1. Jacksonville, October 5 2004 Giacomo Rotoli Superconductivity Group & INFM Coherentia Dipartimento di Energetica, Università di L’Aquila ITALY Unconventional Josephson junction arrays for qubit devices. Collaborations: F. Tafuri, Napoli II A. Tagliacozzo, A. Naddeo, P. Lucignano, I. Borriello, Napoli I

2. Gran Sasso range (2914 m/9000 ft) and L’Aquila Superconductivity Group Applied Physics Division Dipartimento di Energetica L’Aquila We are here

3. 1D open unconventional arrays Building block: the two-junction loop conventional loop for small  use  Eq)  -loop for small   dia,  (0)=0  + para,  - dia, moreover there are spontaneous currents for f going to zero, i.e.,  + (0)=1 and  - (0)=-1 ++++ ---- 001103

4. Model: 1D GB Long Josephson Junction with presence of  -sections alternanting with conventional sections. This is equivalent to have localized  -loops in a 1D array Quest: what is the fundamental state in zero field ? 0   Chain of ½ Flux quanta or Semi-fluxons (SF) +SF  SF  SF  SF +SF  0 1D open unconventional arrays

5. 0  Quest: what is the effect of the magnetic field ?  0 Two solutions are no longer degenerate! Red ones is paramagnetic and have a lower energy with respect to Blue ones which is diamagnetic and with higher energy… screening current adds 1D open unconventional arrays

6. Total energy Total energy is the sum of Josephson and magnetic energy We can write Moreover, using flux quantization, Magnetic energy is written Where  = 2  I 0 L/  0. With  j =  j -  j-1 +2  n j we obtain

7. The quantum number n j is typically zero for open arrays because the variations of the phases are small if  is not Large. On the other hand, in an annular array the last loop n N =n play the role of winding number of the phase, i.e., the number of flux quanta into the annulus. The winding number

8. Q: How we find phases  i ? A: Solving Discrete Sine-Gordon equation (DSG) We assume f constant, i.e., f i =f, moreover With  N+2 =  0 =0, i + =i,i - =i-1, f N+1 = f 0 =0 (see E. Goldobin et al., Phys. Rev. B66, 100508, 2002; J. R. Kirtley et al., Phys. Rev. B56, 886, 1997) 1D open unconventional arrays

9. 0-  junction (equal length) (a)diamagnetic sol (b)paramagnetic sol N=63,  =0.04 Mean magnetization for different GBLJJs: symmetric 0-  => circles  x 2 / J 2 t ( 1  m) 2 /(5  m) 2 =0.04 Grain size Josephson length dd G. Rotoli PRB68, 052505, 2003 1D open unconventional arrays

10. N=255,  =0.04 with 15  -loops (a)7 dia + 8 para (b)5 dia + 10 para (c)3 dia + 12 para (b) and (c) corresponds to a pre-selection of paramagnetic solutions due to FC (c) (b) (a) (b) (a) (c) FC can be introduced assuming that it flips some SF from dia to para state G. Rotoli PRB68, 052505, 2003 Previous work on 1D open unconventional arrays

11. F. Tafuri and J. R. Kirtley, Phys. Rev. B62, 13934, 2000; Tilt-Twist 45 degree YBCO GB junctions sample diamagnetic with ½ half flux quanta pinned to defects and along GB, paramagnetism only local F. Lombardi et al., Phys. Rev. Lett. in print, 2002; Tilt-Twist GB junctions with angles betw 0 and 90 rich structure of spontaneous currents for 0/90 GB Il’ichev et al., to be subm. Phys. Rev. B, 2002; First paramagnetic signal recorded, very flat GB form 45 deg asymmetric twist junctions, no spontaneous currents have been experimentally observed H. J. H. Smilde et al., Phys. Rev. Lett. 88, 057004, 2002; Artificial “zig-zag” LTC-HTC arrays Other papers in unconv. arrays and junctions

12. Some estimate of demag field:  d H d (a) =7.6 mG H d (b) =36 mG H d (c) =80 mG we use L = c-axis equal to 5  m Note that in (a) fields are of the same order of magnitude cited in Tafuri and Kirtley ( c-axis =5.9  m) 001103 J L 1D open unconventional arrays

