Post on 04-Jan-2016
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 !!!