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Josephson ladders as a model system for 1D quantum

phase transitions

Matthew Bell, Benoît Douçot, Michael Gershenson, Lev Ioffe, Aleksandra

Petković

To cite this version:

Matthew Bell, Benoît Douçot, Michael Gershenson, Lev Ioffe, Aleksandra Petković. Josephson ladders

as a model system for 1D quantum phase transitions. Comptes Rendus Physique, Centre Mersenne,

2018, 19 (6), pp.484-497. �10.1016/j.crhy.2018.09.002�. �hal-01952997�

Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

Quantum simulation / Simulation quantique

Josephson

ladders

as

a

model

system

for

1D

quantum

phase

transitions

Une

chaîne

supraconductrice

comme

simulateur

de

transitions

de

phases

quantiques

en

une

dimension

Matthew

T. Bell

a

,

Benoît Douçot

b

,

Michael

E. Gershenson

c

,

Lev

B. Ioffe

b

,

∗

,

Aleksandra Petkovi ´c

d

a_{DepartmentofElectricalEngineering,UniversityofMassachusettsBoston,Boston,MA02125,USA} b_{LPTHE,UniversitéPierre-et-Marie-Curie,75005Paris,France}

c_{DepartmentofPhysicsandAstronomy,RutgersUniversity,Piscataway,NJ08854,USA} d_{LPT,IRSAMC,UniversitéPaul-Sabatier,31062Toulousecedex4,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline19October2018 Keywords:

Quantumsimulations Quantumphasetransitions ArraysofJosephsonjunctions TransversefieldIsingmodel Mots-clés :

Simulationsquantiques Transitionsdephasequantiques ChaînesdesjonctionsJosephson Chaîned’Isingquantiqueenchamp transverse

Weproposeanovelplatformforthestudyofquantumphasetransitionsinonedimension (1DQPT). ThesystemconsistsofaspeciallydesignedchainofasymmetricSQUIDs;each SQUIDcontainsseveralJosephsonjunctionswithonejunctionsharedbetweenthe nearest-neighborSQUIDs. We develop the theoretical descriptionofthe low-energypart ofthe spectrum.Inparticular, weshowthatthesystemexhibitsaquantumphasetransitionof Isingtype.Inthevicinityofthe transition,thelow-energyexcitationsofthesystemcan bedescribedbyMajoranafermions.Thisallowsustocomputethematrixelementsofthe physicalperturbationsinthe low-energysector.Inthemicrowaveexperimentswiththis system,we exploredthe phaseboundaries between the orderedand disordered phases andthecriticalbehaviorofthesystem’slow-energymodesclosetothetransition.Dueto theflexiblechaindesignandcontroloftheparametersofindividualJosephsonjunctions, futureexperimentswillbeabletoaddresstheeffectsofnon-integrabilityanddisorderon the1DQPT.

©2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Nousproposonsunenouvelleplateformepourl’étudedestransitionsdephasequantiques enunedimension.LesystèmeconsisteenunechaînedebouclesdeSQUIDasymétriques spécialementconfigurées :chaqueSQUIDcontientplusieursjonctionsJosephson,dontune partagéeavecleSQUIDvoisin.Desexpériencessousmicro-ondesélectromagnétiquesnous ontpermisd’explorerleslignesdetransitionentrephaseordonnéeetphasedésordonnée, ainsique lecomportementcritique desétatsexcitésde plusbasseénergieau voisinage decettetransition.GrâceàlaflexibilitédelaconfigurationdesSQUIDSetàlapossibilité decontrôler individuellementles paramètresde chaque jonction Josephson,cesystème

*

Correspondingauthor.

E-mailaddress:ioffe@lpthe.jussieu.fr(L.B. Ioffe). https://doi.org/10.1016/j.crhy.2018.09.002

1631-0705/©2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

permettrad’explorer,lorsdeprochainesexpériences,leseffetsdelanon-intégrabilitéou dudésordresurcettetransitiondephasequantiqueenunedimension.

©2018PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Theideaofquantumsimulationsemergedintheearly1980soutoftherealizationofthefundamentaldifficulty of em-ulatingcomplexquantum systemusingclassicalcomputers. Over time,thisidea hasevolvedintoa sub-fieldofquantum computing[1].Similartothequantumcomputers,quantumsimulatorsarebasedonthenetworksofquantumbits(qubits), but, incontrasttothefully-fledged quantumcomputers, quantumsimulators donotemploy discretegate operationsand error correction codes. The quantum simulators with tunableparameters are designed to emulate only certain types of Hamiltonians.However,itispossibletoshow thatavery generalHamiltoniancanbe simulatedby aseeminglyrestricted classofspinchainswithXXandYYinteractions[2,3].Onehopesthatsuchsimulatorswillfacilitatedesigningnovel quan-tumsystemsandexploringphenomena andregimesinaccessibleinthepast.Furthermore,quantumsimulatorsenablethe experimental studyofquantumannealinginthe contextofadiabatic quantumcomputation. Modernquantum simulators arebasedonseveralplatforms,whichincludecoldatoms[4,5],coldions[6],andsuperconductingqubits[7].

Ourresearch focusesondesigning artificialspinsystems – tunable1D Josephsonarrayswithcontrolled interactions – thatemulatethequantum1Dmodels.Theintegrablemodelofa1DIsingspinchaininthetransversemagneticfieldserves asaparadigminthecontextofnonequilibriumthermodynamicsandquantumcriticalphenomena[8,9].Boththetransverse field IsingandXYmodels,beingrelevanttoabroadrangeofphysicalsystems,playedacrucialrole intheunderstanding of quantum phase transitions [9]. These models have generated a formidable body of theoretical activity over the past fiftyyears.Recently,thesemodelsplayedacrucialroleinthedevelopmentofquantumannealingtechniquesandadiabatic quantumalgorithms[10].

Inthe past,theexperimental studyofquantumspin dynamicsin1D hasbeen largelylimitedto microscopicspinsin condensed-matter systems. It has been demonstrated that such quasi-1D spin materials asLiHoF4 and CoNb2O6 can be

continuously tunedacrossthequantumphase transition(QPT)[11–13].Thoughtheseworksopenedupnewvistasinthe studies oftransverse field Isingmodel, theexperimental realizationof1D quantum spin modelsin well-controllableand tunablesystems remains a challenge.Indeed,asan experimental tool, the quasi-1Dspin systems insolidsare limitedin severalrespects: (i) theinter-chain interactionsare notnegligibly weak,and, thus,thesesystemsare inevitablyquasi-1D, (ii) theexchange interactionsbetweenthenearest-neighborspinscannotbe varied,(iii) theexchangeinteractions arethe sameforallpairofspins,whichdoesnot allowforexploringtheeffectofdisorderandphaseboundarieswithoutadding a significant amount ofimpurities, and(iv) the available experimental tool for thesesystems – scatteringofneutrons – interactsonlywithanarrowclassofexcitations.Flexibilityinthedesignofartificialspinsystems,whicharefreefromthese limitations,facilitatesbridgingthegapbetweenthetheoreticalstudyofidealspinchainsandtheexperimentalinvestigation ofbulkmagneticsamples.Inparticular,thisflexibilityallowsonetoaddressanimportantissueoftheeffectsofdisorderon thestaticsanddynamicsoftransversefieldspinmodels.Recently,thetransverse-fieldIsingmodelwasrealizedinthechain ofartificial andfully-controllable spins – eight flux qubits withtunablespin – spin couplings [10]. We pursuea similar approachusingspeciallydesignedone-dimensionalJosephsonladders.

