The effect of an AC perturbation on the shot noise of a fractional quantum
Hall fluid is studied both in the weak and the strong backscattering regimes.
It is known that the zero-frequency current is linear in the bias voltage,
while the noise derivative exhibits steps as a function of bias. In contrast,
at Laughlin fractions, the backscattering current and the backscattering noise
both exhibit evenly spaced singularities, which are reminiscent of the
tunneling density of states singularities for quasiparticles. The spacing is
determined by the quasiparticle charge $\nu e$ and the ratio of the DC bias
with respect to the drive frequency. Photo--assisted transport can thus be
considered as a probe for effective charges at such filling factors, and could
be used in the study of more complicated fractions of the Hall effect. A
non-perturbative method for studying photo--assisted transport at $\nu=1/2$ is
developed, using a refermionization procedure.Comment: 14 pages, 6 figure

A. Crépieux

A. Kumar

A.A. Kozhevnikov

Adeline Crépieux

C. Bruder

C. de C. Chamon

C.L. Kane

C.W.J. Beenakker

D. Feldman

G. Cuniberti

G. Platero

G.B. Lesovik

H.H. Lin

I. Safi

J. Torrès

K.I. Imura

L. Saminadayar

L.H. Reydellet

L.P. Kouwenhoven

M. Büttiker

M. Reznikov

M. Sassetti

M.H. Pedersen

P. Fendley

Pierre Devillard

R. Aguado

R. de-Picciotto

R.J. Schoelkopf

S.R. Yang

T. Martin

T.H. Oosterkamp

Thierry Martin

V. Rodriguez

V.V. Ponomarenko

X. Jehl

X.G. Wen

Y. Gavish

Y. Levinson

Y. Oreg

English

arXiv

14 pages, 6 figuresThe effect of an AC perturbation on the shot noise of a fractional quantum Hall fluid is studied both in the weak and the strong backscattering regimes. It is known that the zero-frequency current is linear in the bias voltage, while the noise derivative exhibits steps as a function of bias. In contrast, at Laughlin fractions, the backscattering current and the backscattering noise both exhibit evenly spaced singularities, which are reminiscent of the tunneling density of states singularities for quasiparticles. The spacing is determined by the quasiparticle charge $\\nu e$ and the ratio of the DC bias with respect to the drive frequency. Photo--assisted transport can thus be considered as a probe for effective charges at such filling factors, and could be used in the study of more complicated fractions of the Hall effect. A non-perturbative method for studying photo--assisted transport at $\\nu=1/2$ is developed, using a refermionization procedure

Crépieux, Adeline

Devillard, Pierre

Martin, T.

HAL AMU

Photo--assisted current and shot noise in the fractional quantum Hall effect

Proceeding of the 18th International conference on noise and fluctuations, Salamanca, SpainThe effect of an ac perturbation on the shot noise of a fractional quantum Hall fluid is studied at finite temperature. For a normal metal, it is known that the zero-frequency noise derivative exhibits steps as a function of bias voltage. In contrast, at Laughlin fractions, the backscattering noise exhibits evenly spaced singularities, which are reminiscent of tunneling density-of-states singularities for quasiparticles. The spacing is determined by the quasiparticle charge ne and the ratio of the dc bias with respect to the drive frequency. Photo-assisted transport can thus be considered as a probe for effective charges of the quantum Hall effect

