1. D.Giuliano (Cosenza), P. Sodano (Perugia) Local Pairing of Cooper pairs in Josephson junction networks Obergurgl, June 2010

2. Plan of the talk: 1. Josephson junction network interferometer as a model of a boundary double Sine-Gordon Hamiltonian; 2. Boundary interaction periodicity and coherent tunneling of pairs of Cooper pairs; 3.Probing the effective tunneling charge via dc Josephson effect; 5. Conclusions, possible applications, perspectives. 4. Phase diagram and dual interaction;

3. 1 The network; A circular Josephson junction array, pierced by a magnetic flux φ, connected to two 1-d JJ “leads. Charging energy of SC grains Josephson energy

4. Effective spin-1/2 Hamiltonian Setting φ≈π-> near by degenerate eigenstates of Hc Projection onto low energy subspace

5. We have singled out an effective spin ½ degree of freedom, controlled with (at least one) tunable parameter How to either set up, or probe, or even further control) the state of S G ? Connect it to two one-dimensional JJ arrays working as leads

6. The leads: Effective field theory of a JJ-chain (L. I. Glazman and A. I. Larkin, PRL 79, 3736 (1997); D.G., P. Sodano, NPB 711, 480 (2005)) Mapping onto spin chain+Jordan-Wigner fermions+Bosonization  Luttinger liquid (LL) effective Hamiltonian (N=n+1/2)

7. LL parameters Connection between the central region and the leads Summing over the central region states->boundary degrees of freedom interacting with a localized spin-1/2

8. 2. Boundary interaction periodicity and coherent tunneling of pairs; B z measures the “detuning” of the degeneracy point due to a displacement from φ=π: B z =Jsin[(φ-π)/4] B x measures the detuning due to an applied gate voltage: B x ≈(λ 2 /J)(V g -N-1/2) The control parameters: Using B z to “tune” the effective charge tunneling across the device: (See below for technical details)

9. In this case, a simplified model may be used for performing calculations Charge difference operator between the two leads An harmonics of Φ(0) of period 2π/a varies the relative charge by ae *, that is, it lets a total charge ae * tunnel across the central region H B allows for direct tunneling of single Cooper pairs (charge e * ), as well as of pairs of Cooper pairs (charge 2e * )

10. It is the only term that survives when cos(θ)=0 Discrete symmetry B=0->enhanced (τ 1 ) discrete symmetry Usually, charge 2e* tunneling is a higher-order process and is neglected, BUT …

11. Technicalities: 1.Introduce two pairs of Dirac fermions a,a + ;b,b +, and represent the effective spin operators as S z = a + a-b + b; S x =a + b+b + a 2.Use the following (euclidean) action for the fermion operators (β=1/k B T): 3.Integrate over the fermion fields according to the recipe (Λ=√(B z 2 +B x 2 ))

12. 3.Probing the charge tunneling across the central region via a dc Josephson current measurement; Inducing a dc Josephson current-> Connecting the outer boundaries to two bulk superconductors at fixed phase difference α->Dirichlet-like boundary conditions at the outer boundary (x=L) (that is, the plasmon phase field has to smoothly adapt to the phase difference between the bulk superconducting leads) Dynamical boundary conditions at x=0

13. Both boundary conditions may be accounted at weak coupling at x=0 (i.e., g1 ≈g2 ≈0), by taking Vacuum expectation values of vertex operators

14. Computing the dc Josephson current Partition function at weak coupling As B z =0

15. Josephson current for various values of B z : B z decreases counterclockwisely from the top left panel and is =0 at the top right panel

16. The two harmonics in I J correspond to tunneling of singe CPs and coherent tunneling of pairs of CPs, respectively. The ratio between the contributions of the two processes to the total current may be tuned by acting on B z, that is, on the flux φ

17. 4. Phase diagram and dual interaction; All the previous results rely on the assumption that the Josephson coupling between leads and central region λ<<E J,J How reliable is this assumption? As the size of the system (L) increases, low-energy, long wavelength collective plasmon modes of the leads may get entangled with the isolated “boundary” degrees of freedom. This may lead to a final state that is nonperturbative in H B. This happens if the boundary couplings scale slower than 1/L

18. “Running” couplings Flow equations for the running couplings

19. For g 1 ≠0, the boundary interaction is a relevant operator (and, accordingly, the perturbative approach is nor reliable), as soon as g>1. The second harmonics is nonperturbatively renormalized, as well

20. Strong limit for the boundary coupling Φ(0) is “pinned” at a minimum of the boundary potential->Dirichlet boundary condition Non τ 1 -symmetric case τ 1 -symmetric case

21. Boundary potential and instanton trajectories

23. Leading boundary interaction at the SCP “Jumps” between the minima of the boundary potential->shifs of the eigenvalues of P ->dual vertex operators

24. “Dual” boundary interaction “Short” instantons “Long” instantons Short instantons exist, as boundary excitations, as a consequence of τ 1 –symmetry. Breaking τ 1 –symmetry implies short instanton confinement on a scale that depends on B

25. When short instanton exist at any scale L, they “destabilize” the SFP. The SFP-picture is not consistent anymore and the IR behavior of the system is driven by a finite coupling FP. Short instantons static solitons in the double Sine-Gordon model Instanton trajectory -> P →P(τ) Integrating on the oscillator modes ->Effective (Euclidean) action for P(τ) ->Equation of motion in the inverted potential

26. Effective instanton action “Equation of motion” =(apart for the finite-size term proportoanal to 1/L) to the equation for static solitons in the DSG model

27. Two short instantons → one long instanton Separation between short instantons The short-instanton scaling (of μ) stops at a scale L≈uR(φ). If τ 1 -symmetry holds (i.e., short instantons are deconfined: R(φ)→∞), scaling does not stop and the system is attracted by a FFP

28. φ=π,1.01π,1.1π,2π

29. 5. Conclusions and (possible) further perspectives; a. Possibility of acting on the external control parameters of the JJN to trigger the opening of an exotic phases, corresponding to an IR attractive FFP; b. FFP corresponds to a “4-e” phase, with frustration of decoherence. At the FFP an effective, 2-level quantum system emerges in the device, with enhanced quantum coherence between the states; c. Making the experiment work !!!