The paperis organized asfollows.In Section 2,we introduce theJosephson ladder asa novelplatform forthe study of the 1D quantum phase transitions. The theory of these quantum systems is presented in Section 3. The microwave experimentsonthecharacterizationofthespectraofthesenovelquantumsystemsaredescribedinSection4.Theoutlook isprovidedinSection5.

2. 1DJosephsonladders

TheJosephsonladdersdesignedforthestudyofthe1DQPTrepresenta1DchainofcoupledasymmetricSQUIDs(Fig.1). The unit cell of the ladders is similar to that of the Josephson arrays developed as superinductors (the elements with the microwaveimpedancemuch greater thanthe resistancequantum h

/

e2_{) [14]. Eachunit cell contains asingle smaller}

junctionwiththeJosephsonenergyEJSinonearmandthreelargerJosephsonjunctionswiththeJosephsonenergy EJL in

theotherarm(Fig.1a).Theadjacentcellsarecoupledviaonelargerjunction.

Theladdersare characterizedby theratioofJosephson energiesofthe largerandsmallerjunctions,r

≡

EJL

/

EJS.Forr

greater thanthe criticalvalue r0,the Josephsonenergyof theladder asafunction ofthe phase

ϕ

across theladder has

onlyoneminimumat

ϕ

=

0 regardlessofthemagnitudeoftheexternalmagneticfieldperpendiculartotheladder’splane. Thevalueofr0 dependsontheladderlengthandonthestrengthofquantumfluctuations(inthequasiclassicalcase,r0

=

5

foran infinitelylongladder) –seeSection 3.Theregime r

>

r0,wherethe laddercan becharacterized bythe Josephson

inductanceLJ

∝ (

d 2_E

J(ϕ)

dϕ2

)

−1[15],wascomprehensivelystudiedin[14].

Ifr

<

r0,thedouble-minimadependence EJ

(

ϕ

)

isrealizedoverarangeofthemagneticflux



C

<



<



0

− 

C where



C

(

r

)

isthe critical flux,



0

= π

h

¯

/

e is the flux quantum.Figs. 1c, d show that ata fixed r

<

r0,the spontaneous

sym-Fig. 1. Theunitcells,potentialenergy,andthephasediagramoftheJosephsonladder[14].Panel (a):theunitcellsoftheladderincludesmallerandlarger JosephsonjunctionswithJosephsonenergiesEJSandEJL,respectively.Allcellsarethreadedbythesamemagneticflux;thephasedifferenceacrossthe ladderisϕ2−ϕ1=ϕ.Panel (b):thedesignoftheladder.BottomelectrodesoftheJosephsontunneljunctionsareshowninblue,topelectrodesinred. Panel (c):theJosephsonenergy EJ(ϕ)ofaladderwiththeratioEJL/EJS≡r<r0calculatedforthreevaluesoftheflux:=0 (dashedcurve),= C (solidcurve),and= 0/2 (dash-dottedcurve).Theenergyisperiodicinphasewiththeperiod2π,hereweshowonlyasinglebranch.Panel(d):the phasediagramoftheladdersonther-plane.Forr<rc(),thebroken-symmetryphaseisformedatC<<0− Casaresultofthephasetransition oftheIsingtype.Theboundaryofthisordered-stateregionisshownasthesolidblackcurve.Theextensionofthefluctuationalregimeisoftheorderof

δr≈0.1 forrealisticarrays.

metry breaking occursat a criticalvalue ofthe flux



C

(

r

)

and the single-minimum energy EJ

(

ϕ

)

is transformed into a

double-minimafunction.TheJosephsoninductancedivergeswith



approaching



C

(

r

)

(hence,theterm“superinductance”);

quantumphasefluctuationseliminatethedivergency.

In thedouble-minima regime, the directionofcurrentsin theladder unit cells induced byan external magneticfield canbe viewedastwostatesofthe1/2pseudo-spins.Asthepotentialbarrierbetweenthesestatesincreases,thequantum phasefluctuationsdecreaseandtheglobalbroken-symmetrystateemerges.Theappearanceofthedouble-wellpotentialin asinglecelldoesnotimplyaglobalphasetransitioninthewholesystem.Thelarge-scalequantumfluctuationsdestroythe globalorderparameterifthebarrierbetweentwominimaistoosmall.Theglobalorderparameterisformedatrc

()

<

r0

asaresultofthephasetransitionoftheIsingtype(rc

()

correspondstothesolidblackcurveinFig.1d).Theextensionof

the fluctuationalregimeisoftheorderof

δ

r

≈

0

.

1 forrealistic arrays.Notethat themagneticfieldthat drivesthe ladder acrosstheQPT,B

<



0

/

A,where A istheareaoftheunitcell,isveryweak(

∼

1 G inourexperiments).

We emphasizethatJosephsonladderswith EJL

/

ECL



1 haveexponentiallysmallprobabilityofphaseslipprocessesin

which thephaseacross theladderchanges by

∼

2

π

.Thisimpliesthatthestaticoffsetchargeson individualislands have noeffectonthequantumstatesoftheladder.Weshallneglecttheeffectsofthesechargesandphaseslipprocessesinthe followingdiscussion.Theestimateforthephaseslipratecanbefoundin[14].

The low-energyphysicsofthe ladderscloseto theQPTcanbe mappedon the

ϕ

4 _{model,which isrelevanttoa wide}

rangeofphysicalphenomena,fromquark confinementtoferromagnetism.Nearthecriticalpoint,thismodelisequivalent totheintegrablemodelof1DIsingspinchaininthetransversemagneticfield.Flexibilityofthearraydesignandtunability of theparametersofindividual Josephsonjunctions offerseveraluniqueopportunitiesforexploration ofthe1D quantum phasetransitions.Forexample,thisnovelplatformfacilitatesthestudyofeffectsofnon-integrabilityanddisorderonQPTs andtheemergenceofergodicbehaviorinalmostintegrablequantumsystems.

3. TheoryoflongJosephsonladders

3.1. Effectiveactionofthelow-energymodesofthelongchains

As we showbelow, thelongchaindisplays thephase transitionthatoccursasa functionofthe ratior

=

EJL

/

EJS ata

fixed fluxthroughaunit cell,oratafixedr asafunctionofthefluxthrougheachunitcell.Asusualinphasetransitions, thecriticalpropertiesarenotsensitivetotheparameterthatdrivesthetransition,sowefocusonthetransitiondrivenby r below.Smallmodificationsintroducedbytransitiondrivenbymagneticfieldwerediscussedin[14].Themostimportant isthefactthat,closetor

=

r0,theeffectivedistancefromthecriticallineisaquadraticfunctionof



− 

0

/

2.

We start byreviewingthe propertiesofshortchains inwhichone can neglectthephase variations betweenadjacent elements anditsvariationintime.ConsiderasmallportionofthefullyfrustratedladdershowninFig.2.Thetranslational invarianceimpliesthat thesolutionisdescribedbytwophasesacrosslargerjunctions:

α

atthe“vertical”junctionsand

β

atthe“horizontal”ones.Theenergyperunitcellthatcontainstwolargerandonesmallerjunctionsisgivenby

E

(

α

, β)

= −

EJL

(

cos

α

+

cos

β)

−

EJScos

(

π −

2

α

− β)

(1)

At EJL



EJSthefirstterminEq. (1) dominatesandE

(

α

,

β)

hasaminimumat

(

α

,

β)

= (

0

,

0

)

.Theexpansionnearthis

pointgives E(2)

(

α

, β)

=

EJS 2

[

α

β

]



r

−

4

−

2

−

2 r

−

1

 

α

β



(2)

