Interplay between Coulomb blockade and Josephson effect in a topological superconductor-quantum dot device

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by  Yu-Li Lee

Interplay between Coulomb blockade and Josephson effect in a topological
superconductor-quantum dot device
Yu-Li Lee
∗
Department of Physics, National Changhua University of Education, Changhua, Taiwan, R.O.C.
Yu-Wen Lee
†
Department of Applied Physics, Tunghai University, Taichung, Taiwan, R.O.C.
(Dated: January 5, 2016)
6
1 Westudythebehaviorofatopological Josephsonjunctioninwhichtwotopologicalsuperconduc-
0 tors are coupled through a quantum dot. We focus on the case with the bulk superconducting gap
2 being thelargest energy scale. Two parameter regimes are investigated: a weak tunnelingbetween

n thedotandthesuperconductors,withthedotnearitschargedegeneracypoint,andastrongtunnel-
a ingregimeinwhichthetransmissionbetweenthedotandthesuperconductorsisnearlyperfect. We
J show that in the former situation, the Andreev spectrum for each sector with fixed fermion parity

3 consists of only two levels, which gives rise to the nontrivial current-phase relation. Moreover, we
studytheRabioscillation between thetwolevels andindicates that thecorresponding frequencyis

] a4π-periodicfunctionofthephasedifferencebetweenthetwosuperconductors,whichisimmuneto
n thequasiparticle poisoning. In thelatter case, we find that theCoulomb charging energy enhances
o
the effect of backscattering at the interfaces between the dot and the superconductors. Both the
c
temperatureand thegate-voltage dependenceof thecritical Josephson current are examined.
-
r
p PACSnumbers: 74.50.+r73.63.Kv74.45.+c
u
s
t. I. INTRODUCTION lands etc., which have to be taken into account4,15–19.
a
Therefore, it is extremely important to understand how
m
this 4π Josephson effect survives these complications.
Over the past few years, the Majorana bound states
-
d (MBSs) in topological superconductors (TSCs) have at- Thereareseveralrecentattemptsinthisresearchdirec-
n tracted a lot of attention in the condensed matter re- tion. Forexample,bycouplingthetopologicalJosephson
o search community1–4. These exotic objects are basi- junction to an external normal-metalprobe, the dynam-
c
cally the zero-energy quasiparticles which are the equal- ics of the fermion-parity switch through the ejection of
[
amplitude superposition of particles and holes in TSCs. quasiparticles via the probe is analyzed in Ref. 20. In

1 One surprising characteristic of these MBSs is that they Ref. 21, the effects of electromagnetic environments and
v obey non-Abelian braiding statistics5. Furthermore, a anadditionalquantumdot(QD)onthechargetransport
5
widelyseparatedpairofMBSsformsanonlocalfermionic betweentheTSCandanormal-metalleadareconsidered.
6
2 state which is immune to local sources of decoherence. Interestingly, in the latter case, it is claimed that when
0 Both features provide a potential for future applica- two TSCs are coupled via a QD, the resulting system is
0 tions in quantum computing and quantum information equivalent to a resonant-level model at low energy from
. processes6–8. Uptonow,thereareseveralcandidatesys- which a Coulomb oscillation in the conductance follows
1
0 temsthatarepromisedtorealizetheseMBSs. Typically, in both the weak and the strong tunneling regime. It is
6 s-wavesuperconductorsareputinproximitytootherma- not clear how the Josephson current, which is supposed
1 terials with strong spin-orbital coupling, in the presence toexistinthe absenceofthe QD,isaffectedbythepres-
: or absence of external magnetic fields1–3. There have al- ence of the QD.
v
i readybeenseveralrecentexperimentsinvestigatingthese The interesting situation where the TSC is coupled
X kinds of systems9–12. to a metallic lead through a QD has already been
ar OnewaytorevealtheexistenceoftheMBSsisthrough studied22–25. There, the point is to analyze the compe-
the 4π (or fractional) Josephson effect in a topologi- tition between the Andreev reflection and the Coulomb
cal Josephson junction6,13–15, where the DC Josephson blockade/Kondo effect. In the present work, we would
current is a 4π-periodic function of the superconduct- liketostudytheeffectsofCoulombblockade,introduced
ing (SC) phase difference between the two TSCs. This by a QD, on the DC Josephson current. The TSC itself
doubling of the period is tightly related to these zero- forms a ring with a gap, and the ring is threaded by a
energy MBSs. Roughly speaking, the existence of zero- magnetic flux Φ which determines the SC phase differ-
B
energy MBSs allow the coherent single-particle tunnel- ence φ between the two ends of the TSC. The two ends
ing between TSCs. Since the electron carries half of ofthe TSCis coupledthrougha QDtoforma TSC-QD-
the SC phase, this results in the doubling of the period. TSCjunction. AschematicsetupisshowninFig. 1. We
However,inrealexperimentalsituations,therearemany focusonthecasewherethebulkSCgapisthelargesten-
subtle effects, such as disorder, multiple bands, quasi- ergy scale. In the weak tunneling limit, we consider the
particle poisoning, Coulomb charging energy for SC is- case where the dot is near the charge degeneracy point.
2
section is devoted to a conclusive discussion.
II. THE WEAK TUNNELING REGIME
A. The model
We first consider the weak tunneling regime, i.e.
