Pairing of Cooper pairs in a Josephson junction network containing an impurity

0.16 MB·English

by  D. Giuliano

epl draft
Pairing of Cooper pairs in a Josephson junction network contain-
ing an impurity
D. Giuliano1 and P. Sodano2
0
1 1 Dipartimento di Fisica, Universita` della Calabria and I.N.F.N., Gruppo Collegato di Cosenza, Arcavacata di Rende,
0 I-87036, Cosenza, Italy
2
2 Dipartimento di Fisica, Universita` di Perugia and I.N.F.N., Sezione di Perugia, Via A. Pascoli, I-06123, Perugia,

n Italy
a
J

4
1 PACS 74.20.Mn–Nonconventionalmechanisms (spin fluctuations, etc.)
PACS 74.81.Fa–Josephson junction arrays and wire networks
]
l PACS 75.10.Pq–Spin chain models
e
- Abstract. - We show how to induce pairing of Cooper pairs (and, thus, 4e superconductivity)
r
t as a result of local embedding of a quantum impurity in a Josephson network fabricable with
s
. conventionaljunctions. WefindthataboundarydoubleSine-Gordonmodelprovidesanaccurate
t
a description of the dc Josephson current patterns, as well as of the stable phases accessible to
m the network. We point out that tunneling of pairs of Cooper pairs is robust against quantum
fluctuations, as a consequence of the time reversal invariance, arising when the central region of
-
d thenetwork is pierced by a dimensionless magnetic fluxϕ=π. We findthat, for ϕ=π, a stable
n attractive finite coupling fixed point emerges and point out its relevance for engineering a two
o level quantumsystem with enhanced coherence.
c
[

3
v
3
1 Thereisalargenumberofphysicalsystemsthatcanbe has to resort to nonperturbative methods, to study the
9 mapped onto quantum impurity models in one dimension system and the impurity as a whole. Such nonperturba-
0 [1]. Embedding a quantum impurity in a condensed mat- tivetoolsarenaturallyprovidedbyboundaryfieldtheories
.
8 ter system may alter its responses to external perturba- (BFT)[1]: BFTsallowforderivingexact,nonperturbative
0 tions[2],and/orinducetheemergenceofnonFermiliquid, informations from simple, prototypical models which, in
9
strongly correlated phases [3]. In quantum devices with many instances, provide an accurate descriptionof exper-
0
tunable parameters impurities may be realized by means iments on realistic low dimensional systems [12]. In par-
:
v of point contacts, of constrictions, or by the crossing of ticular, BFTs have been succesfully used to describe the
i
X quantum wires or Josephson junction chains [4–7]. For dc Josephson current pattern in Josephson devices, such
instance, novelquantum behaviors have been recently ev- as chains with a weak link [13,14], and SQUIDs [15,16].
r
a idencedintheanalysisofY-junctionsofquantumwires[6]
In this letter, we analyze a Josephsonjunction network
and of Josephson junction (JJ) chains [7]. Here, we show
(JJN),whoseBFTdescriptionisgivenbyaboundarydou-
how embedding a pertinent impurity in a JJ-chain may
ble Sine-Gordonmodel (DSGM) [17]. The device is made
lead to the emergence of nontrivial symmetry protected
by two half JJ chains, joined through a weak link to a
quantum phases associated [8–11] with the emergence of
centralAharonov-BohmcageC[18],piercedbya(dimen-
4e superconductivity in the network.
sionless) flux ϕ = Φ/Φ (Φ = hc/(2e)) (see Fig.1). For
∗0 ∗0
While a standard perturbative approach works fine simplicity, we connect the outer ends of the chains to two
when impurities are weakly coupled to the other modes bulk superconductors at fixed phase difference α.
