Stretching the electron as far as it will go

1. Stretching the electron as far as it will go Gordon W. Semenoff University of British Columbia PITP Decoherence at the Crossroads, Vancouver, February 2006

2. This is a lecture about coherence rather than decoherence ! Use an idea of A.Kitaev cond-mat/0010440 ``Unpaired Majorana fermions in quantum wires’’ An isolated state of a Majorana fermion is insensitive to quantum fluctuations -- Isolated, localized state, suppresses transitions to other states -- Majorana fermion has real wave-function  no phase fluctuations G.W.S. and P. Sodano cond-mat/0601261, ``Teleportation by a Majorana medium’’ Interesting non-local phenomena -- Isolated state has wave-function with two spatially separated peaks

3. The idea is to arrange a quantum system where the wave-function of a particle has two peaks in different locations The state can be populated – a transition from unoccupied to occupied – by interacting with the system in the region of one of the peaks. Then, a measurement of the position of the particle has some probability of finding it in either place. Such states of the Schrödinger equation are difficult to find because of degeneracy.

4. Double-well potential V(x) + +

5. Fractional Charges in Polyacetylene H C C H Cis-polyacetylene has two degenerate ground states. Each Carbon atom has three valence electrons. Two of the valence electrons form covalent bonds with neighboring atoms. The third is less bound and would occupy a conduction band With one electron per site – i.e. ½-filled band. However, the Pierls instability – the interaction of the electron with the lattice phonon – leads to a dimerization where the extra electrons form alternating covalent bonds as depicted above. There are two possibilities for the orientation of these bonds and thus two ground states.

6. Ignore spin: removing an electron creates pair of domain walls each with charge e/2 +e +e/2 +e/2 Adding an electron creates a pair of domain walls each with charge -e/2 -e -e/2 -e/2

7. By moving some bonds, we can create a pair of domain walls with charge e/2 and -e/2 -e/2 +e/2

8. 1 2 +e/2 +e/2 hole -e/2 -e/2 electron R. Jackiw, A.K. Kerman, Igor R. Klebanov, G.W. Semenoff Nucl.Phys.B225:233,1983 R. Jackiw, C. Rebbi, J.R. Schrieffer cond-mat/0012370

9. P-wave superconductor Consider a simple model of a quantum wire embedded in a p-wave superconductor n=1 n=L Describe in the tight-binding approximation by the Hamiltonian (for spinless electrons) With electron creation and annihilation operators Assume that the system is in some quantum state At an initial time, we insert an electron at n=1. This creates the state What is the quantum probability amplitude that it appears at n=L, that is in the state , after time T has elapsed?

10. Bogoliubov de-Gennes equation: E(k) Has continuum solutions with k

11. And a pair of (Andreev) bound states with E

12. Second quantization of the electron The electronic ground state is 2-fold degenerate. The ground states have different fermion parity Spin superselection rules:

14. If we insert an electron at position n=0 and ask for the amplitude for it to appear at n=L after a time T has elapsed,

15. conclusions • Teleportation or EPR? • Relativistic limitations? • A more thorough analysis of the inclusion of spin is ongoing. • Other systems like vortices, edge states in p-wave superconductors • Decoherence?