G e2/h, where G denotes the tunneling conduc-
l/r l/r
≪
tancefortheleft/rightcontact. Attheenergyscalemuch
lower than the bulk SC gap ∆ , the system can be de-
0
scribed by the Hamiltonian H =H +H , where
D T
H = ǫ d d +E (Nˆ N )2 , (1)
D s †s s c − g
s
X
FIG. 1: (Color online) A schematic setup of the TSC-QD-
TSC junction. The TSC itself forms a ring threaded by a is the Hamiltonian of the dot with E =e2/(2C) and
c
magnetic fluxΦB. The energy levelsin thedot can betuned
tbwyeaencatphaeciQtiDvelayncdoutphleedTgSaCtecvaonltbaegeaVdgju.sTtehdebcyouapdlidnigtisobnae-l HT= Γrse−iχr/2γrds−iΓlse−iχl/2γlds+H.c. ,
gate voltages, which are not shown in thefigure. Xs (cid:16) (cid:17)
(2)
describesthetunnelingbetweenthetwoendsoftheTSC
and the QD, with the real tunneling amplitudes Γ and
ls
Γ . WeshallassumethatE , Γ ∆ . Hereγ and
rs c l(r)s 0 l
This is because in this case the Andreev spectrum, aris- γ , obeying the algebra γ2 =|1 = γ|2≪and γ ,γ = 0,
ingfromthehybridizationbetweentheMBSsattheends r l r { l r}
denote the twoMajoranafermions atthe ends ofthe left
oftheTSCandthechargedegreesoffreedomoftheQD,
and the rightSC electrodes, χ and χ , obeying the con-
l r
showsasimplestructure,i.e. twoAndreevlevelsforeach
straintχ χ =2eΦ φ,aretheSCphasesofthe left
l r B
sector with fixed fermion parity. This results in a non- − ≡
andtherightSCelectrodes,ǫ denotesthesingle-particle
s
trivialcurrent-phase relationwith a 4π periodicity. This
energyofthedotwiththecompletesetofgoodquantum
simple structure of the Andreev spectrum and its 4π de-
numbers s, the constant N is proportional to the gate
g
pendence on φ can be revealed as sharp peaks in the
voltage V , the operators d and d obey the canonical
g s †s
spectral function of the dot, which could be measured
anticommutation relations, and the Hermitian operator
by STM26 or microwave-optical experiments27. More- Nˆ, having integer eigenvalues, satisfies the commutation
over,bychangingtheflux throughtheringabruptly,the relations [Nˆ,d ]= d .
s s
Josephson current exhibits the behavior of Rabi oscilla- Let N betheeig−enstateoftheoperatorNˆ witheigen-
tion with the frequency being a 4π-periodic function of | i
values N = 0, 1, 2, . Following Ref. 28, for large
φ. This φ dependence of the Rabi frequency is immune ± ± ···
charging energy, i.e. E Γ , only two charge
to quasiparticle poisoning. Both features, the 4π depen- c ≫ | l(r)s|
states N and N + 1 give dominant contributions
dence of the peaks in the spectral function of the dot | 0i | 0 i
to the low-energy properties near the degeneracy point
and the Rabi frequency, can be viewed as the hallmark
N =N +1/2,whereN is someinteger. Toa goodap-
for the presence of the MBSs. In the strong tunneling g 0 0
proximation, we can truncate the Hilbert space keeping
regime where the transmission between the QD and the
only these two states. Within this truncated space, one
SCelectrodesisnearlyperfect,thecurrent-phaserelation
mayintroduceasetoffermionoperatorsf andf defined
is the usual one, i.e. I = I sin(φ/2), provided that the †
c by f N = 0 and f N = N +1 . Moreover, they
fermion parity is conserved. We calculate the tempera- | 0i †| 0i | 0 i
obey the canonicalanticommutationrelations. Then, we
ture and the gate-voltagedependence of the critical cur- have Nˆ N =f f, and H becomes
rent Ic by assuming that the temperature is much larger − 0 † D
than the average level spacing in the dot. Due to the
H =ǫ f f , (3)
D d †
Coulomb charging energy of the dot, the backscattering
at the interfaces between the QD and the SC electrodes
up to a constant term, where ǫ accounts for the en-
d
isenhanced,whichisrevealedinthe temperaturedepen-
ergy difference between the states N + 1 and N ,
0 0
dence of I . Our main results are summarized in Figs. 2 | i | i
c and can be written as ǫ = ǫ 2E (N N 1/2).
d 0 c g 0
and 3. − − −
Here the constant ǫ N + 1 ǫ d d N + 1
0 ≡ h 0 | s s †s s| 0 i −
The rest of the paper is organized as follows. We N ǫ d d N is independent of N N . On the
present our analysis on the Andreev spectrum and the hoth0e|r hsansd,†sHs| c0ain be written as P g − 0
T
DC Josephson current in the weak and the strong tun- P
neling regimes in Sec. II and III, respectively. The last HT =tre−iχr/2γrf itle−iχl/2γlf +H.c. , (4)
−
3
where t Γ N d N +1 . Without loss of
l(r) ≡ s l(r)sh 0| s| 0 i
generality, we may set t >0.