Fig. 2. Arrayschematics.Panel(a):exactlyatfullfrustration,theclassicalgroundstateofthearraycorrespondstoatranslationallyinvariantpatternof phasedifferencesα=α,β= β.Awayfromthefullfrustration,theperiodisdoubledandphasesα =α,β = β.Intheabsenceofthegroundcapacitance, thekineticenergyofthearraydependsonlyonlocalphasedifferencesαandβ.Foragenericfluctuation,thephasesindifferentunitcellsbecome different,αj =α,etc.Panel (b):thepresenceofanon-zerogroundcapacitanceleadstoasmalltermthatdependsontheglobalphase=j<i(αj+ βj) thatbecomesrelevantforlongchains.Panel (c)showstheenergyasafunctionofthecoordinatesα,β.Closetotheminimumtheenergychangesslowly inthedirectionindicatedbytheblackline.

whichisaquadraticformwitheigenvalues EJL and

(

r

−

5

)

EJS.Thesecondeigenvaluechangessignastheratior

=

EJL

/

EJS

decreases,signalingtheinstabilityandappearanceofanontrivialgroundstate.Thelongchainsdisplaythepropertiessimilar tothecriticalIsingmodelatr slightlylessthanthecriticalvalueoftheratio,r0

=

5,asexplainedbelow.1 Exactlyatr0

=

5,

thefunction E

(

α

,

β)

becomes flatalongtheeigenvector

(

2

,

1

)

,butremains steepintheorthogonal direction(seeFig. 2c). Weintroducenewcoordinatesinthe“flat”and“steep”directions:

γ

= (

2

α

+ β)/

√

5,

δ

= (

α

−

2

β)/

√

5.Fig.1cshowsthe energyplottedalong the flatdirection. Neglectingthe phase deviationsin thesteep direction, thetotal phaseacross the unitcellisgivenby

ϕ

eb

=

3

/

√

5

γ

.

Atr

>

r0,thegroundstate valuesofphases

γ

and

ϕ

,

γ

0,and

ϕ

0,are zero.Deviationsofthe totalphase from

ϕ

0

=

0

resultinthequadraticincreaseoftheenergy

E(2)

(

ϕ

)

=

EJL

2

δ

r

γ

2

where

δ

r

= (

r

−

r0

)/

r. Attheoptimalpoint(

δ

r

=

0),thephasestiffnessvanishes andthephasefluctuationsarelimitedby

thenext-orderquarticterm:

E(4)

(

γ

)

=

1 24



25

−

17 r 25



EJS

γ

4

≈

0

.

9 EJS

γ

4 (3)

whereinthelastequalitywe replacedthenumericalcoefficientby itsvalue atr

=

r0.Atsmallerr

<

r0,thegroundstate

corresponds toanon-zerovalueof

|

γ

0

(

r

)

|

∝

√

δ

r. Theappearanceofanon-zero

γ

0 impliesa phasetransitionthatbreaks

Z2 symmetry

γ

→ −

γ

.

Closetothephasetransition,thephasefluctuationsbecomerelevant.Wederivetheactionofthesefluctuationsassuming thattheyareslowinspaceandtime.Webeginwithspatialfluctuations.Forageneralspace-varyingphase,onegets

E

= −

EJL



i

(

cos

α

i−1/2

+

cos

α

i+1/2

+

cos

β

i

)

−

EJS



i

cos

(

π −

α

i−1/2

−

α

i+1/2

− β

i

)

1 _{Short-scalequantumfluctuationsmayrenormalizethevalueofr}

0byshiftingitdown:r0=5− δrsh.Thiseffectisnumericallysmall(δrsh0.1)for thechainsstudiedexperimentallyandqualitativelyirrelevant,soweneglectitinthefollowingdiscussion.

where we labelthephasesof each wireby thecoordinates oftheir centers, sothat the phasesofhorizontalwires have integer coordinatesandthecentersofthe verticaloneshavehalf-integercoordinates.Expandingin thephases

α

i

,

β

i



1

andtakingtheFouriertransform,wegetthequadraticpartoftheenergyofthefluctuationwithmomentumk:

E(2)

(

k

)

=

EJS 2



k [

α

k

, β

k]



r

+

2

ξ

k

ξ

k

ξ

k r

−

1

 

α

₋k

β

₋k



(4)

where

ξ

k

= −

2cos

(

k

/

2

)

. Expanding it for small k and leavingonly the contribution of the slowmode

α

=

2

/

√

5

γ

and

β

=

1

/

√

5

γ

,wegetfortheenergyperunitcell

E(2)

(

k

)

=

1 2EJL



3 5 r

(

∇

γ

)

2

_{+ δ}

_r

_γ

2



(5)

Thekineticenergyperunitcellisduetothecapacitancesofthelargerandsmallerjunctions.WeassumethatCL

=

rCSand

expresstheresultintheunitsofECL

= (

2e

)

2

/

CL:

T

=

1 2 ECL



_d

α

dt



2

+



d

β

dt



2

+

1 r



d

(

2

α

+ β)

dt



2

=

(

r

+

5

)

2 r ECL



d

γ

dt



2 (6)

Awayfromthecriticalpoint,theinteractionbetweenfluctuationscanbeneglected.Inthisregime,thespectrumofthe slowfluctuationscharacterizedby(5),(6) isgivenbytherelativisticdependence

ω

2

= (

c k

)

2

+

m2 (7)

c2

=

3

5

(

r

+

5

)

EJLECL

m2

=

r

δ

r

5

(

r

+

5

)

EJLECL

Notethatatallr,thevelocityisrelativelysmallbecauseitisduetotheJosephsonenergyofthesmallerjunctionsand thechargingenergyofthelargerones, sothatthewavedispersionisalwayssmallintheJosephsonchainofthistype.As theoptimalpoint

δ

r

=

0 isapproached,thegapinthespectrumclosesandthefluctuationsbecomemorerelevant.Atlarge EJL

/

ECL



1,thewidthofthefluctuationalregime,

δ

r∗,wheretheeffectsoftheinteraction(3) arerelevant,issmall.Inthe

followingdiscussionofthisregime,weassumer

≈

r0

=

5,sothatthefullactionbecomes

S

=

L dt dx L

=

1 ECL



d

γ

dt



2

+

EJL 2



3 25

(

∇

γ

)

2

_{+ δ}

_r

_γ

2

₊

9 25

γ

4



Rescalingthevariables

x

=

√

3 5 s t

=



2 EJLECL

τ

γ

=

5 3

√

2

η

wereduceittoafullydimensionlessform

S

=

1 2g



_d

_η

d

τ



2

+ (∇

η

)

2

+ δ

r

η

2

₊

1 2

η

4

d

τ

dx (8) where g

=

3

√

6 5



ECL EJL (9)

Weestimate thewidthofthefluctuationalregimewherethespectrum (7) isnotvalidbycomputingthefluctuational correctionstotheaction(8).Theleadingcorrectiontothecoefficientof

η

4 _termis

9 g 8

π δ

r

sothatthewidthofthefluctuationalregime

δ

r∗

≈

g. 3.2. MappingtotheIsingmodel

Thetransitiontotheorderedstate breaks Z2 symmetryandbelongsto thesameuniversalityclass astheIsing model.

Intheorderedstate,thefield

η

acquiresaverage

η

= ±

η

0 where

η

0

= (−δ

r

)

1/2;inthisregimethelow-energyexcitations

aredomainwallswiththestaticenergy

0

=

1

√

2 1 g

δ

r 3/2

Becausethetimeneededtocreateanddestroyadomainwallis

τ

= δ

r−1/2,theactioncorrespondingtothisprocessis S

∼ δ

r

/

g,whichagaintellsusthatthewidthofthefluctuationalregimeis

δ

r∗

≈

g.

