D.Giuliano (Cosenza), P. Sodano (Perugia)

Local Pairing of Cooper pairs in Josephson junction

networks

Obergurgl, June 2010Obergurgl, June 2010

Plan of the talk:

1. Josephson junction network interferometer as a model of a boundary double Sine-Gordon

Hamiltonian;2. Boundary interaction periodicity and coherent

tunneling of pairs of Cooper pairs;

3.Probing the effective tunneling charge via dc Josephson effect;

5. Conclusions, possible applications, perspectives.

4. Phase diagram and dual interaction;

1 The network;

A circular Josephson junction array, pierced by a magnetic flux φ, connected to two 1-d JJ “leads.

3

01,

2 cos22j

jjJjc

c EQE

H gj

j eViQ

4/11, jjjj

)1(2/1 hhNeVg

Charging energy of SC grains

Josephson energy Jc EE

Effective spin-1/2 Hamiltonian

3

01

3

0

4/ .).(j

zjj

jj

ic ShchSSeJH

Setting φ≈π-> near by degenerate eigenstates of Hc

6

1

iiii6

1

Projection onto low energy subspace

)1,(

3

0

nnn

nn

j

jj

nnP

nF F

jF

zj PNiPS

2

1

FjFj PiPS ]exp[

We have singled out an effective spin ½ degree of freedom, controlled with (at least one) tunable

parameter

aGS

How to either set up, or

probe, or even further control) the state of SG?

Connect it to two one-dimensional JJ arrays working as leads

The leads: Effective field theory of a JJ-chain

(L. I. Glazman and A. I. Larkin, PRL 79, 3736 (1997);

D.G., P. Sodano, NPB 711, 480 (2005))

,

)(1

)()(1

)(

2

)(0 cos2k j

kj

kjZ

j

kj

kjJ

jk

j

C nnEENiE

H

Mapping onto spin chain+Jordan-Wigner fermions+Bosonization Luttinger liquid (LL)

effective Hamiltonian

, 0

2)(

2

2)( 1

4k

L kk

LL dxtux

ug

H

(N=n+1/2)

LL parameters

42

42

ggv

ggvg

F

F

24

22 )( ggvu F

)]2cos(1[442 akagg F

)16

3(

2

C

Jz

E

EE

Connection between the central region and the leads

..' 3

)0(2

0

)0(2 chSeSeH

ii

Summing over the central region states->boundary degrees of freedom interacting with a localized

spin-1/2 x

GxzGz

zGB SBSBgSgH )]0(2cos[)0(cos 21

JEg /21

342 / JEg

]sin[ zB22 / JhBx )2()1(

2

1

2. Boundary interaction periodicity and coherent tunneling of pairs;

Bz measures the “detuning” of the degeneracy point due to a displacement from φ=π: Bz=Jsin[(φ-

π)/4]Bx measures the detuning due to an applied gate voltage: Bx ≈(λ2/J)(Vg-N-1/2)

The control parameters:

Using Bz to “tune” the effective charge tunneling across the device:

))cos((00 zGz SB (See below for

technical details)

In this case, a simplified model may be used for performing calculations

)]0(2cos[)0(cos)cos( 21 ggH B Charge difference operator between the two

leads

tu

gedxQ

L

0*

)0(*)0( ],[ iaia eaeeQAn harmonics of Φ(0) of period 2π/a varies the

relative charge by ae*, that is, it lets a total charge ae* tunnel across the central region

HB allows for direct tunneling of single Cooper

pairs (charge e* ), as well as of pairs of Cooper pairs (charge 2e*)

It is the only term that survives when cos(θ)=0

Discrete symmetry

)(,2 Zkk

B=0->enhanced (τ1) discrete symmetry

kzG

zG SSk )1(,

Usually, charge 2e* tunneling is a higher-order process and is neglected, BUT …

Technicalities:

1.Introduce two pairs of Dirac fermions a,a+;b,b+, and represent the effective spin operators as Sz =

a+a-b+b; Sx=a+b+b+a

2.Use the following (euclidean) action for the fermion operators (β=1/kBT):

0

0 abbaBbBibaBiadS xzz

3.Integrate over the fermion fields according to the recipe

)sin();cos( xz SS

'||222

'||222

)(cos)(sin)]'()([

;)(sin)(cos)]'()([

eSST

eSST

xx

zz

(Λ=√(Bz2+Bx

2))

3.Probing the charge tunneling across the central region via a dc Josephson current measurement;

Inducing a dc Josephson current-> Connecting the outer boundaries to two bulk superconductors at fixed phase difference α->Dirichlet-like boundary

conditions at the outer boundary (x=L) (that is, the plasmon phase field has to smoothly adapt to the

phase difference between the bulk superconducting leads)