l/r
P
We see that near the degenerate point, the Hamilto-
nian H = H + H is quadratic in the fermion op-
D T
erators f and γ , and thus the whole problem can
l/r
be solved exactly. To proceed, we introduce another
set of fermion operators: a = (γ + iγ )/2 and a =
l r †
(γ iγ )/2, which obey the canonical anticommutation
l r
−
relations. Then, we may choose the set of orthonormal-
ized states 0,0 , 1,1 , 0,1 , 1,0 as the basis of the
truncated H{il|berti s|pacie,|wheire| thei}state m,n contains FIG.2: (Coloronline)TheDCJosephsoncurrentI0atT =0
| i for the TSC-QD-TSC junction in the weak-tunneling regime
m f-fermions and n a-fermions. In terms of this choice
of the basis, the Hamiltonian H can be written as wI0itihnNugni≈tsNo0f+et12//E2.c.WeOsneltyttlh=etrres=ultt,sǫf0o=r t0h,eansedctmoreawsuitrhe
even fermion parity are shown here. Left: the current-phase
h 0
H = 0e h , (5) relation at Ng−N0−1/2=±0.1 for t/Ec =0.03 (solid line)
(cid:20) o (cid:21) and t/Ec = 0.1 (dashed line). We also plot I0 in the limit
where t/Ec →0 to show the deviation from the one I0 ∝sin(φ/2).
Right: I0 versus Ng−N0−1/2 at φ=0.3π for t/Ec =0.03
0 i(∆l+∆r) (solid line) and t/Ec =0.1 (dashed line).
−
h = ,
e
 
i(∆ +∆ ) ǫ
l r ∗ d
 
0 −i(∆l−∆r) is the DC Josephson current at T =0. We see that Ie is
ho = , (6) a 4π-periodic function of φ, a characteristic of the topo-
 
i(∆l ∆r)∗ ǫd logical Josephson junction. In contrast with the short
−
  topological Josephson junction in which the DC Joseph-
with ∆l/r ≡ tl/re−iχl/r/2. We see that H is block di- son current is independent of T when T ≪∆013, Ie is a
agonal due to the fermion-parity symmetry of H. The function of T here.
Hermitian matrices he and ho describe the dynamics of Forthesubspacewithoddfermionparity,theAndreev
the states in the subspaces with even and odd fermion spectrum is given by
parity, respectively.
1
E˜ (φ)= ǫ ǫ2+4∆ ∆ 2 , (10)
± 2 d± d | l− r|
B. The Andreev spectrum and the DC Josephson (cid:18) q (cid:19)
current where ∆ ∆ 2 = t2+t2 2t t cos(φ/2). Hence, the
| l− r| l r − l r
DC Josephson current is of the form
WefirststudytheAndreevspectrumandcalculatethe
resultingDCJosephsoncurrent. Theenergyspectrumof E˜ (φ) E˜ (φ)
+
he is determined by its eigenvalues, which are given by Io(φ)=Io0(φ)tanh" 2−T − #, (11)
1
E (φ)= ǫ ǫ2+4∆ +∆ 2 , (7) where
± 2 d± d | l r|
(cid:18) q (cid:19) 2et t sin(φ/2)
l r
with the corresponding eigenstates denoted by ;φ , Io0(φ)= , (12)
where|∆l+∆r|2 =t2l +t2r+2tltrcos(φ/2). TheAn|±dreeiv − ǫ2d+4|∆l−∆r|2
spectrum will exhibit sharp peaks in the spectral func- is the DC Josephson cuprrent at T =0.
tionofthedot,whichisa4π-periodicfunctionofφ. This
A few comments on the above results are in order.
structure could be measured by STM26 or microwave-
First of all, due to the presence of more than a single
optical experiments27.
Andreev level for each sector with fixed fermion par-
In terms of the formula I = 2e∂F /∂φ, where F is
e e e ity, the current-phase relation at T = 0 deviates from
−
the free energy of the junction with even fermion parity
the usual one proportional to sin(φ/2). Next, for given
atfinitetemperatureT,wegettheDCJosephsoncurrent
φ, I and I have opposite signs, which is similar to
e0 o0
the short topological Josephson junction13,15. Finally,
E (φ) E (φ)
+
Ie(φ)=Ie0(φ)tanh − − , (8) the Andreev spectra for the sectors with even and odd
2T
(cid:20) (cid:21) fermion parity are related to each other through the re-
lation: E˜ (φ 2π)=E (φ). Moreover, they cross each
where
otheratφ±=π±, leadingt±othe fermion-parityanomalyas
2etltrsin(φ/2) the short topological Josephson junction does. The be-
I (φ)= , (9)
e0
ǫ2d+4|∆l+∆r|2 haviourof Ie0 as a function ofφ and Ng is shownin Fig.
