Splitting Kramers degeneracy with superconducting phase difference

Transcription

1 Splitting Kramers degeneracy with superconducting phase difference Bernard van Heck, Shuo Mi (Leiden), Anton Akhmerov (Delft) arxiv: ESI, Vienna, 11 September 2014

2 Plan Using phase difference in a Josephson junction as a means of breaking time reversal symmetry. What does breaking time reversal mean? Why it won t work. How to make it work (and why 3 is much better than 2)?

3 Time reversal breaking in a mesoscopic JJ Several manifestations: Splitting of Kramer s degeneracy (Chtchelkatchev&Nazarov, Béri&Bardarson&Beenakker) Closing of the induced gap Protected zero energy level crossings (switches in the ground state fermion parity) P = Pf(iH) Spectral peak in the DOS (Ivanov, Altland&Bagrets) ( ρ(e) = ρ sin(2πe/δ) ) 2πE/δ

4 Setup and formalism Scattering matrices of electrons and holes: S h ( E) = S e (E) Andreev reflection matrix: r A = ie iφ i Bound state condition: S e (E)r A S h (E)r A ψ = e 2i arccos(e/ ) ψ

5 Setup and formalism Scattering matrices of electrons and holes: S h ( E) = S e (E) Andreev reflection matrix: r A = ie iφ i Bound state condition: S(E)r A S ( E)r A ψ = e 2i arccos(e/ ) ψ

6 Short junction limit S(E) S(0) Lowest density of Andreev states, strongest effect phase difference on a single state. Due to unitarity and time reversal symmetry of S the energies are given by E n = ± 1 T n sin 2 (φ/2) (Beenakker)

7 Why it won t work Splitting of Kramers degeneracy Closing of the gap Protected zero energy level crossings (switches in the ground state fermion parity) Spectral peak in the DOS

8 Why it won t work Splitting of Kramers degeneracy :-( δe E 2 /E T < 2 /E T Closing of the gap Protected zero energy level crossings (switches in the ground state fermion parity) Spectral peak in the DOS

9 Why it won t work Splitting of Kramers degeneracy :-( δe E 2 /E T < 2 /E T Closing of the gap :-( E cos(φ/2) DOS 0 1 E/ Protected zero energy level crossings (switches in the ground state fermion parity) Spectral peak in the DOS

10 Why it won t work Splitting of Kramers degeneracy :-( δe E 2 /E T < 2 /E T Closing of the gap :-( E cos(φ/2) DOS 0 1 E/ Protected zero energy level crossings (switches in the ground state fermion parity) :-( Spectral peak in the DOS

11 Why it won t work Splitting of Kramers degeneracy :-( δe E 2 /E T < 2 /E T Closing of the gap :-( E cos(φ/2) DOS 0 1 E/ Protected zero energy level crossings (switches in the ground state fermion parity) :-( Spectral peak in the DOS :-(

12 A big improvement All the special properties of the spectrum originate from the small number of leads!

13 Kramers degeneracy splitting Take a Rashba quantum dot with E E SO, R l SO, and λ R

14 Kramers degeneracy splitting Take a Rashba quantum dot with E E SO, R l SO, and λ R Calculate the splitting between the lowest two Andreev levels

15 Kramers degeneracy splitting Take a Rashba quantum dot with E E SO, R l SO, and λ R Calculate the splitting between the lowest two Andreev levels 2π 0.2 φ 2 π δe/ 0 π 2π φ 1 Kramers degeneracy is strongly broken. 0

16 Protected level crossings Once again, try a random quantum dot: 2π 1 φ 2 π E min 0 π 2π φ 1 0

17 Protected level crossings Once again, try a random quantum dot: 2π 1 φ 2 π E min Level crossings are allowed. 0 π 2π φ 1 0

18 Protected level crossings Are level crossings allowed for any (φ 1, φ 2 )?

19 Protected level crossings Are level crossings allowed for any (φ 1, φ 2 )? 2π 0.2 φ 2 π P odd 0 π 2π φ 1 0 No: the gap may only close when all the clockwise phase differences are smaller (or larger) than π. (Note that this result holds for any junction)

20 Proof 1. The expression for Andreev spectrum: Sr A S ra ψ = ω2 ψ, E = Im ω

21 Proof 1. The expression for Andreev spectrum: Sr A S ra ψ = ω2 ψ, E = Im ω 2. The simplified expression for Andreev spectrum: (Sr A r A S T )ψ = 2e iα E ψ

22 Proof 1. The expression for Andreev spectrum: Sr A S ra ψ = ω2 ψ, E = Im ω 2. The simplified expression for Andreev spectrum: (Sr A r A S T )ψ = 2e iα E ψ 3. S = S T due to time reversal.

23 Proof 1. The expression for Andreev spectrum: Sr A S ra ψ = ω2 ψ, E = Im ω 2. The simplified expression for Andreev spectrum: (Sr A r A S T )ψ = 2e iα E ψ 3. S = S T due to time reversal. 4. This means Sψ ψ, S(r A ψ) = 2 E eiα ψ (r Aψ )

24 Proof 1. The expression for Andreev spectrum: Sr A S ra ψ = ω2 ψ, E = Im ω 2. The simplified expression for Andreev spectrum: (Sr A r A S T )ψ = 2e iα E ψ 3. S = S T due to time reversal. 4. This means Sψ ψ, S(r A ψ) = 2 E eiα ψ (r Aψ ) 5. The necessary and sufficient condition for existence of a unitary S: ψ, ψ : ψ r A ψ + ψ r A ψ = 2 E eiχ ψ ψ.

25 Proof II 1. We get: E 1 2 ψ r A ψ + ψ r A ψ.

26 Proof II 1. We get: 2. Graphical solution: E 1 2 ψ r A ψ + ψ r A ψ. 3. The lower bound on the gap: E min i,j cos φ i φ j 2

27 Gap closing and the spectral peak Both phenomena are visible In the ensemble (averaging over random antisymmetric S) RMT ρ(ɛ) ɛ/

28 Gap closing and the spectral peak Both phenomena are visible In the ensemble (averaging over random antisymmetric S) In a single realization (averaging over chemical potential) RMT 20 Rashba dot, l so/r = 0.2, l/r = ρ(ɛ) ɛ ρ(ɛ) [a.u.] ɛ/ 0 0 π 2π φ 2 2π φ 1

29 Conclusions Superconducting phase difference can strongly break time reversal symmetry in a Josephson junction. This requires more than two superconducting leads. Spin degeneracy is split by a large fraction of. The induced superconducting gap only closes in a a finite subregion of the phase space.

30 Conclusions Thank you all. The end.