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Unconventional Josephson junction arrays for qubit devices.

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

We are here Gran Sasso range (2914 m/9000 ft) and L’Aquila

C4 C3 C5 C6

  J 0   0 0   0 0 

C2 CB

0 0 

(a)

CF C8

= J J (b) 0   0 0   0 (c) Superconductivity Group Applied Physics Division Dipartimento di Energetica L’Aquila

1D open unconventional arrays Building block: the two-junction loop g conventional loop for small b ( use g   sin 1   b 2 cos   Eq) dia, g (0)=0  -loop for small b g   sin   2  

  1  b 2 cos    2  

   g + para, g dia, moreover there are spontaneous currents for

going to zero, i.e., g + (0)=1 and g (0)=-1 001 103 g + g -

1D open unconventional arrays 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     0 Quest: what is the fundamental state in zero field ?

0  Chain of ½ Flux quanta or Semi-fluxons (SF) +SF  SF + SF  SF +SF  SF

1D open unconventional arrays Quest: what is the effect of the magnetic field ?

screening current adds  0 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…

Total energy

Total energy is the sum of Josephson and magnetic energy

 

j I

 0 2  0

( 

, 

 + 

1 2

( 

 

ext

2  We can write

( 

, 

  1 2 

2  2

+ 1  (  1 )

(

) cos 

Moreover, using flux quantization, Magnetic energy is written 2 

0  0

E M

 1 2 b



( 

 2 

f j

 2  1 2 b



( 

 

 1 + 2 

n j

 2 

f j

2  Where b = 2  I 0 L /  0 . With  j =  j  j-1 +2 

j we obtain 2 

0  0

 

(

1  cos 



+ 1 2 b   

2 + 2  2

b 2  2    b

The winding number

The quantum number because the variations of the phases are small if Large. On the other hand, in an annular array the last loop

n N =n n j

is typically zero for open arrays i.e., the number of flux quanta into the annulus. b is not play the role of winding number of the phase,

1D open unconventional arrays Q: How we find phases  i ?

A: Solving Discrete Sine-Gordon equation (DSG)  

+   

+ ( 

sin 

 1 b ( 

+ 1  2 

+ 

 1  + 2 b  (

f i

+ 

f i

  With  N+2 =  0 =0, i + =i,i =i-1,

N+1 =

0 =0 We assume

constant, i.e.,

i =

, moreover   2  b 1 2

(see E. Goldobin et al., Phys. Rev. B66, 100508, 2002; J. R. Kirtley et al., Phys. Rev. B56, 886, 1997)

1D open unconventional arrays G. Rotoli PRB68, 052505, 2003 b x 2 / l J 2 t ( 1 m m) 2 /(5 m m) 2 =0.04

Grain size Josephson length  d 0  junction (equal length) (a) diamagnetic sol (b) paramagnetic sol N=63, b =0.04

Mean magnetization for different GBLJJs: symmetric 0  => circles

Previous work on 1D open unconventional arrays G. Rotoli PRB68, 052505, 2003 N=255, b =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 FC can be introduced assuming that it flips some SF from dia to para state (c) (c) (b) (a) (b)

Other papers in unconv. arrays and junctions 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

1D open unconventional arrays Some estimate of demag field:  d

H d

  2 l

0 l



001 l J

H d (a) =7.6 mG H d (b) =36 mG H d (c) =80 mG

we use l L = l c-axis equal to 5 m m l L 103 Note that in (a) fields are of the same order of magnitude cited in Tafuri and Kirtley ( l c-axis =5.9 m m)

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    0 0  0  1 qubit 0  0  0 0   0 2 qubit 0  0 0   0

C4 C5



0 

0 0 



Annular arrays A practical layout

CB C1 CF

N = 8 array, with CF (control field) CB (control barrier) CN (control loop N)

0-



Annular JJA DSG

Q: How we find phases  i ?

A: Solving Discrete Sine-Gordon equation (DSG) for the ring  

+   

+ ( 

sin 

 1 b ( 

+ 1  2 

+ 

 1  + 2 b  (

f i

+ 

f i

  With  N+1 =  1 +2 

is the winding number i + =i,i =i-1 A

constant do no longer apply,

have to be not uniform to have effect on a 0  AJJA 

 2  b 1 2

f i

Fundamental states in AJJA

(a) 0.04

0.02

0.00

-0.02

-0.04

0 2 4 6 loop number j 8 0.0

0.5

b 1.0

(b) 1.0

0.9

0.8

0.1

0.0

1.5



Spin notation



2.5

2.0

1.5

1.0

0.5

1 2.5

2 3 4 5 6 2.0

1.5

1.0

0.5

1 2.5

2 3 4 5 6 2.0

1.5

1.0

0.5

1 2 3 4 5 junction 6 0.3

0.2

0.1

0.0

-0.1

-0.2

7 -0.3

1 0.3

0.2

0.1

0.0

-0.1

-0.2

7 -0.3

1 0.3

0.2

0.1

0.0

-0.1

-0.2

7 -0.3

1 2 2 2 3 3 4 4 5 6 5 6 3 4 loop 5 6 7 7 7

AJJA arrays (excited states)

N = 2 & 4 

 0 1000 

 1 

 0 

 0 

 1 

 2 (a) 0.4

FF 0.2

100 n=1 0.0

-8 -4 log 2 b 0 N = 6 

 0 

 0 

 0 

 1 

 2 

 3 (b) FF n=2 10 N = 2 N = 4 AF n=0 n=1 0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