p
4
2. Although our results hold only when N N +1/2, this case, the dot is connected to the SC electrodes
g 0
≈
the DC Josephson current already shows the oscillatory through a narrow constriction. We assume that the
behaviour due to the charging energy. width of the constriction at its center allows only a sin-
gle transverse state below the Fermi level. Since the
constriction is formed electrostatically, its boundary is
C. quenched dynamics smooth and do not scatter electrons29. Following the
work of Furusaki and Matveev30, the whole system at
Nowweconsideranabruptchangeoftheflux,i.e. φ= l/r = 1 amounts to a superconductor-normal-metal-
T
φ and φ +φ for t < 0 and t > 0, respectively. This superconductor(SNS)junctionwithafinitechargingen-
0 0 1
amounts to giving an initial state, say |Ψ(0)i = |+;φ0i, ergy. Hence, the Hamiltonian at Tl/r =1 can be written
which is not the eigenstates of H(φ0 +φ1). We would as H0+Hc, where at the energy scale much lower than
liketocalculatetheresultingJosephsoncurrentatt>0. the bulk Fermi energy, H0 can be written as31
The current operator Iˆis defined as
H = vF ∞dx (∂ Θ)2+(∂ Φ)2
Iˆ≡−2e∂∂Hφ . (13) 0 2 Z−∞0 (cid:2) x x (cid:3)
+V dxcos √4πΘ(x)+χ
p l
With the choice of the initial state given above, the
Z−∞ h i
Josephson current at T = 0 for the sector with even +
fermion parity exhibits the behavior of Rabi oscillation: +Vp ∞dxcos √4πΘ(x)+χr , (15)
ZL h i
I (t)=I cos(Ω t+α) , (14)
e 0 e with V =2∆ sin(k a )/(πa ), and
p 0 F 0 0
where I = e(∆ ∆ )c c is the current amplitude,
φqΩue=en=φcyE,+0α+φ≡−. aWE|r−ge[(nl=∆o−tlip−cerǫ∆t2dhr+∗+a)tc4∗+−I|c∆|,−lΩ]+, c,∆±anr=|d2hαi±sa;rtφeh|e+al;lRφpa0ebir,iioafdrneidc- is the charging enHercgy=, wEict(hNˆt−heNegx)a2ct, form of Nˆ sp(e1c6i)-
0 1 0 e fied later. In the above, v is the Fermi velocity, k is
functions of φ. Especially, the periodfor Ω is 4π, which F F
e the Fermi momentum, a is a short-distance cutoff, and
0
is a signature of the zero-energy MBSs.
We would like to emphasize a few points on the above ∆l(r) = ∆0eiχl(r) denotes the p-wave pairing amplitude
in the left (right) SC electrode with ∆ > 0. Moreover,
0
results. First of all, the Josephson current will still ex-
wetakethecenteroftheleftcontactandthatoftheright
hibit the Rabi oscillation even if the change of Φ has
B contact to be located at x = 0 and x = L, respectively.
a finite duration τ. The requirement is that τΩ 1.
e ≪ The bosonic fields Φ and Θ, which obey the commuta-
Next,bytakingintoaccountthedissipationthatwehave
tion relation [Φ(x),Θ(y)] = iθ(y x) with θ(x) being
ignored, the current will eventually decay into its equi- −
the Heaviside unit step function, are related to the right
librium value given by Eq. (9). However, the oscilla-
mover ψ and the left mover ψ in the normal region
tory behavior still exists before the equilibrium value is through+the bosonization formula−:
reached,andthefrequencyΩ canbe measured. Finally,
e
inthepresenceofquasiparticlepoisoning,thestateswith 1
ψ = exp i√π(Φ Θ) .
eJvoesnepahnsdonodcudrfreernmt.ionSipnacreittyhewicllurbroetnhtcoopnetrraibtourteIˆtocotmhe- ± √2πa0 ± ∓
(cid:2) (cid:3)
mutes with the fermion-parity operator, the contribu- This model holds when E v /L, where v /L is the
c F F
≫
tions from the states with different fermion parity will average level spacing in the QD. When are close to
l/r
T
not mix, i.e. no interference appearing in the Josephson one, we have to add a term H to H +H to describe
bs 0 c
current. Therefore,the resulting currentstill exhibits an the weak backscattering at the contacts, where
oscillatorybehaviorwithtwoRabifrequenciesΩ andΩ ,
e o
w4πh-epreeriΩodoic=fuE˜n+ct−ionE˜s−of=φ. Inǫ2dth+is4s|∆enls−e,∆thre|2d.epBeontdhenacree Hbs =vF |rl|ψ+†(0)ψ−(0)+|rr|ψ+†(L)ψ−(L)+H.c.
of the Rabi frequencies onpφ is immune to quasiparticle = vF hr sin[√4πΦ(0)]+ r sin[√4πΦ(L)] .(1i7)
l r
−πa | | | |
poisoning. 0
n o
andr isreflectionamplitudeattheleft(right)contact.
l(r)
WenoticethatH [Eq. (15)]hasbeenusedinRef. 31to
0
III. THE STRONG TUNNELING REGIME
describe the topological SNS junction. In our case, the
Hamiltonianconsistsofanadditionalterm–thecharging
A. The model energy H . We shall see later that the presence of H
c c
will enhance the effects of H , by turning it from an
bs
Now we turn into the strong tunneling regime, i.e. irrelevant perturbation to a marginal one.
G e2/h or 1, where denotes the Withoutlossofgenerality,wemayshiftthe valueofΘ
l/r l/r l(r)
≈ T ≈ T
transmission coefficient of the left (right) contact. In by Θ(x) Θ(x) χ /√4π such that H can be written
l 0
→ −
5
as with q = nπ/L. In the above, a and a obey the
n n †n
canonical commutation relations. Moreover, both com-
H = vF ∞dx (∂ Θ)2+(∂ Φ)2 mutewithqˆandJˆ. InsertingEqs. (23)and(24)intoEq.