Outsideofthefluctuationalregime,thekineticenergyofthedomainwallmovingwithvelocity v is

v2

2

√

2 g

δ

r

3/2

implyingthemassm

= δ

r3/2

_/

√

_{2 g.TheLorentzinvariantequationcompatiblewiththeselimitsforthedomain-wallenergy}

spectrumis

(

k

)

= (

m2

+

k2

)

1/2

Intheorderedphaseandinsidethecriticalfluctuationalregime,themodelshouldbesimilartothequantumIsingmodel

H

=

T



i

σ

_ix

+

J



i

σ

_iz

σ

_iz

wheret and J aresmoothfunctionsof

δ

r.ComparingthespectraofthedomainwallintheIsingmodelandaction(8) we seethat

J

≈

m

/

2

T

≈

1

2ma2

intheorderedphaseat J



T

wherea isthedistancebetweentheIsingspins.Thedistancea issetbythedomainwall sizea2

₌

₁

_/

_|δ

_r

_|

_.

TheIsingmodelhastransitiontotheorderedstateat J

=

T

.Bytranslatingitintotheparametersoftheoriginalmodel, weseethatthetransitionhappensat

δ

rc

= −

c g

wherec

∼

1.

Closetothetransitionintotheorderedstate,theexcitationsoftheIsingmodelareMajoranafermions

c†p

(

k

)

= (−

1

)

 j<knp(j)

_σ

+ p

(

k

)

np

(

j

)

=



σ

z j

(

p

)

+

1



/

2

Qualitatively, these excitationsdescribe the domain walls in theIsing modelthat propagate witha constant velocity exactlyatthetransition,i.e.theyhaverelativisticspectrum(Fig.3).Thefermionicnatureoftheseexcitationsdescribesthe exclusionprincipleofdomainwalls.Generally,awayfromthetransitionpoint,thespectrumoftheseexcitationsisthesame asforfreebutmassiveonedimensionalfermions:

(

k

)

=

2



T

2

₊

_J2

₋

₂

_T

_{J cos}

₍

_ka

₎

₍₁₀₎

We now discuss mapping ofthe matrix elements of the physicaloperators to the operators of the Ising model. The physicalcurrentsthroughtheindividualjunctionsare J

=

EJLsin

α

, J

=

EJLsin

β

and J

= −

EJSsin

(

2

α

+ β)

.Thus,thecurrent

Fig. 3. CharacteristicspectraoftheexcitationsintheIsingmodelatandslightlyawayfromthetransition(blueandyellowcurves,respectively).Atlow energies,thespectrabecomerelativisticwiththemassthatvanishesatthetransition.Inthepresenceofanon-zerogroundcapacitance,thespectraasa functionofk remainqualitativelysimilar.

operator isgenerallygiven by J

=

cJEJL

η

,wherecJ

∼

1 isa numericalcoefficient that dependsonthe particularcurrent

studied.Inthecriticalregimeinwhichlocally

η

= ±

η

0,thecurrentoperatorisdirectlyrelatedtotheIsingvariable

ˆ

Jeff

=

cJEJL

η

0

σ

z (11)

External flux noises interactdirectly withthe current operators by Hnoise

= δ

ext

ˆ

J . Belowwe discussthe effect ofthis

interactiononthedecayanddephasingofMajoranamodeswithspectrum(10).

The charge operator associatedwithlarger andsmallerjunctions are QL1

=

id

/

d

α

, QL2

=

id

/

d

β

, QS

=

id

/

d

(

2

α

+ β)

.

These charges interactwith thepotentials created by stray chargeson each island. As before, we focus on theeffect of this interaction with the soft modesdescribed by the Ising variables. Consider a single unit cell described by the local Hamiltonian H

= −

ECL 4



d d

γ



2

+

1 2EJL



+δ

r

γ

2

+

9 25

γ

4



This Hamiltonianhastwo low-energystates



±

(

γ

)

thatcorrespond tosymmetric andantisymmetricstates thatbecome eigenstates ofIsing’s operator

σ

x after mappingto the Ising model. Thesestates are split by the energy

δ

E

=

E−

−

E₊, whichbecomes

δ

E

=

2t intheIsingmodel.Assumingthat

δ

E



√

ECLEL,i.e.thattheIsingmodesarewellseparatedfrom

therestofthespectrum,wecanevaluatethematrixelementofthechargeoperator Q γ

=

id

/

d

γ

betweenthesestates:



+



Qγ

−=

i

T

Ec

η

0

Thesymmetryofthestatesguaranteesthatallothermatrixelementsofthechargearezero,thus

ˆ

Qeff

=

T

Ec

η

0

σ

y (12)

Notethatthematrixelementofthecharge operatorcontainstwofactorsthataresmallintheregimewherethemodel (8) canbemappedtoIsing’smodel:

η

0



1 and

T /

Ec



1.

3.3. Matrixelementsofthephysicaloperatorsclosetothecriticalpoint

Close tothecriticalpoint, thematrixelements ofboththeexternalcharge andfluxare givenby operatorsthatact on theeffectiveIsingvariables(11),(12).Weevaluatetheeffectoftheseoperatorsonthelow-energy(Majorana)modesofthe criticalIsingmodel.Tocomputethematrixelementsofthe

σ

iy operatorbetweentheground-stateandlow-energyexcited

states,itisconvenienttousetheJordan–Wignertransformationintheform:

σ

x i

=

1

−

2 c † ici (13)

σ

_iz

=

⎛

⎝



i−1 j=1

(

1

−

2 c†_jcj

)

⎞

⎠ (

ci

+

c†_i

)

(14)

σ

_iy

=

⎛

⎝



i−1 j=1

(

1

−

2 c†_jcj

)

⎞

⎠

i

(

ci

−

c†_i

)

(15)

HereitisassumedthatwehavealadderwithN unitcells,sotheindexi runsfrom1toN.NotethattheIsingHamiltonian ischangedintoitsoppositeafterperforminga

π

rotationaroundthex axisofthespinsononesublattice(e.g.,fori even). Doingthischangesthedispersionrelationinto:

(

k

)

=

2



T

2

₊

_J2

₋

₂

_T

_{J cos}

₍

_ka

₎

₍₁₆₎

whichis convenient,becausethe minimumofthisdispersion relationoccursatk

=

0.Introducing Ai

=

ci

+

c†i and Bi

=

c†_i

−

ci allowsusthentowritetheIsingHamiltonianas:

H

= −

T

N



i=1 AiBi

−

J N−1



i=1 BiAi+1 (17)

Thesingle-excitationcreationandannihilationoperators

γ

_k†,

γ

k arewrittenas:

γ

k

=

N



j=1

(φ

k_jAj

+

χ

kj Bj

)

γ

_k†

=

N



j=1

(φ

k_jAj

−

χ

kj Bj

)

(18)

wherethecoefficients

φ

k_j and

χ

k

j arereal.Theseoperators aredeterminedbythecondition that

[

H

,

γ

k

]

= −

(

k

)

γ

k,which

gives:

2 t

φ

k_j

−

2 J

φ

k_j+1

= −

(

k

)

χ

k_j (19)

2 t

χ

k_j

−

2 J

χ

k_j−1

= −

(

k

) φ

k_j (20)

Withtheopen boundaryconditions,wehavetosolvetheseequationstogetherwiththeconstraints

χ

k

0

=

0 and

φ

k N+1

=

0.

