Jacksonville, October 5 2004Jacksonville, October 5 2004

Giacomo RotoliGiacomo Rotoli

Superconductivity Group & INFM Coherentia Superconductivity Group & INFM Coherentia Dipartimento di Energetica, Università di L’AquilaDipartimento di Energetica, Università di L’Aquila

ITALYITALY

Unconventional Josephson junction Unconventional Josephson junction arrays for qubit devices.arrays for qubit devices.

Collaborations: Collaborations: F. Tafuri, Napoli IIF. Tafuri, Napoli IIA. Tagliacozzo, A. Naddeo, P. Lucignano, I. Borriello, Napoli IA. Tagliacozzo, A. Naddeo, P. Lucignano, I. Borriello, Napoli I

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

Superconductivity GroupSuperconductivity GroupApplied Physics DivisionApplied Physics DivisionDipartimento di EnergeticaDipartimento di EnergeticaL’AquilaL’Aquila

We are hereWe are here

0

0

0

0

CB

CF

C1

C8C7

C6

C5

C4 C3

C2

JJ

J=

0

0

0

0

0

0

0

0

(a)

(b)

(c)

1D open unconventional arrays 1D open unconventional arrays

Building block: the two-junction loopBuilding block: the two-junction loop

conventional loop for small conventional loop for small use use Eq) Eq)

ff cos1sin 2

-loop for small -loop for small

dia, dia, (0)=0(0)=0

ff

2cos

21

2sin

++ para, para, -- dia, moreover there are dia, moreover there arespontaneous currents for spontaneous currents for ff going to zero, going to zero,i.e., i.e., ++(0)=1 and (0)=1 and --(0)=-1 (0)=-1

++

--

001 103

Model: 1D GB Long Josephson Junction with presence of Model: 1D GB Long Josephson Junction with presence of -sections alternanting with conventional sections. -sections alternanting with conventional sections. This is equivalent to have localized This is equivalent to have localized -loops in a 1D array-loops in a 1D array

Quest: what is the fundamental state in zero field ?Quest: what is the fundamental state in zero field ?

0

Chain of Chain of ½ Flux quanta ½ Flux quanta or Semi-fluxons (SF)or Semi-fluxons (SF)

+SF+SF

SFSF SFSF

SFSF

SFSF

+SF+SF

0

1D open unconventional arrays1D open unconventional arrays

0

Quest: what is the effect of the magnetic field ?Quest: what is the effect of the magnetic field ?

0

Two solutions are no longer degenerate!Two solutions are no longer degenerate!RedRed ones is paramagnetic and have a lower energy ones is paramagnetic and have a lower energywith respect to with respect to BlueBlue ones which is diamagnetic and with ones which is diamagnetic and with higher energy…higher energy…

screening current addsscreening current adds

1D open unconventional arrays1D open unconventional arrays

ffE

I ii 22

2

1cos1

2 222

00

Total energy

2,,00

2

1,

2 extjjj j

tjj LV

IE

Total energy is the sum of Josephson and magnetic energy

We can write jjk

tjJtjjV cos)1(12

1, )(2

,2

,

Moreover, using flux quantization, Magnetic energy is written

Where = 2I0L/0 . With j=j-j-1+2nj we obtain

212

00

222

12

2

12

jjjjj

jjjM fnfE

I

The quantum number nj is typically zero for open arrays because the variations of the phases are small if is notLarge. On the other hand, in an annular array the last loop nN=n play the role of winding number of the phase, i.e., the number of flux quanta into the annulus.

The winding number

Q: How we find phases Q: How we find phases ii ? ?A: Solving Discrete Sine-Gordon equation (DSG)A: Solving Discrete Sine-Gordon equation (DSG)

iiiiiiik

ii ff

2

21

sin1 11

21

2

f

We assumeWe assume ff constant, i.e.,constant, i.e., ffii==ff , , moreovermoreover

With With N+2N+2==00=0, i=0, i++=i,i=i,i--=i-1,=i-1,ffN+1N+1==ff00=0=0

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

1D open unconventional arrays1D open unconventional arrays

0-0- junction (equal length) junction (equal length)(a)(a)diamagnetic soldiamagnetic sol(b)(b)paramagnetic solparamagnetic solN=63, N=63, =0.04=0.04