0 x x
2 (19) gives
Z−∞0 (cid:2) (cid:3)
+Vp dxcos √4πΘ(x) H =+∞v q a a + πvF Jˆ φ 2. (25)
Z−∞ h i 0 F n †n n 2L − 2π
+V +∞dxcos √4πΘ(x) φ . (18) nX=1 (cid:18) (cid:19)
p
ZL h − i The second term in Eq. (25) describes the dynamics of
the zeromodes,while the firsttermdescribesthe excita-
At the energy scale much smaller than the SC gap ∆ ,
0 tions in the QD.
thepairingtermssuppressthefluctuationsofΘ(x)inthe
regions x < 0 and x > L. Thus, for E ∆ , one may
0
≪
integrateoutthedegreesoffreedomintheseregions,and
B. The effective action of the zero modes
the resulting H becomes
0
v L From Eq. (25), we notice that φ appears only in the
H = F dx (∂ Θ)2+(∂ Φ)2 , (19) zero modes. Thus, to study the Josephson effect, we
0 x x
2 Z0 would like to integrate out the Φ¯ field to get an effective
(cid:2) (cid:3)
Hamiltonianforthezeromodes. Wedothisbytwosteps.
with the boundary conditions
First of all, we notice that H and H depend only on
c bs
φ Φ(0)andΦ(L). Thisobservationsuggestsustointegrate
Θ(0)=0 , Θ(L)= √4π −√πJˆ. (20) φout=Φ¯Φ¯((x0))wanitdhφx =6=Φ¯0(,LL).to get an effective theory for
l r
InEq. (20),JˆisaHermitianoperatorwithintegereigen- To achieve this goal, we turn into the path-integral
values. The boundary conditions (20), in fact, describe formulation. Since the action is Gaussian in Φ¯(x) with
the perfect Andreev reflections at x = 0,L, which is the x=0,L,theintegralcanbe doneexactlyandthe result-
6
characteristicfortheinterfacebetweenthenormalregion ing effective action for φl and φr is given by
and the TSC33.
1 β
tiaNnoowpewraetcoarnNˆw,rwitheicdhowisngitvheenebxypression of the Hermi- Seff = 2Z0 dτ1dτ2φI(τ1)∆−I1(τ1−τ2)φI(τ2)
1 β
Nˆ = 1 Ldx∂xΦ= Φ(L)−Φ(0) . (21) +2Z0 dτ1dτ2φC(τ1)∆−C1(τ1−τ2)φC(τ2)
√πZ0 √π β
λ dτsin[√2π(φ φ ) 2q]
l I C
Hence, Hc becomes − Z0 − −
β
Hc = Eπc[Φ(L)−Φ(0)−√πNg]2 . (22) −λrZ0 dτsin[√2π(φI +φC)−2q]
2E β π 2
Our working Hamiltonian is H = H +H +H with + c dτ φ N , (26)
0 c bs C g
π − 2
H0, Hc, and Hbs given by Eqs. (19), (22), and (17), Z0 (cid:18) r (cid:19)
respectively. Moreover,we shall assume that r 1.
| l/r|≪ where φl = (φI φC)/√2, φr = (φI +φC)/√2, λl/r =
ThemodeexpansionsofΦ(x)andΘ(x)consistentwith −
v r /(πa ), and
F l/r 0
the boundary conditions (20) are of the forms | |
∆ (τ) = D(τ;0,0)+D(τ;0,L) ,
I
qˆ
Φ(x) = +Φ¯(x) , ∆ (τ) = D(τ;0,0) D(τ;0,L) .
C
−√π −
φ x In the above,
Θ(x) = √πJˆ +Θ¯(x) , (23)
(cid:18)√4π − (cid:19)L D(τ;x1,x2) τ Φ¯(τ1,x1)Φ¯(τ2,x2) 0 ,
≡hT { }i
where the Hermitian operator qˆ is conjugate to Jˆ, i.e. is the free propagator of Φ¯, whereτ τ τ and
1 2 τ
[qˆ,Jˆ]=i, and denotestimeorderingalongtheimagin≡ary-ti−meaxis. FTor
the temperature T v /L, we have
F
+ ≫
Φ¯(x) = nX=∞1√qinLcos(qnx)(cid:0)an−a†n(cid:1), ∆˜I(iωl)≈ |ω1l| , ∆˜C(iωl)≈ |ω1l| ,
+
Θ¯(x) = n=∞1√q1nLsin(qnx) an+a†n , (24) wFohuerrieerωtlr=ans2floπrTmwofitAh(lτ)b.eing integer and A˜(iωl) is the
X (cid:0) (cid:1)
6
AttheenergyscalemuchsmallerthanE ,thecharging where
c
energy H suppresses the fluctuations of φ , by setting
c C + β
itsaveragevaluetobe φ = π/2N . Thus,atenergy ∞
h Ci g Q[ρ,q]= exp i dτJ(∂τq ρ)
sflcuaclteusamtiuonchs somf aφlClerbtyharenpElacc,inopgnethmeasycainttteergirnagteteorumtsthine J(τX)=−∞ " Z0 − #
S [Eq. (26)] with its values averaged over the Gaus- + N 1
eff ∞ −
sian fluctuations of φC, yielding the effective action for = expiNlim+ Jj(qj+1−bj+1−qj +bj)
φI JjX=−∞ → ∞Xj=0
N 1 + 
1 β − ∞
S˜ = 2 dτ1dτ2φI(τ1)∆−I1(τ1−τ2)φI(τ2) =Nlim+ eiJj(qj+1−bj+1−qj+bj) ,
Z0β → ∞jY=0JjX=−∞
+ dτ λ˜ei(√2πφI−2q)+H.c. , (27) with Aj = A(τj), ∆τ = β/N, and τj = j∆τ. Moreover,
Z0 h i we have written ρ = ∂τb. By using the Poisson summa-
where γ =0.5772156649 is the Euler’s constant and tion formula
···
λ˜ = 2ieπγ2Ec |rl|e−iπNg +|rr|eiπNg . ∞ δ(x−2πn)= 21π ∞ eimx .