),( tL

Dynamical boundary conditions at x=0

0)]0(2sin[2)]0(sin[)0(

2 21

ggSx

ug zG

Both boundary conditions may be accounted at weak coupling at x=0 (i.e., g1 ≈g2 ≈0), by taking

utnLi

n

en

nxn

Lgtx

)2

1(

21)(

])2

1(cos[

2),(

))](),(([ 1 mnnmn

Vacuum expectation values of vertex operators

iaia ee 00 )0()0(,00)( nn

Computing the dc Josephson current

ZI J ln1

][]][[ 000

)(

0

)(

BB HdHdH eTZeTeTrZ

Partition function at weak coupling

))]2cos()cos()(cos(exp[ 21000

ggZeZZBH

As Bz=0

)]2cos(exp[/ 20 gZZ

Josephson current for various values of Bz: Bz decreases counterclockwisely from the

top left panel and is =0 at the top right panel

The two harmonics in IJ correspond to tunneling of singe CPs and coherent tunneling of pairs of

CPs, respectively. The ratio between the contributions of the two processes to the total

current may be tuned by acting on Bz , that is, on the flux φ

)sin(

)2sin(

4. Phase diagram and dual interaction;

All the previous results rely on the assumption that the Josephson coupling between leads and

central region λ<<EJ,J

How reliable is this assumption?

As the size of the system (L) increases, low-energy, long wavelength collective plasmon

modes of the leads may get entangled with the isolated “boundary” degrees of freedom. This

may lead to a final state that is nonperturbative in HB. This happens if the boundary couplings

scale slower than 1/L

“Running” couplings

2211 ; LgLg

Flow equations for the running couplings

2110

1 11

)/ln(

gLLd

d2

21

20

2 sin2

41

)/ln(

gLLd

d

For g1≠0, the boundary interaction is a relevant operator (and, accordingly, the perturbative

approach is nor reliable), as soon as g>1. The second harmonics is nonperturbatively

renormalized, as well

Strong limit for the boundary coupling

Φ(0) is “pinned” at a minimum of the boundary potential->Dirichlet boundary

condition

nutLi

n

en

nxn

LL

xP

gtx

)(]sin[

2),(

Non τ1 -symmetric case

τ1 -symmetric case

22

2 ng

P

)2

(2

n

gP

Boundary potential and instanton trajectories

P

osc PL

ZZ ]exp[ 2

Leading boundary interaction at the SCP

“Jumps” between the minima of the boundary potential->shifs of the

eigenvalues of P ->dual vertex operators

)0()0( ],[ iaia aeeP

nutLi

n

en

nxn

Li

L

xP

L

vtgtx

)(]cos[2),(

tux

tux

“Dual” boundary interaction

)]0(cos[2

)0(cos

~

BH

“Short” instanton

s

“Long” instanton

s

Short instantons exist, as boundary excitations, as a consequence of τ1 –

symmetry. Breaking τ1 –symmetry implies short instanton confinement on a scale

that depends on B

When short instanton exist at any scale L, they “destabilize” the SFP. The SFP-

picture is not consistent anymore and the IR behavior of the system is driven by a

finite coupling FP.

Short instantons<->static solitons in the double Sine-Gordon model

Instanton trajectory -> P →P(τ)Integrating on the oscillator modes -

>Effective (Euclidean) action for P(τ) ->Equation of motion in the inverted

potential

Effective instanton action

0

2122 ]2cos[]cos[)cos(

42PgPgP

L

guP

MdSEff

“Equation of motion”

0]2sin[2]sin[)cos(2 21 PgPgPL

ugPM

=(apart for the finite-size term proportoanal to 1/L) to the equation for

static solitons in the DSG model

Two short instantons→one long instanton

1))((

2exparctan

2)(

aRa

MP

Separation between short instantons

|)cos(|8)(

2|)cos(|sinh

1

221

2

g

gR

gg

The short-instanton scaling (of μ) stops at a scale L≈uR(φ). If τ1-symmetry holds (i.e., short instantons are deconfined: R(φ)→∞), scaling does not stop and the

system is attracted by a FFP

φ=π,1.01π,1.1π,2π

5. Conclusions and (possible) further perspectives;

a. Possibility of acting on the external control parameters of the JJN to trigger the opening of an exotic phases, corresponding to an IR attractive

FFP; b. FFP corresponds to a “4-e” phase, with frustration of decoherence. At the FFP an

effective, 2-level quantum system emerges in the device, with enhanced quantum coherence

between the states; c. Making the experiment work !!!