ofthe system(the “environment”),there aresituationsin
As we shall see, the dc Josephson current across the
which the impurities are strongly coupled to the environ-
JJN is a periodic function of α of period 2π, given
ment,affectingitsbehaviorthroughachangeofboundary
by I (α) = E¯(2)(ϕ)sin(α) + 2E¯(4)(ϕ)sin(2α). Varying
conditions: when this happens, it is impossible to disen- J W W
tangle the impurity from the rest of the system, the per- ϕ changes the ratio E¯W(4)(ϕ)/E¯W(2)(ϕ); in particular, for
turbative approach breaks down, and, consequently, one ϕ = π, E¯(2)(π) = 0, and, thus, the period of I (α) is
W J
p-1
D. Giuliano 1 P. Sodano 2
S
1 α/2
z
J
Ez Kz J
ϕ π ϕ π
EJ K
ϕ
−α/2 S
3
Fig. 1: The network.
ϕ π ϕ π ϕ π
halved, signalling the tunneling of pairs of Cooper pairs
(PCP)s through the JJN. A semiclassical analysis [19]
already accounts for the two harmonics contributing to
I (α): though, for a generic ϕ, single Cooper pair (SCP)
J
Fig.2: LeadingtunnelingprocessesacrossC:Toppanel: Sin-
tunneling is the dominating process for charge transport
gleCooperpairtunnelingforϕ6=π;Bottom panel: Tunnel-
acrossthe JJN(see toppanelofFig.2), forϕ=π, disrup-
ing of pairs of Cooper pairs for ϕ=π.
tiveinterferenceacrossCforbidsSCPtunneling,andonly
allowsfor PCP tunneling, thus letting 4e superconductiv-
ity to emerge in the JJN (see bottom panel of Fig.2). teraction Hamiltonian H [ Φ , Θ ;S ,S ] =
T a a 1 3
{ } { }
apIptriosawcheellskbnroewaknd[1o5w]nthinatosneem-diicmlaesnsisciaolnaanlJdJ/Nors,mwehanenfitehlde −J a=L,R;s=1,3[e√i2Φa(0)eiǫaϕ4Ss− + h.c.] +
quantum phase fluctuations diverge logarithmically with a=PL,R;s=1,3 √J2zπ∂Θ∂ax(0)Ssz ( ǫL = 1,ǫR = −1), and
the length of the system thus inducing a nonperturbative HPK[S1,S3] − K[S1+S3− + S1+S3−] + KzS1zS3z. S1,S3 are
renormalizationoftheJosephsoncouplings. Here,wepro- defined as in Fig.1, Φa is the collective plasmon field
vide a full quantum treatment of the JJN in Fig.1, based of the half chain a, while Θa(x,t) is its dual field, that
on the BFT approach: We shall derive the dc Josephson is, ∂Φa(x,t) = 1∂Θa(x,t), and ∂Φa(x,t) = 1∂Θa(x,t). The
∂x g u∂t u∂t g ∂x
current patterns and evidence the emergence- for a suit- LL parameters are defined as g = π/[2(π arccos(∆))],
able choice of the control and constructive parameters of − 2
u = v π( 1 (∆)2)/(arccos(∆)), with ∆ = Ez/E ,
ntheewJaJtNtr-aoctfiaverofibxuedstp4oeinstupinertchoenpdhuacsteivditiyagarsasmociaactceedsstioblae vf = 2afE2Jq, wh−ere2EJ and Ez a2re the Josephson energJy
to the network. and the Coulomb repulsion energy of the half chains,
The central region is described by and a is the lattice step [20]. In deriving Eq.(2), it is
assumed that J/E 1 and Jz/Ez 1 (i.e., that
J
E 3 3 3 C is weakly coupled≪to the chains); ≪this allows to
HC = 2c [Qj]2−2J cos(χj,j+1)+Jz QjQj+1 , use Neumann boundary conditions at x = 0, that is
Xj=0 Xj=0 Xj=0 ∂ΦL(0) = ∂ΦR(0) =0.