n= m=
This is, in fact, a pe(cid:0)rturbative expansion i(cid:1)n λ , which X−∞ X−∞
l/r
is justified because r 1 and the fluctuations of φ we get
l/r C
| |≪
around its expectation value are gapped.
N 1 +
NowweareinapositiontointegrateoutφI. Wenotice Q[ρ,q]= lim − ∞δ qj+1−bj+1−qj +bj l .
that ∆˜−I1(iωl) = |ωl|, which implies that the scaling di- N→+∞j=0l= (cid:18) 2π − (cid:19)
mension of the λ˜ term is 1. That is, it is a marginal Y X−∞
perturbation. (In the absence of the charging energy We see that Q[ρ,q] provides each time slice a δ-function
Hc, the scaling dimension of Hbs is 2 so that it is an which imposes the constraints bj+1 −bj = qj+1 −qj −
irrelevant perturbation33. In this sense, the large charg- 2πlj. Theintegrationoverbinthepresenceoftheselocal
ing energy enhances the effects of H on low-energy constraintsleadstoaglobaloneb(β) b(0)=2mπwhere
bs −
physics by freezing parts of the charge fluctuations.) m = − Nj=−11lj is an integer. Since ρ(β) = ρ(0), this is
Since rl/r 1, we may integrate out φI by the per- possible only when b(τ) = q(τ) +2mπτ/β. Because b
turbat|ivee|xp≪ansioninλ˜, yieldingthe effective actionfor appearsPin the action only in the guise of ∂ b, we may
τ
the zero modes: rewrite Q[ρ,q] as
+
= + ∞δ , (28) +∞ 2mπτ
0 n Q[ρ,q]= δ b(τ) q(τ) .
I I I
n=1 − − β
X m= (cid:20) (cid:21)
X−∞
where δ =O(λ˜ 2n) and
n Now we are in a position to integrate out b, and the
I | |
partition function Z becomes
β πv φ 2
F
= dτ J( i∂ q)+ J . (29)
I0 Z0 " − τ 2L (cid:18) − 2π(cid:19) # Z = +∞ D[q]e I˜, (31)
−
The partition function of the TSC-QD-TSC junction is mX=−∞Zq(β)−q(0)=2mπ
then given by
where
+
Z = D[q] ∞ e−I . (30) I˜= βdτ L (∂ q)2 i φ ∂ q ++∞δ . (32)
τ τ n
Z J(τX)=−∞ Z0 (cid:20)2πvF − 2π (cid:21) n=1 I
X
Since the eigenvalues of Jˆtake integer values, q must be
Theeffectsofthe term2mπτ/β havebeentakenintoac-
an angular variable. Without loss of generality, we take
count by changing the boundary condition for q(τ) from
0 q < 2π and the periodic boundary condition, i.e.
the periodic one to q(β) q(0)=2mπ. Equation (31) is
≤
q(β)=q(0) in Eq. (30). −
thedesireddualrepresentationofEq. (30). Weshalluse
Itisconvenienttocalculatethefreeenergyintermsof
both in the following.
a dual representation of Eq. (30). To achieve this goal,
we first perform the summation over J. By using the
Hubbard-Stratonovichtransformation, we find that C. The DC Josephson current
+
∞ e−I0 = D[ρ]e−2πLvFρ2+i2φπρQ[ρ,q] , NowweareabletocomputetheDCJosephsoncurrent.
J(τX)=−∞ Z When λ˜ = 0 or Tl/r = 1, the action I˜ is nothing but
7
the one for a particle, of mass L/(πv ) and unit charge, We see that I(0) and I(0) have opposite signs at given T
F e o
movingona unitcirclethreadbya fluxφ, providedthat and φ.
we treat q as the azimuthal angle. In this sense, the When λ˜ = 0, small reflection amplitudes at the con-
6
integer m is the winding number. Hence the partition tacts will modify the DC Josephson current. We shall
function at =1 can be written as calculate the corrections to (0) by the perturbative ex-
l/r
T pansionin λ˜ 2. TheleadingIcontributionarisesfromδ
1
+∞ which takes| t|he form I
Z0(φ)= e−βǫJ , (33)
β
J=X−∞ δ 1= λ˜ 2 dτ1dτ2V(τ1 τ2)cos 2[q(τ1) q(τ2)] ,
I −| | − { − }
leading to the free energy F = TlnZ , where ǫ = Z0
0 0 J
πvF[J φ/(2π)]2 istheenergyoft−hezeromodewiththe where
2L −
quantumnumberJ. TheresultingDCJosephsoncurrent
is of the form V(τ) = ei√2πφI(τ)e−i√2πφI(0)
= Dexp 2π[∆ (τ) ∆ (0)E] .