Mean magnetizationMean magnetizationfor different GBLJJs:for different GBLJJs:symmetric 0-symmetric 0- => => circlescircles

x2/J2t (1 m)2/(5 m)2=0.04

Grain sizeGrain size Josephson lengthJosephson length

d

G. Rotoli PRB68, 052505, 2003G. Rotoli PRB68, 052505, 2003

1D open unconventional arrays1D open unconventional arrays

N=255, N=255, =0.04=0.04with 15 with 15 -loops-loops(a)(a) 7 dia + 8 para7 dia + 8 para(b)(b) 5 dia + 10 para5 dia + 10 para(c)(c) 3 dia + 12 para3 dia + 12 para

(b) and (c) corresponds to(b) and (c) corresponds toa pre-selection of paramagnetica pre-selection of paramagneticsolutions due to FCsolutions due to FC

(c)(c) (b)(b)(a)(a)

(b)(b)(a)(a)

(c)(c)

FC can be introduced FC can be introduced assuming that it flips someassuming that it flips someSF from dia to para stateSF from dia to para state

G. Rotoli PRB68, 052505, 2003G. Rotoli PRB68, 052505, 2003

Previous work on 1D open unconventional arraysPrevious work on 1D open unconventional arrays

F. Tafuri and J. R. Kirtley, Phys. Rev. B62, 13934, 2000;F. Tafuri and J. R. Kirtley, Phys. Rev. B62, 13934, 2000;Tilt-Twist 45 degree YBCO GB junctionsTilt-Twist 45 degree YBCO GB junctionssample diamagnetic with ½ half flux quanta pinnedsample diamagnetic with ½ half flux quanta pinnedto defects and along GB, paramagnetism only localto defects and along GB, paramagnetism only local

F. Lombardi et al., Phys. Rev. Lett. in print, 2002;F. Lombardi et al., Phys. Rev. Lett. in print, 2002;Tilt-Twist GB junctions with angles betw 0 and 90Tilt-Twist GB junctions with angles betw 0 and 90rich structure of spontaneous currents for 0/90 GBrich structure of spontaneous currents for 0/90 GB

Il’ichev et al., to be subm. Phys. Rev. B, 2002;Il’ichev et al., to be subm. Phys. Rev. B, 2002;First paramagnetic signal recorded, very flat GBFirst paramagnetic signal recorded, very flat GBform 45 deg asymmetric twist junctions, no form 45 deg asymmetric twist junctions, no spontaneous currents have been experimentally spontaneous currents have been experimentally observedobserved

H. J. H. Smilde et al., Phys. Rev. Lett. 88, 057004, 2002;H. J. H. Smilde et al., Phys. Rev. Lett. 88, 057004, 2002;Artificial “zig-zag” LTC-HTC arraysArtificial “zig-zag” LTC-HTC arrays

Other papers in unconv. arrays and junctionsOther papers in unconv. arrays and junctions

Some estimate of demag field: Some estimate of demag field: dd

dJL

dH 2

0

Hd(a)=7.6 mG

Hd(b)=36 mG

Hd(c)=80 mG

we use we use LL==c-axisc-axis equal to 5 equal to 5 mm

Note that in (a) fields are of the same order of Note that in (a) fields are of the same order of magnitude cited in Tafuri and Kirtley (magnitude cited in Tafuri and Kirtley (c-axisc-axis=5.9 =5.9 m)m)

001 103

JJ

LL

1D open unconventional arrays1D open unconventional arrays

0-0- Annular JJ arrays Annular JJ arrays1)1) Have properties similar to the Annular Josephson junctionHave properties similar to the Annular Josephson junctionSo can be thinked are related to “fluxon qubit” (A. Ustinov, So can be thinked are related to “fluxon qubit” (A. Ustinov, Nature 425, 155, 2003)Nature 425, 155, 2003)2) Will have some “protection” from external perturbation 2) Will have some “protection” from external perturbation In the limit of large N (Doucout et al., PRL90, 107003, 2003)In the limit of large N (Doucout et al., PRL90, 107003, 2003)3) Can be build using 3) Can be build using -junctions as in Hilgenkamp et al., -junctions as in Hilgenkamp et al., Nature 50, 422, 2003Nature 50, 422, 2003

Merging together these three ideas we haveMerging together these three ideas we have

