Pair-amplitude dynamics in strongly coupled superconductor–quantum dot hybrids 2022 Markus Heckschen.pdf

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MarkusHeckschen1andBj¨ornSothmann1
1TheoretischePhysik,Universit¨atDuisburg-EssenandCENIDE,D-47048Duisburg,Germany
(Dated:October20,2021)
Weconsiderathree-terminalsystemconsistingofaquantumdotstronglycoupledtotwosu-

areal-timediagrammaticapproach,wecalculatethedynamicsoftheproximity-inducedpairam-

,it

ofthepairamplitudeoccuralsowhenthesystemisperiodicallydrivenbothintheadiabaticand
fast-
currentthroughthedottodemonstratethatitisanexperimentallyaccessiblequantity.

modewasreportedinRefs.[9,10]inmaterialswhere
superconductivitycoexistswithacharge-densitywave.
Superconductivityisanactivefieldofresearchinmod-
ThelattercouplestheHiggsmodetophononssuchthat
erncondensed--
itcanbedetectedasanadditionalpeakintheRaman
damentalperspectivebecauseitprovidesanexampleof

quantumcoherenceatthemacroscopicscalewhichgives
possibletodirectlyexciteandobservetheHiggsmode
risetophenomenasuchastheJosephsoneffectwherea
bypump-probespectroscopy[11].Theexperimentally
dissipationlesschargecurrentflowswithoutanapplied
observedoscillationoftheelectromagneticresponsecan
biasvoltage[1].Furthermore,itprovidesanimpor-
beexplainedtheoreticallyintermsoftheAndersonpseu-
tantingredientforapplicationssuchassuperconducting
dospindynamicsinatwo-dimensionalBCSmodel[12]
qubitsinfuturequantumcomputers[2–4].
andbyagauge-invariantkineticequationofsupercon-
Thetransitionfromanormalmetalintothesuper-ductivity[13].TheHiggsmodehasalsobeenprobedby
conductingstateisasecond-orderphasetransitioninitsthird-ordernonlinearopticalresponsewhichprovides
whichtheU(1)symmetryisspontaneouslybrokenbyacleardistinctionbetweentheHiggsmodeandquasi-
thecondensationofelectronsintospin-singlet,s-waveparticleexcitations[14,15].Recently,theHiggsmode
Cooperpairsashasbeenestablishedbythemicroscopichasalsobeendetectedinastrongly-interactingfermionic
BCStheoryofsuperconductivitybyBardeen,Coopersuperfluid[16].Furthertheoreticalstudiesfocusedon
andSchrieffer[5].TheassociatedorderparameterisatheHiggsmodeinunconventionald-wavesuperconduc-
iφ
complexnumber∆ewhichcorrespondstothemacro-tors[17,18]andonthecouplingoftheHiggsandLeggett
-modesintwo-gapsuperconductors[19].Currentreviews
eterisadynamicquantitywhichcanexhibitcollectiveontheHiggsmodeinsuperconductorscanbefoundin
oscillationssuchastheNambu-GoldstonemodeandtheRefs.[20,21].

superconductingphaseϕwhilethelatterisagappedSofar,theHiggsmodehasmainlybeenstudiedin
excitationoftheabsolutevalue∆ofthesuperconduct-,itwasdemonstrated
ingorderparameterwithminimalexcitationenergy2∆.thattheleakageofquasiparticlesintoanormalconduc-
DuetotheAnderson-Higgsmechanism,thefrequencytorleadstoastrongdampingoftheHiggsmodeina
oftheNambu-Goldstonemodeisshiftedtotheplasmasuperconductor-,the
frequencyinbulksuperconductors[6–8].Therefore,thecoherenttunnelingofelectronsacrossthejunctiongives
Higgsmodeistheonlycollectivelow-energyexcitationofrisetotheoccurrenceoftwonewHiggsmodes[22].Sig-
thesuperconductingorderparameterandstableagainstnaturesoftheHiggsmodeinthechargecurrentofa
-normalmetaltunneljunctionwerere-
arXiv:[cond--hall]19Oct2021TheexperimentaldetectionoftheHiggsmodeischal-portedinRef.[23].Superconductorscaninducesu-
-
neutralandcouplesonlyquadraticallytoexternalelec-
,largefieldstrengthsareinterestingquestionaboutthenatureofthedynamicsof
-inducedpairamplitudeinthenonsuper-
,thisdynamics
THzrangewheresuitablesourceanddetectorshavebeenhasbeenstudiedinaweaklycoupled,temperature-biased
,onehastoexcludesuperconductor-quantumdothybrid[24].Itwasshown
theadditionalexcitationofquasiparticlesbyCooperpairthatboththeabsolutevalueandthephaseoftheinduced
breakingwhichrequiresthesameenergyastheexcita-,astrong
2
dampingduetotunnelingofquasiparticlestypicallyover-
comestheoscillationsunlessthesystemisdrivenexter-Iz