Theallowedvaluesofk aredeterminedby:

sin

(

kaN

)

sin

(

ka

(

N

+

1

))

=

T

J (21)

Notethatk

=

0 and k

= π

arebothexcluded,becausethey leadtovanishingamplitudes

φ

k_j and

χ

k

j.Therefore,wehaveto

searchforrealsolutionsintheinterval0

<

k

<

π

.ThereareexactlyN suchsolutionswhen

T /

J

>

_NN₊₁,whichcorresponds totheparamagneticphase.Intheothercase

T /

J

<

_NN₊₁,wegetonlyN

−

1 realsolutionsandonecomplexsolutionk

=

i

κ

, where

κ

isgivenby:

sinh

(

kaN

)

sinh

(

ka

(

N

+

1

))

=

T

J (22)

andthecorrespondingenergyis:

(

i

κ

)

=

2



T

2

₊

_J2

₋

₂

_T

_{J cosh}

₍

_κ

_a

₎

₍₂₃₎

whichliesinsidethegapoftheinfinitechain.Therealsolutionshavetheform:

φ

k_j

= φ

₀ksin

(

ka

(

N

+

1

−

j

))

sin

(

ka

(

N

+

1

))

(24)

χ

k_j

= −φ

₀k

(

k

)

2 J sin

(

kaj

)

sin

(

ka

)

(25)

andthebound-statewave-functionisobtainedbyanalyticalcontinuationfromk toi

κ

.Canonicalfermionic anticommuta-tionrelationsaresatisfiedprovided:

N



j=1

φ

k_j

φ

k_j

=

N



j=1

χ

k_j

χ

k_j

=

δ

kk 4 (26)

Whenk

=

k,theserelationsfixthevalueof

φ

k₀ intheaboveexpressionsfor

φ

k_j and

χ

k

j.Completenesscanthenbeusedto

Aj

=

2



k

φ

k_j

(

γ

k

+

γ

k†

)

(27) Bj

=

2



k

χ

k_j

(

γ

k

−

γ

_k†

)

(28)

Notethatthek sumshavetoincorporatetheboundstatewhen

T /

J

<

_NN₊₁.

Thegroundstateofthesystem

|

0



isdeterminedbytheconditionthat

γ

k

|

0

 =

0 forallallowedk’s.Asshownearlier,we

havetoevaluatethematrixelementof

σ

_iybetweenthegroundstateandlowlyingstates.TheJordan–Wignerrepresentation for

σ

_iy canalsobewrittenas:

σ

_iy

= −

i

⎛

⎝



i−1 j=1 AjBj

⎞

⎠

Bi (29)

so it isthe product ofan odd numberof fermionic operators. Therefore,

σ

_iy couples the groundstate withthe excited states containingan oddnumberofelementaryfermionic excitations. Thesimplestofthesestatesareofthe form

γ

_k†

|

0



. Evaluation ofthematrixelements isstraightforward,usingWick’s theorem.Forthis,we needthebasiccorrelators, which areeasilydeterminedfrom (27), (28),andthecompletenessrelations:



0

|

AiAj

|

0



= δ

i j (30)



0

|

BiBj

|

0



= −δ

i j (31)



0

|

AiBj

|

0



= −

4



k

φ

k_i

χ

k_j (32)



0

|

Bi

γ

_k†

|

0

 =

2

χ

kj (33)

BecausethecorrelationsofAjoperatorswiththemselvesarepurelylocal,andsimilarlyforthecorrelationsofBj operators,

we seethatupon computing



0

|

σ

_iy

γ

_k†

|

0



,each Aihastobepairedwitha Bj,and

γ

_k† hastobepairedwiththeremaining

Bjoperator.Thisleadstothedeterminantalformula:



0

|

σ

_iy

γ

_k†

|

0

 = −

i DetCk_i (34)

Here,Ck_i isani

×

i matrixgivenby:

Ck_i

=

⎛

⎜

⎜

⎜

⎜

⎝

G1,1 G1,2 G1,3

...

G1,i G2,1 G2,2 G2,3

...

G2,i

...

...

...

...

...

Gi−1,1 Gi−1,2 Gi−1,3

...

Gi−1,i Dk₁ Dk₂ Dk₃

...

Dk_i

⎞

⎟

⎟

⎟

⎟

⎠

(35)

where Gi,j

= 

0

|

AiBj

|

0



and Dk_i

= 

0

|

Bi

γ

_k†

|

0



,which aregivenby Eqs. (32) and (33), respectively.Such determinantal

ex-pressions fortheIsing formfactors arewell known, see,e.g., [16].Fortheir actual evaluationon afinite-size chainwith open boundary conditions,we used numerics. The results are shown in Fig. 4. As one might expect, the flux operator proportional to

σ

z _{acquires large matrixelements inthe ordered state. Becausethe groundstate ina finite systemis a}

symmetriccombinationofupanddownferromagneticallyorderedstates,thenon-zeromatrixelements correspondtothe 01transitions.Similarly,thechargeoperatoracquireslargematrixelementsbetweenthegroundandthefirstexcitedstates in thedisordered state oftheIsing model.Inboth cases,thesematrixelements implythe decayrateofthefirst excited state.Thematrixelementsofthehigherenergystatesdisplaysimilarbehavior.

3.4. Effectofanon-zerogroundcapacitance

In a realistic ladderof Josephsonjunctions, each elementof theladder hasa non-zero groundcapacitance, asshown schematicallyinFig.2b.Forinstance,intheladdersimplementedexperimentally,theratioCG

/

CL

≈

0

.

05–0

.

1.Thoughsmall,

such capacitancehassignificanteffectonthelow-energyspectrumbecauseitintroduces thetermsintheeffectiveaction oftheunitcellthatdependonthetimederivativeofthetotalphase:

δ

L

=

1 2 ECG



d



dt



2 (36)

where ECG

= (

2e

)

2

/

CGandthetotalphase



isrelatedtothelow-energymodevariable

γ

by



=

3

/

√

Fig. 4. MatrixelementsoftherelevantphysicaloperatorsbetweenthelowestenergystatesclosetothequantumcriticalpointoftheIsingmodel;from lefttoright:thematrixelementofthedimensionlesschargeoperatorσy_{betweentwoloweststates,thesamematrixelementbetweenthegroundand} thesecondexcitedstate,andthematrixelementoffluxnoiseσz_{betweenthegroundstateandthelowestexcitedstate.}

Thefullkineticenergyofthelow-energymodesbecomes

T

=

(

r

+

5

)

2 r ECL



1

+

κ

2 k2

 

d

γ

dt



2 (37) where

κ

2

=

9 r 5

(

r

+

5

)

ECL ECG



1 (38)

istheinversecharacteristiclengthassociatedwiththegroundcapacitance.

Theappearanceofthelength

κ

−1 _{(38) impliesthattheeffectiveCoulombinteraction betweenthechargesisscreened}

atdistancesL

>

κ

−1_{,intermsofthechargevariable,qγ conjugatedto}

_γ

_{,theenergy(37) becomes}

T

=

r ECL 2

(

r

+

5

)

k2 k2

₊

_κ

2q 2 γ (39)

Inthelinearregimewherenon-lineareffectscanbeneglected,thisadditionchangesthespectrum(7) tobecome

ω

2

₌

(

c k

)

2

+

m2

1

+

κ2

k2

(40)

that displays additional softening atlow-wavevectorsc k

 (

c

κ

,

m

)

:

ω

≈

m

κ

−1_{k. Thesoftening of thespectrum (40) is}

observableonlyifitoccursoutsideofthecriticalregime,i.e.at

δ

r

>

g,whichcorrespondsto

κ

>

g

≈ (

ECL

/

EJL

)

1/2.