0

0

0

0

0

0

0

0

0

0

0

0

1 qubit 2 qubit

Annular arraysAnnular arraysA practical layoutA practical layout

0

0

0

0

CB

CF

C1

C8C7

C6

C5

C4 C3

C2

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

Q: How we find phases Q: How we find phases ii ? ?A: Solving Discrete Sine-Gordon equation (DSG) for the ringA: Solving Discrete Sine-Gordon equation (DSG) for the ring

iiiiiiik

ii ff

2

21

sin1 11

21

2

i

i

fAA ff constant do no longer apply, constant do no longer apply, ff have to have to

be not uniform to have effect on a 0-be not uniform to have effect on a 0- AJJA AJJA

With With N+1N+1==11+2+2nn, , nn is the winding number i is the winding number i++=i,i=i,i--

=i-1=i-1

0-0- Annular JJA DSG Annular JJA DSG

Fundamental states in AJJAFundamental states in AJJA

1 2 3 4 5 6 70.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1 2 3 4 5 6 7-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1 2 3 4 5 6 70.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 70.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6 7-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

Pha

se

mm

Pha

seP

hase

junction

m

loop

Spin notationSpin notation

0 2 4 6 8

-0.04

-0.02

0.00

0.02

0.04

0.0 0.5 1.0 1.50.0

0.1

0.8

0.9

1.0(a)

mag

netiz

atio

n

loop number j

(b)

mag

netiz

atio

n

Ene

rgy

AJJA arrays (excited states)AJJA arrays (excited states)

-8 -4 00.0

0.2

0.4

-8 -6 -4 -2 0 2

1

10

100

1000

-8 -6 -4 -2 0 2

|m|

log2

N = 2 N = 4

(a)

n=0

n=1

AF

FF

log2

n=0

n=1

n=2

AF

FF(b)

logE

log2

2

1

0

0

1

0

n

n

n

n

n

nN = 2 & 4 N = 6

3

2

1

0

0

0

n

n

n

n

n

n

1 2 3 4 5 6 7

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1 2 3 4 5 6 7

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

1 2 3 4 5 6 7-0.4

-0.2

0.0

0.2

0.4

1 2 3 4 5 6 7

-1

0

1

2

3

4

1 2 3 4 5 6 70

2

4

6

8

10

12

1 2 3 4 5 6 7-0.4

-0.2

0.0

0.2

0.4

mm

loop

Pha

se

junction junction

loop

n = 0

n = 1

AJJA (excited states) (2)AJJA (excited states) (2)

0 4 8 12 16-0.4

-0.2

0.0

0.2

0.4

0 4 8 12 16-0.4

-0.2

0.0

0.2

0.4

0 4 8 12 16-0.4

-0.2

0.0

0.2

0.4

0 4 8 12 16-0.4

-0.2

0.0

0.2

0.4

0 4 8 12 16-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0 16 32 48 64-0.4

-0.2

0.0

0.2

0.4

mm

m

loop loop

0 4 8 12 160

2

4

6

8

10

0 4 8 12 16-0.4

0.0

0.4

0.8

1.2

0 16 32 48 640

2

4

6

8

10

0 16 32 48 64-0.4

0.0

0.4

0.8

1.2

(d)(c)

(b)(a)

Ph

ase

m

Ph

ase

junction

m

loop

Fractionalization phenomenonFractionalization phenomenon

K-AK statesK-AK states

large

small

0 – 0 – Annular long junction Annular long junction

0 0

0

0

0

c)

d)

0 00 c-type

s-type m-type0 30 60 90

0

1

2

3

0 30 60 90-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

0 30 60 900

1

2

3

0 30 60 900

1

2

3

0 30 60 90-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

0 30 60 90-0.12

-0.08

-0.04

0.00

0.04

0.08

0.12

phas

e

m

phas

eph

ase

junction

m

loop

m

Fund. stateFund. state

E. Goldobin et al. PRB66, 100508, E. Goldobin et al. PRB66, 100508, 20022002E. Goldobin et al. PRB67, 224515, E. Goldobin et al. PRB67, 224515, 20032003E. Goldobin et al. cond-mat/0404091 E. Goldobin et al. cond-mat/0404091 (ring)(ring)

k 0- boundariesN/k sections

LJJ case 0-LJJ case 0- JJ JJ

0 32 64 96-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 32 64 96-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 32 64 96-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 32 64 96-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

128 144 160

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

112 128 144 160 176

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

(a)

m

(b)

(c)

m

(d)