thepairamplitudedynamicsinaquantumdotstrongly
coupledtosuperconductorsintheregimewherethesu-Ix

thisparameterregime,tunnelingofquasiparticlesbe-ΓS+UΓS
+iϕiϕ
tweenthedotandthesuperconductorsisexponentially∆e2∆e−2
suppressedwhileatthesametimeastrongproximityef-

weaklycouplingthequantumdottoanadditionalnor-
ΓN

forpulseddrivinghasbeenpreviouslystudiedinlightofµN
potentialapplicationsinquantumcomputing[25].Us-
ingareal-timediagrammaticapproachwhichenablesus
toaccountfortheproximityeffectandstrongCoulomb
interactionsonthedotinanonequilibriumscenario,we
-
-levelquantumdotisstronglycoupledtotwogrounded
Wefindthatinbothscenariospronouncedoscillationssuperconductorswithsuperconductingphasesϕη=±ϕ/2by
ofthepairamplitudeoccur,thusovercomingthelimita-tunnelcouplingstrengthsΓSL=ΓSR=,the
tionsoftheweak-couplinglimit[24].quantumdotisweaklytunnelcoupledtoanormalconductor
withelectrochemicalpotentialµ-

.
,weintroduceourtheoreticaldescriptionbasedon
areal--
-
,
=ϕR=ϕ/
superconductors−areassumedtohavethesamecritical
temperatureandaregrounded,µS=

voltageµN=eVtothenormalreservoir.
Weconsiderathree-terminalsetupcomposedoftwoThesingle-level,spin-degeneratequantumdotwith
conventionalBCSsuperconductorscoupledsymmetri-levelpositionisdescribedbythedotHamiltonian
callytoasingle-levelquantumdotwhichinadditionis
X†††
weaklycoupledtoanormalconductor;=cσcσ+Uc↑c↑c↓c↓,(3)
couplingtothesuperconductorsinducessuperconduct-σ
ingcorrelationsonthequantumdotviatheproximity
effectwhilethecouplingtothenormalmetalallowsforwherethesecondtermsdenotestheon-siteCoulombre-

.
ThesetupisdescribedbythetotalHamiltonianThetunnelingHamiltonianisgivenby
X
XH=ta†c+.,(4)
H=Hη+Hdot+Htun.(1)tunηηkσσ
ηηkσ
ThefirsttermdescribesthethreeleadsηSL,SR,Nwithtunnel-matrixelementtηwhichweassumetobe
intermsofamean-fieldBCS-Hamiltonian∈{}-
mentsarerelatedtothetunnel-couplingstrengthΓη=
X†X
iϕη
2NN
H=aa+∆eaa+..2πtηρη,withdensityofstatesinthenormalstateρη.
ηη,kηkσηkσηηk↑η−k↓||
kσkDuetothesymmetriccouplingtothesuperconductors,
(2)wehaveΓSL=ΓSR=ΓS.
Thefirsttermcorrespondstothekineticenergyofelec-Inthiswork,weconcentrateonthedynamicsofthe
-inducedsuperconductingpairamplitudeon

,∆istheabsolutepairsinsidethesuperconductinggap,ourfocusisonthe
,it
assumetobeequalinbothsuperconductorsandwhichisareasonableapproximationtoconsideroursetupin
3
theinfinite-gaplimitwherethesuperconductinggapisThematrixelementsofthereduceddensitymatrixare