Inorderto evaluatetheeffectofthe groundcapacitance onthelow-energy modesinthe criticalregime,we evaluate the effectofthe additionalkinetic energy (36) on theeffective actionof theMajorana quasiparticles.It is convenientto representtheactionasduetotheinteractionwiththescalarfield

φ

:

δ

L

=

id

η

dt

φ

+

EG 2



(

φ)

2

+

κ

2

φ

2



(41)

where EG

= (

2

/

5

)

ECG.The last termin(41) isdueto thescreeningofthe Coulomb interactionatshortscales discussed

above. Byrepeating the samearguments that led usto (12) for the charge operator, the additional termsin Lagrangian density(41) canbetranslatedinto

δ

H

=

T

η

0

σ

ix

φ (

xi

)

+

EG 2



(

φ)

2

+

κ

2

φ

2



(42)

fortheeffectiveIsingmodelthatdescribesthevicinityofthetransitionpoint.

Inthefermionicrepresentation,thesetermsimplyweakresidualinteractionbetweenfermionicdensities

σ

x

i

=

1

−

2c † ici.

Thisinteraction leads totheself-energy correction,

(

k

)

,tothe fermionicGreenfunction. Theimportant propertyofthe long-rangeinteractionbetweenspinsorfermionicmodes(18) isitsdependenceonthemomentumk butnoton

ω

. Physi-cally,itisduetotheinstantaneousandlong-rangenatureofCoulombinteraction.Suchinteractionleadstotheself-energy correctionsthat dependonthe momentumbutnot onthefrequencyandthus violatethe Lorentzinvariance ofthe low-energyspectra.Qualitatively, theeffectof theCoulomb interactionis toattract thespin flips (fermiondensities)atshort scales.Theappearanceofthefrequency-independent

(

k

)

impliesthattheminimumoftheenergy

ω

k

(

r

)

occursatdifferent

r fordifferent k.

Intheexperimentdiscussed inSection4,we studythespectrumdependenceon theflux,not on r.Thisisequivalent to the motion in the horizontal directionin the phase diagram of Fig. 1d instead of the vertical one. The shift in r of thepositionoftheminimaimpliesthatthecorresponding parabolasatwhich

ω

k

(

r

,

)

hasaminimumareshiftedup for

Fig. 5. Simplifieddiagramofthe microwavesetup [14].Panel (a):schematicdescription oftheon-chipcircuit.TheJosephsonladder,incombination withthecapacitorCK,formsaresonatorthatiscoupledwiththereadoutLC resonatorviathekineticinductanceLCofanarrowsuperconductingwire. Panel (b):theon-chipcircuitlayoutofthetestedcircuitinductivelycoupledwiththemicrowave(MW)feedline.Panel (c):simplifiedcircuitdiagramofthe measurementsetup.

Theanalyticalexpressionsfortheshiftof

ω

k

(

r

,

)

canbeobtainedclosetotheIsingcriticalpoint.Exactlyatthecritical

point ofthe Ising modelwithout corrections(42), thefermionic densityhaspower-lawcorrelators, which translatesinto power-lawcorrelations



σ

₀x

σ

_rx



=

1

π

2_r2

whichcanbedescribedasthecorrelatorofthefreebosonicfields, s.Inthishydrodynamicdescription,theadditionofthe terms(42) doesnotchangethequadraticnatureoftheenergy.TheGreenfunctionD

(

ω

,

k

)

oftheBose fieldacquiresaself energy

(

k

)

=

(

T

η

0

)

2 EG 1 k2

₊

_κ

2

that hasnofrequencydependence.Thisselfenergycorrectionhasexactlytheeffectdiscussedabove:itshiftstheposition of singularityin D

(

ω

,

k

)

away fromthe

ω

=

(

k

)

line, whichimpliesthe shiftinthespectrum ofelementary(fermionic) excitations.Inparticular,thepositionsoftheminima,rmin,of

ω

k

(

r

,

)

asafunctionofr fordifferentk occuratdifferent r.

Theeffectismaximalfork

∼

κ

atwhich



∼

cg thatleadstotheshiftinrmin

∼

g.

4. Experimentaldata

Our microwave measurements were designedtoexplore the spectrum ofthe low-energymodesof Josephsonladders near the 1D QPT andthe “lifetime” of these modes. The experimental set-up for these spectral and time-domain mea-surements hasbeen described inRefs. [17,18]. Briefly, inthese experiments,the Josephsonladder was coupled withthe lumped-elementreadoutresonatorviathekineticinductanceLC ofanarrowsuperconductingfilm(Figs.5a, b).Theladders,

the readoutcircuits, andthe microwave (MW) transmissionline onthe chipwere fabricated using multi-angle electron-beam deposition of Al films through a lift-off mask (for fabrication details, see Refs. [14]). The in-plane dimensions of theJosephsonjunctionsvaried between0

.

1

×

0

.

1

μ

m2 and0

.

3

×

0

.

3

μ

m2; theareaofaunitcellwas 15

μ

m2.Theglobal magneticfield,whichdeterminesthefluxesinallsuperconductingloops,hasbeengeneratedbyasuperconductingsolenoid. Inthedispersivetwo-tonemeasurementsoftheladderspectra,theresonanceofthereadoutresonatorwasmonitoredat theprobefrequency f1,whiletheladderwasexcited bythesecond-tone(“pump”)frequency fS (Fig.5c).Themicrowaves

weretransmittedthroughthemicrostriplinecoupledwiththeJosephsonladderandthe LC resonator.Theamplifiedsignal wasmixedbymixerM1withthelocaloscillatorsignalatfrequency f2.Theintermediate-frequencysignalat



=

f1

−

f2

=

30 MHz was digitized by a 1 GS/s digitizing card.The signal was digitallymultiplied by sin

(

t

)

andcos

(

t

)

, averaged over an integernumberofperiods,anditsamplitude A andphasewere extracted.The referencephase(which randomly changes when both f1 and f2 are varied in measurements) was found usingsimilar processingof the low-noise signal

provided by mixer M2 anddigitized bythe second channel ofthe ADC. Excitation ofthemodes resultedin achange of the ladderimpedance[19];thischangewas registeredasa shiftoftheresonanceofthereadoutresonatorprobed at f1.

Fig. 6. Two-tonespectroscopyofthelow-energyinternalmodesoftheladderswithr≤r0asafunctionofthemagneticfluxintheunitcells.Panel (a):the datafora24-cellladderwithr=3.2.Theresonancemodesareperiodicinthefluxwiththeperiod0.Twocriticalpointsarelocatedsymmetricallywith respectto= 0/2.Atthecriticalpoints,thesingle-minimumdependenceEJ(ϕ)istransformedintoadouble-minimafunction;thiscorrespondstothe QPTbetweenthe“paramagnetic”and“ferromagnetic”phases.Notethedifferenceinthewidthoftheresonancesinthe“paramagnetic”and“ferromagnetic” phases(shownastheerrorbarsfortwodatapoints).Panel (b):thedatafora92-cellladderwithr=3.2.Theresonancefrequenciesofseverallow-energy modesareshownnearthecriticalpoint.Withapproachingthecriticalpointat≈0.450weobservedpronouncedsofteningofthelow-energymodes oftheladder.Theresonancesintheferromagneticphaseareverybroad(seethetext),sotheaccuracyoffindingthecriticalpointsislow.Theminima positionscanbeextractedreliablyfromtheremainingpointsbyfittingwithapolynomialcurve.Arrowsindicatethecriticalpoints,C(k),thatareshifted fordifferentmodesduetothelong-rangeinteractionsbetweenthearray’sunitcells.Inthelimitofaninfinitelylongchain,weexpectthecriticalpointto belocatedat=0.460.

Thelow-energymodesweremeasuredasafunctionofthefluxintheladder’sunitcellswithinthefrequencyrangeofour microwavesetup0.5–20 GHz.

Inthetime-domainexperiments,wehaveobservedRabioscillationsfora“qubit”formedbytheladderandashunting capacitance CK.Theladdermodeswereexcited byashort(

∼

0

.

