qubit splitting to readout the Majorana qubit(s) 2. Control voltage to apply phase gate 3. When two states are degenerate, still sensitive to noise

Better Cooper pair box: a transmon 1. E J E C, E J can be tuned by magnetic flux (Koch et al., 2007)

Better Cooper pair box: a transmon 1. E J E C, E J can be tuned by magnetic flux 2. When E J E C, the splitting between states exponentially small (Koch et al., 2007)

Better Cooper pair box: a transmon 1. E J E C, E J can be tuned by magnetic flux 2. When E J E C, the splitting between states exponentially small 3. All the required measurements were implemented (without Majorana yet) (Koch et al., 2007) (Schreier et al., 2008)

Conclusions part II 1. Coherent manipulation of charge (fermion) parity of Majoranas makes a protected quantum computer 2. Transmon qubit fits very well for this manipulation (Majorana interaction is switched on/off exponentially, not algebraically as with flux qubit)

Conclusions Thank you all. The end.