13. 0-  Annular JJ arrays 1)Have properties similar to the Annular Josephson junction So can be thinked are related to “fluxon qubit” (A. Ustinov, Nature 425, 155, 2003) 2) Will have some “protection” from external perturbation In the limit of large N (Doucout et al., PRL90, 107003, 2003) 3) Can be build using  -junctions as in Hilgenkamp et al., Nature 50, 422, 2003 Merging together these three ideas we have 1 qubit2 qubit

14. Annular arrays A practical layout N = 8 array, with CF (control field) CB (control barrier) CN (control loop N)

15. Q: How we find phases  i ? A: Solving Discrete Sine-Gordon equation (DSG) for the ring A f constant do no longer apply, f have to be not uniform to have effect on a 0-  AJJA With  N+1 =  1 +2  n, n is the winding number i + =i,i - =i-1 0-  Annular JJA DSG

16. Fundamental states in AJJA Spin notation

17. AJJA arrays (excited states) N = 2 & 4N = 6 n = 0 n = 1

18. AJJA (excited states) (2) Fractionalization phenomenon K-AK states large  small 

19. 0 –  Annular long junction Fund. state E. Goldobin et al. PRB66, 100508, 2002 E. Goldobin et al. PRB67, 224515, 2003 E. Goldobin et al. cond-mat/0404091 (ring) k 0 -  boundaries N/k sections

20. LJJ case 0-  JJ K = 2,4 N=32,64k=6N=96 l / k=1 l / k=2 (nor. length of sections)

21. Annular arrays in magnetic field I Single loop (Cn) frustation on an N=16 array Frustation over loops 10-16 On an N=16 array

22. Annular arrays in magnetic field II Frustation applied via CF is independent of N and induce a flip between para-dia sol. at  =2.1 Critical field for flip between fund. states Effect of frustation applied via a single loop, say C1, decrease with N

23. Magnetic behavior of annular 0-  LJJ The effect of field in LJJ case is very similar

24. Variation of fundamental state energy for different values of  and Magnetic field In the N=16 and N=64 AJJA Top: magnetic field in a single loop Bottom: magnetic field over 7 loops Magnetic behavior for different spatial configuration

25. Annular arrays: flip dynamics N = 16 array via C1 N=256, k =16 array via s-type control

26. The process (classical) Classically it is possible to flip an half-flux quantum adding it a full flux quantum (fluxon) E. Goldobin et al. cond-mat/0404091 Successive time plot of annihilation of a fluxon on a 0-  boundary where a positive half-flux quantum was localized. Annihilation ends in a negative half flux quantum + radiation motion direction

27. The process (quantum) Calculation for quantum process in collaboration With A. Tagliacozzo, A. Naddeo and I. Borriello (Napoli I) is in progress… The flip process is approximated summing up the analytical expression for fluxon (kink) and a localized half-flux quantum with kink velocity As free parameter to be used in a variational approach. Next step is the calculation of euclidean action for the flip, its minimization will give the result.

28.  -Junction realization There are essentially three way to fabricate  -junctions: dId dId YBCO made have the best performances in dissipation and recently show also MQT effect (collaboration Napoli II, F. Tafuri + Chalmers, T. Cleason) dissipation are good (100  ) control of currents and capacity not so easy dIs dIs used by Hilgenkamp et al. in “zigzag” arrays, are YBCO-Nb ramp edge junctions dissipation are intermediate (20  ), control on other parameters is good SFS SFS these are Nb-(Ni-Cu)-Nb junctions which show a phase shift depending on F barrier thickness dissipation is high at moment, critical currents and capacitance can be controlled in a fine manner

29. Conclusion Part of results shown here will be submitted to ASC04 conference, Jacksonville, FL USA 3-8 october 2004 session 3EI01 1)Annular unconventional arrays and their LJJ counterpart the annular 0-  junction are very interesting physical object condensing the properties of half-flux quantum arrays and annular junction together with some energy and topological protection properties 2)It is conceivable to think to a protected qubit made of unconventional arrays, which will be the simplest topologically not trivial system showing the above properties and realizable with present tecnology (conventional ring array was realized for study breather solutions, see PRE 66, 016603, 2002) 3)A quantum description of flip process between half-flux quantum is in progress

30. We would like to thank F.Tafuri, A. Tagliacozzo, I. Borriello, A. Naddeo for helpful discussions and suggestions. This work was supported by Italian MIUR under PRIN 2001 “Reti di giunzioni Josephson quantistiche: aspetti teorici e loro controparte sperimentale”. Contact: e-mail => rotoli@ing.univaq.it web => http://ing.univaq.it/energeti/research/Fisica/supgru.htm Acknowledgements