02–10

μ

s)MWpulseatthesecond-tonefrequency fS and

the populationof theexcited level was recordedat theendof each pulse;thedata were averaged overmany repetitive measurements.TheRabitimeforthelowest-energymodeexceeded2

μ

s forthearrayswith92unitcells.Thisobservation demonstrates that themulti-junction arrayscan be considered asquantum (notlimited by decoherence)systems over a relativelylongtimescale.Thesemeasurementswillbediscussedelsewhere;belowwefocusonthespectroscopicdata.

The resultsofmeasurements oftheresonancefrequencies forseverallow-energymodesoftheladder withr

<

r0 are

showninFig.6.Thenominalparameters2 _{ofthejunctionsinthischainwere}

EJS

=

4

.

8 K, EC S

=

0

.

41 K

EJL

=

15

.

6 K, ECL

=

0

.

14 K (43)

Fortheseparameters,weestimateg

≈

0

.

1.ThegroundcapacitanceinthisexperimentwasCG

∼

0

.

05CL,sothecharacteristic

lengthduetoCoulombinteractionsis

κ

−1

_≈

₅

_×(

_{unit celllength}

₎

_{.Fortheseparametersoneexpectstoobservetherelativistic}

spectrumofthe low-energymodeswithc

∼ (

10–20

)

[

GHz

]

×

a0,wherea0

∼

3

× (

unit celllength

)

; thiswouldleadtothe

frequencydifference betweenthelowest modes



f

=

1–2GHz, ingood agreementwithour observations.The frequency shift(definedasthedifferencebetweentheminimalfrequencyofamodeanditsfrequencyat



C

(

0

)

)duetotheCoulomb

interactionsisapproximately

∼

0

.

3



f ,whichiscompatiblewiththesmallvalueof

κ

≈

0

.

2

× (

unit celllength

)

−1.

The resonance frequencies depend on the flux periodically with the period



= 

0. We have observed the mode

softening with approaching the quantum criticalpoints, symmetrically positioned withrespect to the flux



0

/

2. Thisis

the key feature of the transverse-field Ising model. According to this model, the phaseson both sides of the quantum criticalpoint(thequantumparamagnetandferromagnet,intheIsingspinterminology)arecharacterizedbydifferenttypes of excitations. Inthe paramagnetic phase, therelevant excitationsare flips of individual spins. These excitationsbecome gaplessexactlyatthecriticalpoint.Incontrast,theexcitationsintheordered(ferromagnetic)phaseare thedomainwalls (ordescribedaskinks)betweendifferentgroundstates(“vacua”).Eventhoughanysuperconductorisabosonicsystem,the lowest-energyexcitationsintheladdersareexpectedtohavefermionicnature;thehigher-energyexcitationsarecomposite particlesmadeofthesefermions(similartotheappearanceofquarksandmesons).Theemergenceofnon-trivialexcitations nearthequantumphasetransitionandtheirpropertiesisoneofthemainthemesofthepresentandfutureresearch.

ThegradualshiftoftheminimaofdifferentmodesshowninFig.6bisduetoanon-zerogroundcapacitanceoftheunit cellsthatresultinweakbutnon-negligiblelong-rangeinteractionsbetweenthepseudo-spinsinagreementwiththetheory presentedinSection 3.The non-zero CG isnot a“bug” buta“feature”:variation ofthegroundcapacitance ofindividual

cells enables fabrication ofthe devicescharacterized by differenttypes ofcouplingbetween theunit cells, which makes possibletheemulationofdifferentquantummodelssuchasIsingandXYmodelsintransversemagneticfields.Interactions

2 _{Thejunctionenergiesgivenbelowwerecomputedbyusingthedataforthejunctionarea,measuredresistanceofthetestjunctionsandAmbegaokar–} Baratoffrelation.Thiscomputationgivesreliablevaluesfor EJLand ECL,butsignificantlyoverestimatesthevalue oftheJosephsonenergyfor smaller junctions.

Fig. 7. Thesingle-tonemeasurementsforalong(92-unit-cell)chain[panel (b)istheblow-upofthedatawithintherange[0.4</0<0.6].Theavoided crossingsbetweenthelow-energymodesandthereadoutresonatormodeareclearlyobservedinthe“paramagnetic”phase(/0<0.45,/0>0.55), wherethereadoutresonanceissharp(theresonancewidthf<1 MHz).Inthe“ferromagnetic”phase(0.45</0<0.55),theresonanceissmeared (f≈4 MHz)duetotransitionsbetweenthelow-energymodesinducedbythefluxnoise(seethetextformoredetails).

betweenthecellsofJosephsonarrays(“spins”)canbetunedbychangingtheJosephsonenergiesofthelargerandsmaller junctions.Thistuningremainsachallengeforothersolid-statesystems[8,9].

The accuracyoftracingthelow-energymodesinthe“ferromagnetic”phaseismuchlower thanthatinthe “paramag-netic” phase (see Fig.6b).Fig. 7showsthat inthesingle-tone measurements withalong 92-unit-cellchain,the readout resonanceisdramaticallybroadenedbetweenthecriticalpoints.Forthisreason, inthetwo-tonemeasurements,wecould not accurately determine thepositions oflow-energy modesforthischain inthe ferromagneticphase.We attribute this resonance“smearing” toastrongmagnetic (flux)noiseinoursystem. Asshowninsection3.3,intheordered (ferromag-netic)phase,theeffectofthechargenoiseonthetransitionsbetweenthelow-energystatesisweaker,whereastheeffect ofthefluxnoiseismuchstrongerthanthatinthedisordered(paramagnetic)phase.Notethatthedecayisproportionalto thesquare ofthematrixelementsshowninFig.4,sothedecayduetothefluxnoiseisenhancedbyafactorof

∼

106 _in

theorderedphase,whilethedecayduetothechargenoiseisreducedbyafactorof

∼

103.Theeffectofthecharge noise isfurthersuppressedbyasmallfactorthattranslatesthematrixelementoftheIsingoperatorsintothematrixelementof thephysicalcharge(theeffectisoppositeforthefluxnoise).Bycomparingequations(11),(12) relatingthedimensionless matrixoperators showninFig.4tothephysicalnoise,we seethatthedecayratedueto thechargenoise isadditionally decreasedbyafactor

(

t

/

Ec

η

0

)

2

 δ

rc,whereasthedecayrateofthefluxnoiseisenhancedbythefactor

(

EJL

η

0

/

ω

)

2,where

ω

isthemodefrequency.Thismakestheeffectofthefluxnoiseontheorderedsideofthetransitionverylarge,whilstthe effectofthechargenoiseremainsmoderate.

5. Conclusionandoutlook

Controllable-by-design interactions between the pseudospins implemented astwo statesof the unit cells ofperiodic Josephson laddersenabled usto studythephaseboundariesbetweenthe orderedanddisorderedphasesandthe critical behaviorclosetothistransition.Inparticular,thestudyofmicrowavepropertiesallowedustocharacterizethelow-energy spectrumofthissystem.

Close to thetransition, the low-energy degreesofthissystemare described by the

ϕ

4 _{model[9], which displays the}

transitionbetweentheordered anddisordered phasesthatbelongstothe Isinguniversalityclass. IntheIsing model,the low-energyexcitationsareMajoranaquasiparticles,thusourobservationofthelow-energymodesofthelongladderscan beviewedasadirectprobeofMajoranaexcitationsinthissystem.Notethat,incontrasttothespinchainswhereonlypairs ofMajoranaparticles canbe producedandstudiedbyscatteringtechniques,Josephsonladderswill allowustostudy the full low-energyspectrum,inparticularthefermionicexcitations. Anappealinganalogyisprovidedbythe particlephysics inwhichfermionicexcitationsarequarks,whilepairofthosecorrespondtopions.Inthisanalogy,thespectrumshownin Fig.6correspondstoquarkswhilethespectrumstudiedinspinchains –topions.