1 2 3 0.4

0.2

0.0

-0.2

-0.4

1 4 3 2 1 0 -1 1 2 2 3 4 loop 5 3 4 junction 5 4 5 6 6

n = 0

0.3

0.2

0.1

0.0

-0.1

-0.2

-0.3

6 7 1

n = 1

0.4

2 3 4 5 6 0.2

0.0

-0.2

7 -0.4

1 7 12 10 8 6 4 2 0 1 2 2 3 4 loop 5 6 7 3 4 junction 5 6 7 7 1 AF n=0 -8 -6 -4 -2 log 2 b 0 2 -8 -6 -4 -2 log 2 b 0 2

-0.2

0.0

0.2

-0.2

0.0

0.2

AJJA (excited states) (2)

0.4

   0.4

K-AK states 10 (a) 1.2

8 0.2

0.8

6 0.0

0.4

4 -0.2

0.0

2 -0.4

0 0.4

4 8 12 16 -0.4

0 0.4

4 8 12 16 0 0 10 (c) 4 8 8 12 16 -0.4

0 1.2

0.2

0.8

6 0.0

0.4

4 -0.2

0.0

2 -0.4

0 0.6

0.4

0.2

0.0

-0.2

-0.4

4 8 12 16 -0.4

0 0.4

0.2

0.0

-0.2

-0.6

0 4 8 12 16 -0.4

0 loop    4 16 8 32 loop 12 48 16 64 (b) (d) 4 8 12 16 0 0 16 32 junction 48 64 -0.4

0 16 48 64 32 loop 



 large b 



 b Fractionalization phenomenon

0 –



Annular long junction

E. Goldobin et al. PRB66, 100508, 2002 E. Goldobin et al. PRB67, 224515, 2003 E. Goldobin et al. cond-mat/0404091 (ring) 0 c)  d) s-type 0  0  m-type 0  3 2 1 0 0 3 2 1 0 0 3 2 1 0 0 30 60 30 30 60 60 junction 0.12

0.08

0.04

0.00

-0.04

-0.08

90 -0.12

0 0.12

0.08

0.04

0.00

-0.04

-0.08

90 -0.12

0 0.12

0.08

0.04

0.00

-0.04

-0.08

90 -0.12

0 30 60 90 30 60 90 30 loop 60 90  0 Fund. state c-type 0 0   

0 

N/k

boundaries sections

(a)

LJJ case 0-



JJ

3.0

2.4

1.8

n = 1 1.2

n = 0 0.6

AF FF 0.4

0.3

0.2

0.1

0.0

-8 -6 -4 -2 0 log 2 b 2 N = 2 N = 4 3.0

2.4

n = 2 1.8

n = 1 1.2

n = 0 AF 0.6

FF n = 1 (unstable) 0.0

0.0

-8 -6 -4 -2 Log 2 b 0 2 -8 -6 -4 -2 Log 2 b 0 2

2 (nor. length of sections)

1 K = 2,4 N=32,64 k=6 N=96 0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

0 0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

0 1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

128 32 32 b = 5.96

b = 4.00

b = 2.88

(a) (c) 144 loop 64 (e) 0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

96 0 0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15

96 0 32 32 (b) 64 64 (d) 1.2

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

160 b = 1.97

b = 0.77

b = 0.25

(f) 112 128 144 loop 160 176 96 96

Annular arrays in magnetic field I

0.08

0.04

0.00

-0.04

-0.08

-0.12

0 2 4 6 8 loop 10 12 14 16 0.10

0.05

0.00

-0.05

-0.10

0 2 4 6 8 loop 10 12 14 16 Single loop (Cn) frustation on an N=16 array Frustation over loops 10-16 On an N=16 array

Annular arrays in magnetic field II

Critical field for flip between fund. states 4

1.8

3 N=8 dia N=8 par N=32 dia N=32 par 1.6

N=8 dia N=8 par N=32 dia N=32 par 1.4

2 (a) 1.2

(b) 1 1.0

-3 -2 -1 0 1 2 3 -3 -2 -1  Frustation applied via CF is independent of N and induce a flip between para-dia sol. at  =2.1

0  1 2 3 Effect of frustation applied via a single loop, say C1, decrease with N

Magnetic behavior of annular 0-



LJJ

0.02

0.01

0.00

-0.01

-0.02

-0.03

0 0.02

0.01

0.00

-0.01

-0.02

-0.03

0 0.10

0.05

0.00

-0.05

-0.10

-0.15

0 50 (b) (c) 100 150 200 250 50 100 150 200 250 (a) 5 10 15 loop 20 25 30 1.5

1.0

0.5

-3 -2 -1 0  1 2 3 The effect of field in LJJ case is very similar

Magnetic behavior for different spatial configuration

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

Annular arrays: flip dynamics

N = 16 array via C1 N=256,

=16 array via s-type control

Classically it is possible to flip an half-flux quantum adding it a full flux quantum (fluxon) E. Goldobin et al. cond-mat/0404091 2 1 4 3 0 -1 -2 -3 -4 0 motion direction 16 32 48 64 X 80 96 112 128 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

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.



-Junction realization

There are essentially three way to fabricate  -junctions: 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 W ) control of currents and capacity not so easy dIs SFS used by Hilgenkamp et al. in “zigzag” arrays, are YBCO-Nb ramp edge junctions dissipation are intermediate (20 W ), control on other parameters is good 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

Conclusion

1) Annular unconventional arrays and their LJJ counterpart the annular 0 properties  junction are very interesting physical object condensing the properties of half-flux quantum arrays and annular junction together with some energy and topological protection 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 Part of results shown here will be submitted to ASC04 conference, Jacksonville, FL USA 3-8 october 2004 session 3EI01

Acknowledgements 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”.

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