(1) ∂x ∂x
The couplings K,K in H [Φ ,Φ ,Θ ,Θ ;S ,S are
with Q = i ∂ V , and χ = χ χ + ϕ. z T L R L R 1 3
j h− ∂χj − gi j,j+1 j − j+1 4 dynamically generated by the interaction between C and
χj isthephaseofthesuperconductingorderparameterat thehalfchains,asitmaybeeasilyinferredfromtherenor-
theislandj,Vg isagatevoltageappliedtoeachsupercon- malization group (RG) equations for the dimensionless
ductingisland,J istheJosephsonenergyofeachjunction, couplings = LK, z = LKz, and = L1−21gJ, z =
Ec isthechargingenergyateachsite,andJz accountsfor Jz. IndeedK, employiKng standard BFTJtechniques [2J1,22],
Coulomb repulsion between charges on nearest neighbor- one obtains d = +[1+cos(ϕ)]( )2 , d z =
awinnrgditjetuunHnciCtnioganVssg.asIosnpttihnha-e1tc/Vh2gaXr=gXiNnZg+-mre12og,diwmeilet,[h1t4ihn]a,ttewgishe,rosENJec,,HoEJanzcme≪miltao1y-, iKnzg+th(aJtzb)o2,thadnlnd(KaLdLn0ln)dd(JLL0z)Ka=reh1dy−na21mgiicJal,lydldnJgJ(eLnzL0e)ra≈tdel0dnK,(wLsL0hh)oewn--
nian is given by HC = −J 3j=0{eiϕ4Sj+Sj−+1 + h.c.} + ever ϕ6=π. K K
Jz 3 SzSz . Using thePstandard bosonization ap- Integrating the RG equations one sees that (L)
j=0 j j+1 J ∼
proPach [12], the JJN Hamiltonian may be described by L1−21g, while (L) L2−g1, that is, for ϕ = π, is al-
K ∼ 6 K
two spinless Luttinger liquids (LL), interacting with iso- ways more relevant than . At variance, for ϕ = π, no
J
lated spin-1/2 variables on C, with Hamiltonian given by K-coupling is generated and is the only relevant cou-
J
pling strength. For g > 1/2, the BFT description of the
H =H [ Φ ]+H [ Φ , Θ ;S ,S ]+H [S ,S ],
JJN LL { a} T { a} { a} 1 3 K 1 3 JJNallowstomakeverygeneralstatementsregardingthe
(2)
regimes accessible to a network of finite size L. Namely,
with the Luttinger liquid Hamiltonian H [ Φ ] =
LL { a} therewillbeaperturbativeweakcouplingregime,accessi-
g Ldx 1 ∂Φa 2+u ∂Φ 2 , the in- bleforsmall and ,andanon-perturbativestrongcou-
4π R0 Pa=L,Rhu(cid:0) ∂t (cid:1) (cid:0)∂x(cid:1) i K J
p-2
Pairing of Cooper pairs in a Josephson junction network containing an impurity
pling regime, accessible when or becomes 1. Most
K J ∼
importantly, there will be a renormalizationgroup invari-
1
ant length scale L∗ ∼ J−2g−1, such that for L < L∗, the 0
JJNisintheperturbativeweakcouplingregime,whileitis
inthenonperturbativestrongcouplingregimeforL>L .
