We study the current produced in a Tomonaga-Luttinger liquid by an applied bias and by weak, point-like impurity potentials which are oscillating in time. We use bosonization to perturbatively calculate the current up to second order in the impurity potent

FIG.2:(Coloronline)DCpartofthebackscatteredcurrentasafunctionofthebiasω0fortwoimpurities,whenω±arenotsmall.Thered(dot),magenta(dash),blue(dash-dot)andblack(solid)linesshowtheresultsforK=1/4,1/2,3/4and1respectively.WehavetakenU1=U2,2kFx12=π/2,φ12= π/4,andωx12/v=1.

Fornon-interactingfermionswithK=1,we ndfromEqs.(27-28)thatinthesingleimpuritycase,

Ippp

bs,dc=

qU2

4πv2ω0cos(2ωt+2φp).(34)

F

ThetotalcurrentisgivenbyI= I0+Ipp+Ipp

bs,dcbs,ac,

2I=

qω0

vF

.(35)

Thisisconsistentwiththefactthatthetransmission

probabilityacrossastaticpoint-likebarrierofheightUis1 (U/vF)2uptoorderU2.Forthecaseofseveralimpurities,we ndfromEqs.(29-30)that

IprUr

bs,dc=

qUp2πv2ω0cos(2kFxrp)cos(2ωt+φp+φr).

F

(37)

Notethatthedcpartofthecurrentisgivenbyalinearcombinationofthepurebiaspartandthepurepumpingpart,anditagreeswiththeexpressiongiveninEq.(13).ForK=1/2,wecanobtainthedi erentpartsofthecurrentsbytakingthelimitK→1/2inEqs.(27-28)

7

(29-30).We ndthat

ppqUp

2=

4παv

[sgn(ω+)cos(2kFxrp+φrp)+sgn(ω )cos(2kFxrp φrp)],

ppqUp

2=

ω+

π

ln|

4παv

[(sgn(ω+)+sgn(ω ))cos(2ωt+φp+φr)+

2

ω

|sin(2ωt+φp+φr)].

(38)

ThustheDCpartsofthecurrentsdonotdependontheprecisevaluesofωandω0iftheyareunequal,andtheyhavea nitediscontinuitywhenωcrosses±ω0.

Toconclude,weseethatthedcpartsofthecurrentsarelinearfunctionsofω0,ωforK=1,andarepiecewiseconstantfunctionsofω0,ωforK=1/2.

F.

Extendedimpurities

TheanalysisinSubsec.III.Dcanbereadilygener-alizedtothecasewherethereisanextendedregionof

oscillatingpotentials[50].LetusreplacethediscretesetofpotentialsgiveninEq.(3)byanoscillatingpotentialofthefollowingform

U(t)=

dxU(x)cos[ωt+φ(x)].(39)WethenseefromEq.(31)thatthedcpartofthe

backscatteredcurrentisgivenby

Iq

bs,dc=

v

×[sgn(ω+)|ω+|2K 2K 2

1

| dxU(x)ei[2kFx+φ(x)]|2

+sgn(ω )|ω |2K 1

| dxU(x)ei[2kFx φ(x)]|2].

(40)

tosecondorderinU(x).Forthepurepumpingcasewithω0=0,we ndthat

I

q

bs,dc=× v

dxdx′U(x)U 2K 2

ω2K 1(x′)sin[2kF(x x′)]

×sin[φ(x) φ(x′)].

(41)

Eq.(41)impliesthatthechargepumpedpercycle,

Q=(2π/ω)Ibs,dc,scalesasω2K 2;forK<1,this