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Unconventional Josephson junction arrays for qubit devices.

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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

001

103

Building block: the two-junction loop

conventional loop for small b (use g Eq)

- dia, g(0)=0

g+

p-loop for small b

- g+ para, g- dia, moreover there are
spontaneous currents for f going to zero,

i.e., g+(0)=1 and g-(0)=-1

g-

p

p

p

0

p

0

p

+SF

+SF

+SF

Chain of

½ Flux quanta

or Semi-fluxons (SF)

-SF

-SF

-SF

Model: 1D GB Long Josephson Junction with presence of

p-sections alternanting with conventional sections.

This is equivalent to have localized p-loops in a 1D array

Quest: what is the fundamental state in zero field ?

0

p

0

p

Quest: what is the effect of the magnetic field ?

screening current adds

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 is the sum of Josephson and magnetic energy

We can write

Moreover, using flux quantization, Magnetic energy is written

Where b = 2pI0L/F0 . With Djj=jj-jj-1+2pnj we obtain

The quantum number nj is typically zero for open arrays because the variations of the phases are small if b is not

Large. 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.

Q: How we find phases ji ?

A: Solving Discrete Sine-Gordon equation (DSG)

With jN+2=j0=0, i+=i,i-=i-1,fN+1=f0=0

We assumefconstant, i.e.,fi=f , moreover

(see E. Goldobin et al., Phys. Rev. B66, 100508, 2002;

J. R. Kirtley et al., Phys. Rev. B56, 886, 1997)

b=Dx2/lJ2t(1 mm)2/(5 mm)2=0.04

Grain size

Josephson length

G. Rotoli PRB68, 052505, 2003

hd

- 0-p junction (equal length)
- diamagnetic sol
- paramagnetic sol
- N=63, b=0.04

Mean magnetization

for different GBLJJs:

symmetric 0-p => circles

(c)

(b)

(a)

(a)

(b)

(c)

G. Rotoli PRB68, 052505, 2003

- N=255, b=0.04
- with 15 p-loops
- 7 dia + 8 para
- 5 dia + 10 para
- 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

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

001

103

lJ

lL

Some estimate of demag field: hd

Hd(a)=7.6 mG

Hd(b)=36 mG

Hd(c)=80 mG

we use lL=lc-axis equal to 5 mm

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

- 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 p-junctions as in Hilgenkamp et al.,
- Nature 50, 422, 2003

Merging together these three ideas we have

1 qubit

2 qubit

N = 8 array, with

CF (control field)

CB (control barrier)

CN (control loop N)

Q: How we find phases ji ?

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

With jN+1=j1+2pn, n is the winding number i+=i,i-=i-1

Afconstant do no longer apply, f have to

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

Spin notation

N = 2 & 4

N = 6

n = 0

n = 1

K-AK states

large b

small b

Fractionalization phenomenon

E. Goldobin et al. PRB66, 100508, 2002

E. Goldobin et al. PRB67, 224515, 2003

E. Goldobin et al. cond-mat/0404091 (ring)

Fund. state

k0-p boundaries

N/k sections

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

l/k=1

K = 2,4

N=32,64

k=6

N=96

Single loop (Cn) frustation on

an N=16 array

Frustation over loops 10-16

On an N=16 array

Critical field for flip

between fund. states

Frustation applied via CF is

independent of N and induce

a flip between para-dia sol. at h=2.1

Effect of frustation applied

via a single loop, say C1,

decrease with N

The effect of field in LJJ

case is very similar

Variation of

fundamental

state energy

for different

values of b and

Magnetic field

In the N=16 and

N=64 AJJA

Top: magnetic

field in a single loop

Bottom: magnetic

field over 7 loops

N=256, k=16 array

via s-type control

N = 16 array

via C1

Classically it is possible to flip an half-flux quantum adding it

a full flux quantum (fluxon) E. Goldobin et al. cond-mat/0404091

motion direction

Successive time

plot of annihilation

of a fluxon on

a 0-p 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.

There are essentially three way to fabricate p-junctions:

dIdYBCO 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 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

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

- Annular unconventional arrays and their LJJ counterpart the annular 0-p 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
- 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)
- 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

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 => [email protected]

web => http://ing.univaq.it/energeti/research/Fisica/supgru.htm