∗
To account for the last term of Eq.(2), one may resort
to a Schrieffer-Wolff transformation [23], which amounts
toprojectingoverthegroundstateofC,bysummingover
its high energy states. At ϕ = π, the ground state of C
0
is twofolddegenerate,while sucha degeneracydisappears
at ϕ = π. As a result, for ϕ = π, the central region C
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is effectively describedby the boundary HamiltonianHB,
given by −2π 0 2π −2π 0 2π
HB[ϕ]=−EW(2)(ϕ)cos[Φ−(0)]−EW(4)(ϕ)cos[2Φ−(0)], (3) Fig. 3: Josephson current vs. α for different values of ϕ and
with E(2)(ϕ) = cos(ϕ) 2J2 +sin2(ϕ)cos(ϕ) 2J4 , forJ2/K2 ≈0.3: Top left panelϕ=π,Bottom left panel
W 2 K+Kz 2 2 K(K+Kz)2
E(4)(ϕ) = sin2(ϕ) 2J4 , while Φ = [Φ Φ ]/√2. ϕ=1.01π,Bottom right panel ϕ=1.1π, Top right panel
W 2 K(K+Kz)2 − L− R ϕ=2π
At variance, for ϕ=π, HB[π] (EJ)4 cos[2Φ (0)]
∼− J3 −
I (α) may be perturbatively computed as I (α) =
J J
lim 1∂ln ,where isthepartitionfunctionofthe where the “instanton mass” M ln(u /a), being
β→∞−β ∂αZ Z β the “instanton size”, while V(x)∼= ET(2)(ϕ)cTos(x)
JJN,givenbylnZ ≈lnZ0− 0 dτhHB(τ)i, HB(τ)being E(4)(ϕ)cos(2x). From the effective −actWion in Eq.(5−),
istheboundaryinteractionHRamiltonianinthe(imaginary W
time) interactionrepresentation,while is the partition one gets the equation of motion MP¨ gπuP +
function at HB =0. From Eq.(3), one oZb0tains πE(2)(ϕ)sin[πP]+2πE(4)(ϕ)sin[2πP]=0whi−ch,2nLeglect-
W W
ing the “inductive term” 1, describes soliton solutions
I (α)=E¯(2)(ϕ)sin(α)+2E¯(4)(ϕ)sin(2α) , (4) ∝ L
J W W of the double-Sine Gordon model [17]. These are given
with E¯W(2)(ϕ) = La g1 EW(2)(ϕ), E¯W(4)(ϕ) = La g4 EW(4)(ϕ). byP(τ)= a=±1 π2 tan−1hexp(cid:16)√2Mπ (aτ +R(ϕ))(cid:17)i, with
The ratio E¯W(2)(ϕ(cid:0))/E¯(cid:1)W(4)(ϕ), measures the r(cid:0)ela(cid:1)tive weight R(ϕ) definePd by 1sinh2 E(2)(ϕ) + EW(4)(ϕ) R(ϕ) =
of SCP vs. PCP tunneling rate. One notices that, for 4 (cid:20)(cid:18)| W | 2 (cid:19) (cid:21)
ϕ = π, E¯(2)(π) = 0, while E¯(4)(π) = 0. Thus, at ϕ = π 2E(4)(ϕ)/E(2)(ϕ).
W W 6 W | W |
PCP tunneling is the only allowed mechanism for charge In Fig.4, we plot HB[Φ] vs. Φ for various values of
transferacrossC.InFig.(3),weplotIJ(α)vs. αfordiffer- ϕ [π,2π] and for 2π Φ(0) 2π. We see that, for
∈ − ≤ ≤
ent values of ϕ. For ϕ=π, IJ(α) is periodic, with period ϕ = π, the minima are separated by a distance 2π. The
6 6
∆α=2π,while, for ϕ π, the periodshrinksto ∆α=π. correspondinginstantontrajectoriescorrespondtoa“long
∼
For g > 1, HB is a relevant operator. Thus, when jump” from P( )= 1 to P( )=1 (representedby a
−∞ − ∞
L/L 1, the boundary Hamiltonian in Eq.(3) is solidarrowinFig.4). Thismaybe regardedasasequence
∗ ≥
the dominating potential term, and the field Φ (0,τ) of two “short instantons” (dashed arrows), separated by
−
takes values corresponding to minima of HB; it obeys adistance 2R(ϕ)(see Fig.5). As E(2)(ϕ) becomes smaller
W
Dirichlet boundary conditions at both boundaries, yield-
(that is, as ϕ gets closer to π), R(ϕ) increases, and even-
ing the mode expansion Φ (x,τ) = α + LL−x πP + tually diverges, as ϕ π+.
