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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.

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

Giacomo Rotoli

Superconductivity Group & INFM Coherentia

Dipartimento di Energetica, Università di L’Aquila



F. Tafuri, Napoli II

A. Tagliacozzo, A. Naddeo, P. Lucignano, I. Borriello, Napoli I

Jacksonville, October 5 2004

Superconductivity group applied physics division dipartimento di energetica l aquila
Superconductivity GroupApplied Physics DivisionDipartimento di EnergeticaL’Aquila

We are here

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

1d open unconventional arrays
1D open unconventional arrays



Building block: the two-junction loop

conventional loop for small b (use g Eq)

  • dia, g(0)=0


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


1d open unconventional arrays1
1D open unconventional arrays











Chain of

½ Flux quanta

or Semi-fluxons (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 ?

1d open unconventional arrays2
1D open unconventional arrays





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
Total 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 winding number
The winding number

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.

1d open unconventional arrays3
1D open unconventional arrays

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)

1d open unconventional arrays4
1D open unconventional arrays

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

Grain size

Josephson length

G. Rotoli PRB68, 052505, 2003


  • 0-p junction (equal length)

  • diamagnetic sol

  • paramagnetic sol

  • N=63, b=0.04

Mean magnetization

for different GBLJJs:

symmetric 0-p => circles

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







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

Other papers in unconv arrays and junctions
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


H. J. H. Smilde et al., Phys. Rev. Lett. 88, 057004, 2002;

Artificial “zig-zag” LTC-HTC arrays

1d open unconventional arrays5
1D open unconventional arrays





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)

0 p annular jj arrays
0-p Annular JJ arrays

  • 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

Annular arrays a practical layout
Annular arraysA practical layout

N = 8 array, with

CF (control field)

CB (control barrier)

CN (control loop N)

0 p annular jja dsg
0-p Annular JJA DSG

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

Ajja arrays excited states
AJJA arrays (excited states)

N = 2 & 4

N = 6

n = 0

n = 1

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

K-AK states

large b

small b

Fractionalization phenomenon

0 p annular long junction
0 – p 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)

Fund. state

k0-p boundaries

N/k sections

Ljj case 0 p jj
LJJ case 0-p JJ

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


K = 2,4




Annular arrays in magnetic field i
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

Annular arrays in magnetic field ii
Annular arrays in magnetic field II

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

Magnetic behavior of annular 0 p ljj
Magnetic behavior of annular 0-p LJJ

The effect of field in LJJ

case is very similar

Magnetic behavior for different spatial configuration
Magnetic behavior for different spatial configuration

Variation of


state energy

for different

values of b and

Magnetic field

In the N=16 and


Top: magnetic

field in a single loop

Bottom: magnetic

field over 7 loops

Annular arrays flip dynamics
Annular arrays: flip dynamics

N=256, k=16 array

via s-type control

N = 16 array

via C1

The process classical
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

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

The process quantum
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


Next step is the calculation of euclidean action for the flip, its minimization will give the result.

P junction realization
p-Junction realization

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 => rotoli@ing.univaq.it

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