+ I I
I(0)(φ)= evF ∞ J φ P , (34) { − }
L − 2π J Using the Baker-Hausdorffformula, one may verify that
J= (cid:18) (cid:19)
X−∞
[Jˆ,eiαqˆ]=αeiαqˆ .
wathtehreeePiJge=nstZ1a0tee−βJǫJ. iWs tehneoptircoebtahbaitlittyheofwthhoelezseproecmtroudme This equation indicates that the operator eiαqˆ shifts the
is 2π-periodic in|φ,iand thus I(0) is a 2π-periodic func- eigenvalue of Jˆby α. Thus, the effects of δ 1 is to shift
tion of φ. Moreover, I(0)( φ)= I(0)(φ), which implies the eigenvalue of Jˆby 2. In general, the tIerms in δ n
that I(0)(0) = 0. The eig−envalu−e of Jˆ is the difference willshift the eigenvalue±of Jˆby 2n. As a result, alltIhe
±
between the numbers of the right- and the left-movers. terms δ n do not change the fermion parity.
I
Hence, even and odd values of J correspond to different Ingeneral,aperturbative expansionofthe freeenergy
fermionparities. Bytakingintoaccounttheconservation F can be written as
of fermion parity, the allowed values of J become either +
odd or even integers. Consequently, I(0) turns into a F =F + ∞F ,
0 n
4π-periodic function of φ, a characteristic of topological
n=1
X
Josephson junctions.
To proceed, it is more convenient to use the dual where Fn =O(λ˜ 2n). In particular, we have
| |
representation of Z [Eq. (31)]. By writing q(τ) =
0 β
q¯(τ)+2mπτ/β with q¯(β) = q¯(0) and then performing F =T δ = T λ˜ 2 dτ dτ V(τ)C(τ;φ) ,
1 1 0 1 2
the integration over q¯, we obtain Z0 h I i − | |Z0
where denotesthe averagewithrespecttoZ , τ =
Z0(φ)= 2L +∞ eimφ−2πvLFβ(2mπ)2 . (35) τ1−τ2h,·a·n·id0 0
svFβ
mX=−∞ C(τ;φ)= cos 2[qˆ(τ1) qˆ(τ2)] 0 .
h { − }i
Since T v /L, the dominant contributions arise from
≫ F When T vF/L, C(τ;φ) can be approximated as
the terms with m = 0 and m = 1. Hence, we may ≫
±
approximate Z0 as C(τ;φ) 1 4e−v2FπLβ cosφsin2 2πτ
≈ − β
(cid:20) (cid:18) (cid:19)(cid:21)
Z0(φ)≈sv2FLβ 1+2e−v2FπLβ cosφ . (36) ×exp −2βπLvFτ(β−τ) ,
(cid:16) (cid:17) (cid:20) (cid:21)
Consequently, the DC Josephson current is given by for 0 τ β.
≤ ≤
Collecting the above results, F is given by
1
I(0)(φ) 4eTe−v2FπLβ sinφ , (37)
≈ β 2πv
F 2T λ˜ 2 dττV(τ)exp Fτ(β τ)
which holds when vF/L≪T ≪Ec. 1 ≈ − | | Z0 (cid:20)− βL − (cid:21)
In the above calculation, we do not take into account
the conservation of fermion parity so that I(0) is a 2π- 1 4e−v2FπLβ cosφsin2 2πτ ,
× − β
periodic function of φ. The constraintfrom the fermion- (cid:20) (cid:18) (cid:19)(cid:21)
parity conservation can be included by assuming that J when v /L T E , which leads to the leading cor-
F c
must be even (or odd) integers. With the similar proce- rection to th≪e DC≪Josephson current
dure, we find that
Ie(0)(φ)≈2eTe−2vπFLβ sin(φ/2)=−Io(0)(φ) . (38) I(1) =−16π2EecT2|λ˜|2e−v2FπLβ sinφ .
8
In the above calculation, we do not take into account
theconservationoffermionparity. Forfixedfermionpar-
ity,theonlychangeinF liesinthefunctionC(τ;φ),and
1
we obtain
I =I sin(φ/2)= I , (41)
e 4π o
−
where
πL 4π2 λ˜ 2
FIG. 3: (Color online) The critical current Ic [Eq. (40)] for I4π =2eTe−2vFβ 1− E|c2| !, (42)
aTSC-QD-TSCjunction inthestrong-tunnelingregime. We
set |rl| = |rr| = |r| and measure Ic in units of evF/(2πL). is the critical current with fixed fermion parity. We see
Left: Ic versus the temperature T (in units of vF/(2πL)) at thatIe andIo haveoppositesignsatgivenT,φ,andNg.
Ng =0.2for|r|=0.03(solidline)and|r|=0.1(dashedline). Moreover,the ratio I4π/I2π depends on the temperature
Right: Ic versus Ng at T = 5vF/(2πL) for |r| = 0.03 (solid T in a nontrivial way, but is independent of the gate
line) and |r|=0.1 (dashed line). voltage.