(e) = 5.96 = 4.00 = 2.88

m

loop

(f) = 1.97 = 0.77 = 0.25

loop

-8 -6 -4 -2 0 20.0

0.1

0.2

0.3

0.4

-8 -6 -4 -2 0 20.0

0.6

1.2

1.8

2.4

3.0

-8 -6 -4 -2 0 20.0

0.6

1.2

1.8

2.4

3.0

m

log2

N = 2 N = 4

En

erg

y

(a)

n = 1

n = 0

AF

FF

Log2

n = 1 (unstable)

n = 0

n = 1

n = 2

(b)

FF

AF

Log2

K = 2,4K = 2,4N=32,64N=32,64

k=6k=6N=96N=96

l/k=1

l/k=2 (nor. length of sections)

Annular arrays in magnetic Annular arrays in magnetic field Ifield I

0 2 4 6 8 10 12 14 16

-0.12

-0.08

-0.04

0.00

0.04

0.08

m

loop

Single loop (Cn) frustation on Single loop (Cn) frustation on an N=16 arrayan N=16 array

0 2 4 6 8 10 12 14 16

-0.10

-0.05

0.00

0.05

0.10

mag

net

izat

ion

m

loop

Frustation over loops 10-16Frustation over loops 10-16On an N=16 arrayOn an N=16 array

Annular arrays in magnetic Annular arrays in magnetic field IIfield II

Frustation applied via CF is independent of N and inducea flip between para-dia sol. at =2.1

Critical field for flip Critical field for flip between fund. statesbetween fund. states

-3 -2 -1 0 1 2 3

1.0

1.2

1.4

1.6

1.8

-3 -2 -1 0 1 2 31

2

3

4

(b)

C1 N=8 dia N=8 par N=32 dia N=32 par

Ene

rgy

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

(a)

CF

Effect of frustation applied via a single loop, say C1, decrease with N

Magnetic behavior of annular 0-Magnetic behavior of annular 0- LJJLJJ

0 5 10 15 20 25 30-0.15

-0.10

-0.05

0.00

0.05

0.10

0 50 100 150 200 250-0.03

-0.02

-0.01

0.00

0.01

0.02

0 50 100 150 200 250

-0.03-0.02-0.010.000.010.02

(a)

m

loop

(b)m

(c)

m

-3 -2 -1 0 1 2 30.5

1.0

1.5

Ene

rgy

The effect of field in LJJcase is very similar

Variation offundamentalstate energyfor differentvalues of andMagnetic fieldIn the N=16 andN=64 AJJATop: magneticfield in a single loopBottom: magnetic field over 7 loops

Magnetic behavior for different spatial Magnetic behavior for different spatial configurationconfiguration

Annular arrays: flip dynamicsAnnular arrays: flip dynamics

N = 16 arrayvia C1

N=256, k=16 arrayvia s-type control

The process (classical)The process (classical)Classically it is possible to flip an half-flux quantum adding itClassically it is possible to flip an half-flux quantum adding ita full flux quantum (fluxon) E. Goldobin et al. cond-mat/0404091a full flux quantum (fluxon) E. Goldobin et al. cond-mat/0404091

0 16 32 48 64 80 96 112 128-4

-3

-2

-1

0

1

2

3

4

Pha

se

X

Successive timeplot of annihilationof a fluxon on a 0- boundarywhere a positivehalf-flux quantumwas localized.Annihilation endsin a negative halfflux quantum + radiation

motion direction

The process The process (quantum)(quantum)

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

-Junction -Junction realizationrealization

There are essentially three way to fabricate -junctions:dIddId YBCO made have the best performances in dissipation

and recently show also MQT effect (collaborationNapoli II, F. Tafuri + Chalmers, T. Cleason)dissipation are good (100 ) control of currentsand capacity not so easy

dIsdIs used by Hilgenkamp et al. in “zigzag” arrays,are YBCO-Nb ramp edge junctionsdissipation are intermediate (20 ), control onother parameters is good

SFSSFS these are Nb-(Ni-Cu)-Nb junctions which showa phase shift depending on F barrier thicknessdissipation is high at moment, critical currentsand capacitance can be controlled in a fine manner

ConclusionConclusion

Part of results shown here will be submitted to Part of results shown here will be submitted to ASC04 conference, Jacksonville, FL USA 3-8 october 2004 session ASC04 conference, Jacksonville, FL USA 3-8 october 2004 session 3EI013EI01

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 studybreather solutions, see PRE 66, 016603, 2002)

3) A quantum description of flip process between half-flux quantum is in progress

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

AcknowledgementsAcknowledgements