Thedevelopedexperimentalplatformcanhelptoaddressvariousfundamentalissues.First,disordercanbecontrollably introducedinthis1Dsystem,andthisenablesthestudyoftheappearanceofintermediateglassynon-ergodicphases. Sec-ond, theplatformallows forengineeringinteractions betweenpseudospins that violateexact integrability oftheeffective Ising modelthat describesthesystematlarge scales.Forinstance, thegroundstate capacitanceresults inlong-range at-tractions betweenkinksintheIsingmodelintheorderedstate.Thisallows onetostudyhowergodicbehaviorreappears inalmostintegrablequantumsystems.

WeenvisionthefutureworkthatwilladdresstheemergenceofthesymmetriesnearthecriticalpointintheJosephson ladders.Bymeasuringmicrowaveresponsesoftheladdersindifferentphasesandextractingtheexcitationspectraandtheir matrix elements,one can fullyexplorethe behaviorofthissystemnearthe criticalpointandextract thecriticalindices. Byintroducing controllablescatteringinthejunctionparameters, onecan exploretheeffectsofdisorderonthequantum phasetransitionsin1D.

Anotherexcitinglineofresearchistheeffectofnon-integrabilityontheexcitationdecayanddephasing.Because,inthe criticalregime,thelowestenergyexcitationsareequivalenttotheonesoftheeffectiveIsingmodelatthecriticalpoint,the integrabilityimpliesthattheseexcitationshaveaninfinitelifetime.Inrealisticarrays,theintegrabilitybecomesapproximate

andexcitationsacquire afinite,albeit long,lifetime. Intheextensivetime-domainmeasurements,thedecoherenceratein the 1D Josephson ladders can be extracted fromRabi oscillations and Ramsey fringesas a function of proximity to the critical point. We expect that the residual interaction betweenquasiparticles leads to much faster decoherencerates at higherenergy.Thesemeasurementsaddressanimportantissueoftheemergenceofclassicalityinclosedquantumsystems. Apotentialextensionoftheseexperimentsistheintentionalgradualvariationoftheunitcellparametersintheladders. Theidea istoproduceladderscharacterized byasmooth transitionregionbetweenthestateswithbrokenandunbroken symmetry.Thepositionofthephaseboundary(ordistancebetweentwoboundaries)withinaladderwithgradualchange intheposition-dependentparameterr

(

x

)

canbevariedbythemagneticfieldthatcontrolstheproximityofindividualcells tothecriticalpoint.Weexpectthattheboundarybetweensingle-well(“paramagnetic”)anddouble-well(“ferromagnetic”) phasessupportsthezero-energy(Majorana)modethatcanbeprobedbyspectroscopicmethods.A crucialquestionthatwe plantoaddressisthemechanismsofthedecoherencerelevantforthismode.

Finally,bycombiningfewladderstogether,onecangettheexperimentalrealizationofveryexoticquantumobjectssuch asthetwo-channelKondoproblemwithits1

/

2ln2 entropyduetoexactlydegenerateMajoranaquasiparticles [20].

Acknowledgements

The work at Rutgers was supported by the NSF award DMR-1708954 and ARO awards W911NF-13-1-0431 and W911NF-17-C-0024. The work at the University of Massachusetts Boston was supported in part by NSF awards ECCS-1608448andDUE-1723511.

References

[1] Naturephysics,Naturephysicsinsight –quantumsimulation,2012.

[2]G.DelasCuevas,T.S.Cubitt,Simpleuniversalmodelscaptureallclassicalspinphysics,Science351 (6278)(2016)1180–1183. [3]T.Cubitt,A.Montanaro,S.Piddock,UniversalquantumHamiltonians,arXiv:1701.05182,2017.

[4]I.Bloch,J.Dalibard,S.Nascimbène,Quantumsimulationswithultracoldquantumgases,Nat.Phys.8(2012)267–276. [5]A.Mazurenko,etal.,Acold-atomFermi–Hubardantiferromagnet,Nature545(2017)462.

[6]R.Blatt,C.F.Roos,Quantumsimulationswithtrappedions,Nat.Phys.8(2012)277.

[7]A.A.Houck,H.E.Tureci,J.Koch,On-chipquantumsimulationswithsuperconductingcircuits,Nat.Phys.8(2012)292.

[8]A.Dutta,G.Aeppli,B.K.Chakrabarti,U.Divakaran,T.F.Rosenbaum,D.Sen,QuantumPhaseTransitionsinTransverseFieldSpinModels:FromStatistical PhysicstoQuantumInformation,CambridgeUniversityPress,Cambridge,UK,2015.

[9]S.Sachdev,QuantumPhaseTransitions,CambridgeUniversityPress,Cambridge,1999.

[10]M.W.Johnson,M.H.S.Amin,S.Gildert,T.Lanting,F.Hamze,N.Dickson,R.Harris,A.J.Berkley,J.Johansson,P.Bunyk,E.M.Chapple,C.Enderud,J.P. Hilton,K.Karimi,E.Ladizinsky,N.Ladizinsky,T.Oh,I.Perminov,C.Rich,M.C.Thom,E.Tolkacheva,C.J.S.Truncik,S.Uchaikin,J.Wang,B.Wilson,G. Rose,Quantumannealingwithmanufacturedspins,Nature473(2011)195–198.

[11]D.Bitko,T.F.Rosenbaum,G.Aeppli,Quantumcriticalbehaviorforamodelmagnet,Phys.Rev.Lett.77(1996)940–943.

[12]H.M.Ronnow,J.Jensen,R.Parthasarathy,G.Aeppli,T.F.Rosenbaum,D.F.McMorrow,C.Kraemer,Magneticexcitationsnearthequantumphasetransition intheIsing ferromagnet,Phys.Rev.B75(2007)054426.

[13]R.Coldea,D.A.Tennant,E.M.Wheeler,E.Wawrzynska,D.Prabhakaran,M.Telling,K.Habicht,P.Smeibidl,K.Kiefer,QuantumcriticalityinanIsing chain:experimentalevidenceforemergentE8symmetry,Science327(2010)177–180.

[14]M.T.Bell,I.A.Sadovskyy,L.B.Ioffe,A.Yu.Kitaev,M.E.Gershenson,Quantumsuperinductorwithtunablenon-linearity,Phys.Rev.Lett.109(2012)1–5, 137003.

[15]M.Tinkham,IntroductiontoSuperconductivity,DoverBooks,2004. [16]R.M.Konik,A.LeClair,G.Mussardo,Int.J.Mod.Phys.A11(1996)2765.

[17]M.T.Bell,J.Paramanandam,L.B.Ioffe,M.E.Gershenson,ProtectedJosephson rhombuschains,Phys.Rev.Lett.112(2014)167001.

[18]M.T.Bell,W.Zhang,L.B.Ioffe,M.E.Gershenson,SpectroscopicevidenceoftheAharonov–Cashereffectinacooperpairbox,Phys.Rev.Lett.116(2016) 107002.

[19]A.Wallraff,D.I.Schuster,A.Blais,L.Frunzio,J.Majer,M.H.Devoret,S.M.Girvin,R.J.Schoelkopf,Approachingunitvisibilityforcontrolofa supercon-ductingqubitwithdispersivereadout,Phys.Rev.Lett.95 (6)(5August2005)060501.

[20]M.Pino,A.M.Tsvelik,L.B.Ioffe,UnpairedMajoranamodesinJosephson-junctionarrayswith gaplessbulkexcitations,Phys.Rev.Lett.115(2015) 197001.