g2 n=0sin πLnx αnne−πnLu−τ [14]. At the st(cid:0)rongly(cid:1) inter- Though the degene→racy between the minima is broken
qactinPg fi6xed p(cid:0)oint,(cid:1)instanton trajectories are the leading bythe1/L-term,apertinenttuningofthephasedifference
quantum fluctuations; they are described by imaginary α allows to restore it. Indeed, for ϕ > π, the degeneracy
time trajectories P(τ), with P(τ ) and P(τ ) ofthe minima atΦ (0)=2πn π isrestoredbychoosing
corresponding to nearest neighbor→ing−m∞inima of HB→. ∞ α = 2πn,n ∈ Z, −while, for ϕ±< π, one has to choose
Theinstantonprofileisderivedfrom δSEff[P] =0,where α=π(1+2n) and, at ϕ=π, α=π(12 +2n).
δP(τ)
In a BFT approach,long instantons are represented by
acSlnEiddffe[uaPn,]wa=ictthRio[anQbSnoEDu[nΦαd−na]]refyo−riSnaEte[sΦrpa−icn]tliiesosnsobLHtLaamiwnieiltdthonfprioaamrnamgtihveeteenrEsbuyg- mtheenbsioounnhda1r=y vge,rwtehxiloepsehroarttorisnset±ainΘt(o0n,τs),bwyitthhesocapleinragtodris-
HB,afterintegratingovertheoscilatormodesofΦ (x,τ), e±2iΘ(0,τ), with scaling dimension h12 = g4. For 1 < g < 4
− and ϕ=π, short instantons are relevant perturbations of
α . As a result, one gets
{ n} the strong coupling (Dirichlet) fixed point. As it happens
β M gπu for Y-shaped JJNs, also here a new, time-reversal invari-
S [P]= dτ (P˙)2+ P2+V[πP] , (5)
Eff Z (cid:26) 2 4L (cid:27) ant, attractive finite coupling fixed point emerges in the
0
p-3
D. Giuliano 1 and P. Sodano 2
1 1
0
0
−1
0
1 1
0
0 0
−1
−1
−2π 0 2π −2π 0 2π
Fig. 5: Profile of the instanton excitations P(τ) for different
values of ϕ and for J2/K2 ≈ 0.3, M = 1: Top left panel
Fig. 4: Minima of the boundary potential HB for differ- ϕ=π, Bottom left panel ϕ=1.01π,Bottom right panel
ent values of ϕ and for Λ(2)(π)/Λ(4)(0) ≈ 0.3 (in arbitrary ϕ=1.1π, Top right panel ϕ=2π
units). Thelong (short) instanton trajectories are represented
assolid(dashed)arrows: Top left panelϕ=π,Bottom left
panel ϕ=1.01π,Bottom right panel ϕ=1.1π, Top right Associated with 4e superconductivity there is, for 1 <
panel ϕ=2π g < 4, a new finite coupling attractive fixed point, which
allows for the possibility of using the JJN as a two level
quantum system with enhanced coherence [7,24]. Fur-
boundaryphasediagram[7], asa resultofthe twofoldde-
thermore, PCP tunneling is robust, as a consequence of
generacy of the ground state of C. At variance, for 1<g
the time reversal invariance, realized only for ϕ=π. The
and ϕ = π, no finite coupling fixed point emerges, since,
proposed mechanism exhibits intriguing similarities with
6
now, a possible departure from the Dirichlet fixed point
the electron bunching phenomenon [25], observed in shot
should be due to long instantons, which are an irrelevant
noise measurements [26] on quantum dots in the Kondo
perturbation. For ϕ = π, the twofold degeneracy of the
regime.
groundstateofCisduetoaZ symmetryofV[Φ ]man-
2
−
ifesting its invariance under time reversal ϕ 2π ϕ,
−→ −
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