Inthislimit,wecantakeintoaccountthecouplingbe-
Pχ1χρχ.
tweenthedotandthesuperconductingreservoirsexactlyχ2=1red2(8)
h||i
sinceeven-andodd-paritydotstatesdecoupleandthe
systemcanbemappedontoanoninteractingonewithInordertoproperlydescribethedynamicsofthepair
renormalizedlevelpositions[26].amplitude,wehavetotakeintoaccountthediagonal
Pχ
Theexactresummationresultsinaneffectivedotdensitymatrixelementsχwhichdescribetheprobabil-
χ
HamiltonianH=HHwhereforasymmetricitytofindthedotinthestateaswellastheoffdiagonal
effdotpP−P+
couplingtothesuperconductors−thepairingHamiltonianelements+and−.Thelatterarecrucialtocapture
isgivenbythefastcoherenttunnelingofCooperpairswhichoccurs
onatimescalegivenbyΓS[31].
ϕ††Thetimeevolutionofthereduceddensitymatrixel-
Hp=ΓScos(c↓c↑+cc).(5)
2↑↓ementsisgivenbyageneralizedmasterequationofthe
form
H
Theeigenbasisofeffisgivenby,whichdescribet
|↑i|↓idiZ00
thesingly-occupiedthedotwithanspin-uporspin-downχ1X0χ1χ10χ10
Pχ(t)=(Eχ1Eχ2)+dtW0(t,t)P0(t).
2~χ2χ2χ2
electronanddt−−00−∞
χ1χ2
"#(9)
1rδrδ
=10+1d,(6)Thefirsttermontherightsideofthemasterequation
|±i√2∓∓2A|i±2A|
particular,itaccountsforthecoherentCooperpairdy-
whichdescribeacoherentsuperpositionofemptynamicsduetothecouplingtothesuperconductingleads.
anddouble-occupieddotstates,0,dwithA=Thesecondtermdescribesthedissipativecouplingtothe
q|i|iχχ0
δ
2211
+χ,theleveldetuningδ=2+
2||22
characterizesdeviationsfromtheparticle-holesymmet-givenbyirreducibleselfenergiesofthequantumdotprop-
ricpointδ=0,andeigenenergiesEσ=andE±=agatorontheKeldyshcontour.
δ
2A[26,27].Theexcitationenergiesofthequan-Thegeneralizedmasterequationcanbetransformed
tum±dotaregivenbythedifferencesoftheeigenenergiesintoaphysicallyintuitiveformbyintroducingapseu-
(E±Eσ)andcanbeidentifiedastheAndreevbounddospindegreeoffreedomwhichcharacterizesthecoher-
±stateenergies−intheinfinite-gaplimitentsuperpositionoftheemptyanddoublyoccupieddot
statevia
0U0
E0=γ+γγ,γ1..(7)d0d0
A,γ,γAP0+PdP0PdPdP0
2∈{±}Ix=,Iy=i−,Iz=−.
222
Thesuperpositionofthezero-occupied0anddouble-(10)
occupieddotdbecomeslargestforδ|=-gaplimit,thedynamicsofthepseudospin
detunings,the|statesiareapproximatelygivenbythedecouplesfromthedotoccupationsanditstimeevolution
emptyanddoubly-occupied|±-typeequation
dI
~=ARIBI.(11)
dt
-TIMEDIAGRAMMATICS−·−×
InEq.(11),Adenotestheaccumulationvectorwhich
Inordertodescribethepair-amplitudedynamicsondescribesaccumulationofpseudospinonthequantum
thequantumdot,wemakeuseofareal-timediagram-
maticapproach[28,29]inanextensiontosuperconduct-
ΓSϕP
cos0γf(EA,γ0,γ)
ingreservoirs[30].Thekeyideaofthereal-timedia-A2γγ=±
ΓNΓSϕP
grammaticsistointegrateoutallnoninteractingdegreesAcosγγ0=±γϕ(EA,γ0,γ),
=πA2
−4Pδ1
offreedomandtodescribethequantumdotwithare-01+γf(EA,γ0,γ)
γγ=±2A2
duceddensitymatrixρred..Ourmethodallowsusto−
-−1
wheref(x)=1+exp[(xµN)/kBT]denotesthe
thermore,itcapturesthecouplingbetweenthedotand{−h}i
Fermi