IV. CONCLUSIONS AND DISCUSSIONS
As a result, we find that for v /L T E , the DC
F c
≪ ≪
Josephson current is of the form
We study the DC Josephson effect in a TSC-QD-TSC
I =I sinφ , (39) junction. In the weak-tunneling regime, the Andreev
2π
spectrum becomes simple near the degenerate point of
where the critical current I is given by the QD. This simple structure should appear as sharp
2π
peaks in the spectral function of the dot, which could
2πL 4π2 λ˜ 2 be observed by established experimental techniques. We
I2π =4eTe−vFβ 1− E|c2| !, (40) alsoconsidertheabruptchangeoftheflux,whichresults
in a Rabi oscillation of the Josephson current with fre-
to the leading order in λ˜ 2. We see that the small re- quencies being 4π-periodic functions of φ. We indicate
| | that this feature is immune to the quasiparticle poison-
flection amplitudes at the contacts do not change the
ing. Far away from the degenerate point, the prolifera-
current-phaserelation. Theymanifeststhemselvesinthe
tionofAndreevlevelswillcomplicatethebehaviorofthe
criticalcurrentin two ways. (i) First of all, the smallre-
Josephsoncurrent. Therefore,we do not consider such a
flection amplitudes at the contacts diminish the critical
situation. We notice that the use of the Rabi oscillation
current. (ii) Next, the current acquires the dependence
to extract the signature of the MBSs has been proposed
onthegatevoltagethroughthereflectionamplitudesbe-
inadifferentsetup34. Inthatcase,theQDisside-coupled
cause
to one end of the TSC in a Josephson junction, and it
4π2 λ˜ 2 4eγ 2 is the electron occupation of the dot which exhibits the
| | = r 2+ r 2+2r r cos(2πN ) .
E2 π | l| | r| | l|| r| g Rabi oscillation due to the DC Josephson current.
c (cid:18) (cid:19) In the strong-tunneling regime, we model the dot and
(cid:2) (cid:3)
Hence, at giventemperature, the criticalcurrentreaches the contacts asa one-dimensionalelectronliquid, follow-
the maximal values for half-integer N and the minimal ing the work on the charge transport through an open
g
values for integer N . The behaviours of I versus T QD30. Using the bosonization,wemaycalculatethe free
g 2π
and N are shown in Figs. 3. By comparing with the energy and the DC Josephson current in a perturbative
g
results in the weak-tunneling regime, we conclude that expansion in the reflection amplitudes. The resulting
thecurrent-phaserelationfortheTSC-QD-TSCjunction current-phase relation is of the form I sin(φ/2), iden-
c
can be changed by varying the values of , which can tical to the short Josephsonjunction. The difference lies
l/r
T
be achievedbyadjusting the appliedgatevoltageswhich at the critical current I , which acquires a nontrivialde-
c
form the constrictions. pendenceonthetemperatureandthegatevoltage. Inthe
We notice that I(1) has the same temperature depen- absence of the backscattering terms, the critical current
denceasI(0). ThisreflectsthefactthatH isamarginal weobtainedisidenticaltotheoneforthelongtopological
bs
perturbation, which follows from the large charging en- SNS junction in the temperature range T v /L35.
F
≫
ergy. If there were no charging energy, H would be an In the absence of the charging energy H , the low-
bs c
irrelevant perturbation such that an additional factor T energy effective Hamiltonian is identical to the one for
would appear in I(1). Consequently, the effects of H the long topological SNS junction31,32. The role of the
bs
would be suppressed at low temperatures33. Our analy- chargingenergyistoenhancethe effectsofthe backscat-
sisindicates thatthe effects ofreflectionsatthe contacts teringatthecontactsonlow-energyproperties. Thishas
are enhanced in the presence of a large charging energy. beenshowninthe study ofthe chargetransportthrough
9
the openQD30. There,the backscatteringterms become the current-phase relation of the TSC-QD-TSC junction
relevantperturbationsandanon-perturbativemethodis depends on the transmission coefficients of the con-
l/r
T
warranted. In the present case, thanks to the supercon- tacts. Since the control of the critical current is impor-
ductivity in the bulk, the backscatteringterms turn into tant for the basic elements in large-scale integrated SC
a marginal perturbation, and a perturbative expansion circuits36, the dependence of the current-phase relation
works. on is deserved to investigate.
l/r
T
Intheweaktunnelingregime,weonlyconsiderthecase
nearthedegeneratepointoftheQDsothatonlytwolev-
els of the dot are involved. It is interesting to study how
the Andreev spectrum and the resulting current-phase
Acknowledgments
relation evolve upon moving far away from the degener-
ate point. On the other hand, our results in the strong
tunneling regime hold only for T v /L. Thus, the The work of Y.-W. Lee is supported by the National
F
≫
DCJosephsoneffectatthetemperaturerangeT v /L Science Council of Taiwan under Grant No. NSC 102-
F
≈
requires further study. Finally, as our analysis suggests, 2112-M-029-002-MY3.
∗ Electronic address: [email protected] 20 B. Tarasinski, D. Chevallier, J.A. Hutasoit, B. Baxevanis,
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