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

currentisbackscatteredtotheright.Thetotalcurrent owingtotherightisgivenbyI= I0+Ibs,whereIbsisthecorrectiontothecurrentduetobackscatteringbytheimpurities.Thebackscatteredcurrentisde nedasI dN bs(t)=q

R2

dxdyρ(x)V(x y)ρ(y),(19)

whereV(x)isarealand

evenfunction

of

x

,

andtheden-sityρ=ψ ψisgiveninEq.(2).WecanwriteEq.(19)inasimplewayifV(x)issoshortrangedthattheargu-mentsxandyofthetwodensity eldscanbesetequal

4

ingtheanticommu-tationrelationsbetweenthefermion elds,weobtain

Hint=g2

dxψ RψRψ

LψL,(20)whereg2isrelatedtotheFourierasg2=V

transformofV(x)(0) V (2kF).De ningaparameterγ=g2/(2πvF),wehavetherelations

K=

1 γ

2v

φ

2

φ

2παη Rη Lei2√

2πα

η Lη R

e

i2

√(2π)2[(xp xr)2 (v(t′ t) iα)2]K

(24)

forallvaluesofK.

Forthenon-interactingcasewithK=1,wecaneval-uatetheabovegroundstateexpectationvaluedirectlywithoutusingbosonization.Weusethesecondquan-tizedexpressionsforthefermion elds,

ψ ∞

dk

R=

∞

ik( x vFt)2π

aLke,

(25)

wherethecreationandannihilationoperatorssatisfythe

anticommutationrelations{aRk,a

k k′).ThegroundstateRk′|0 }is=annihilated{aLk,aLk′2πδ(}by

=aRk,aLkfork>0andbya Rk,a

Lkfork<0.Wethen ndthatthegroundstateexpectationvalueagreeswiththeresultgiveninEq.(24)forK=1andv=vF.

Ingeneral,thebackscatteredcurrenthastwoparts:oneindependentoftimewhichwecallIdc,andtheothervaryingwithtime,withfrequency2ωtosecondorderinUp,whichwecallIac.Iacdoesnotcontributetoany