Martin, Thierry

HAL: Hyper Article en Ligne

Photo-assisted shot noise in the fractional quantum Hall regime

Photo-assisted shot noise in the fractional quantum HallregimeAdeline Cre´pieux, Pierre Devillard, Thierry MartinTo cite this version:Adeline Cre´pieux, Pierre Devillard, Thierry Martin. Photo-assisted shot noise in the fractionalquantum Hall regime. 2005, pp.488, 2005, <10.1063/1.2036799>. <hal-00005121>HAL Id: hal-00005121https://hal.archives-ouvertes.fr/hal-00005121Submitted on 3 Jun 2005HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Photo-assisted shot noisein the fractional quantum Hall regimeAdeline Crépieux∗, Pierre Devillard† and Thierry Martin∗∗Centre de Physique Théorique, Université de la Méditerranée, Case 907, 13288 Marseille, France†Centre de Physique Théorique, Université de Provence, Case 907, 13288 Marseille, FranceAbstract.The effect of an ac perturbation on the shot noise of a fractional quantum Hall fluid is studiedat finite temperature. For a normal metal, it is known that the zero-frequency noise derivativeexhibits steps as a function of bias voltage. In contrast, at Laughlin fractions, the backscatteringnoise exhibits evenly spaced singularities, which are reminiscent of tunneling density-of-statessingularities for quasiparticles. The spacing is determined by the quasiparticle charge νe andthe ratio of the dc bias with respect to the drive frequency. Photo-assisted transport can thus beconsidered as a probe for effective charges of the quantum Hall effect.Keywords: Shot noise, Luttinger liquid, Edge states, Photo-assisted transport.PACS: 73.43.-f; 73.50.Td; 03.65.Ta.INTRODUCTIONIn mesoscopic systems, the measurement of shot noise makes it possible to probethe effective charges which flow in conductors, and opens the possibility for studyingthe role of the statistics in stationary quantum transport experiments. This has beenillustrated experimentally and theoretically where the interaction between electrons isless important[1, 2, 3, 4, 5, 6] or when it is more relevant[7, 8, 9, 10, 11]. The presentwork deals with the study of photo-assisted shot noise in a specific one dimensionalcorrelated system: a Hall bar in the fractional quantum Hall regime, for which chargetransport occurs via two counter-propagating chiral edges states.MODELWe consider the system depicted on Fig. 1 which is described by the Hamiltonian:H =h¯vF4pi ∑r=R,L∫ds(∂sφr(t))2 +A(t)Ψ†R(t)ΨL(t)+A∗(t)Ψ†L(t)ΨR(t) . (1)The bosonic fields φR(L), which describe the right and left moving chiral excitationsalong the edge states, are related to the fermionic fields ΨR(L) through:ΨR(L)(t) =FR(L)√2piaei√νφR(L)(t) , (2)Γ0 I  (t)BV(t)FIGURE 1. Backscattering between edge states in the presence of a bias voltage modulation V (t).where FR(L) is a Klein factor, a, the short-distance cutoff and ν , the filling factorwhich characterize the charge e∗ = νe of the backscattered quasiparticles. The hoppingamplitude between the edge states has a time-dependence due to the applied voltageV (t) = V0 +V1 cos(ωt):A(t) = Γ0+∞∑n=−∞Jn(ω1ω)ei(ω0+nω)t , (3)where we have made an expansion in term of an infinite sum of Bessel functions,which is a signature of photo-assisted processes[12]. It is important to notice that thefrequencies ω0 and ω1 which appear in Eq. (3) are related to the filling factor ν:ω0 ≡ νeV0/h¯ , ω1 ≡ νeV1/h¯ . (4)where ν = 1/(2m+1) with m integer.PHOTO-ASSISTED SHOT NOISEThe symmetrized backscattering current noise correlator is expressed with the help ofthe Keldysh contour:S(t, t ′) = 12〈IB(t)IB(t ′)〉+ 12〈IB(t′)IB(t)〉−〈IB(t)〉〈IB(t ′)〉=12 ∑η=±1〈TK{IB(tη)IB(t ′−η)e−i∫K dt1HB(t1)}〉 , (5)where HB is the sum of the second and the third terms in Eq. (1), and:IB(t) =iνeh¯ A(t)Ψ†R(t)ΨL(t)−h.c. (6)We are interested in the Poissonian limit only, so in the weak backscattering case,one collects the second order contribution in the tunnel barrier amplitude A(t), andthe product of the average backscattering currents can be dropped. The meaning of thePoissonian limit is that quasiparticles which tunnel from one edge to another do so inan independent manner. Yet by doing so they can absorb or emit n “photon” quanta of−3 −2 −1 0 1 2 3−2.5−2−1.5−1−0.500.511.522.5ω0/ωdS(0,0)/dω0FIGURE 2. Noise derivative for a normal metal at different temperatures: kBT/h¯ω = 0.01 (solid line)and kBT/h¯ω = 0.1 (dashed line). We take ω1/ω = eV1/h¯ω = 3/2.ω (n integer). The main purpose of this work is to analyze the double Fourier transformof the noise S(Ω1,Ω2) ∝∫dt∫dt ′S(t, t ′)exp(iΩ1t + iΩ2t ′) when both frequencies Ω1and Ω2 are set to zero. Indeed, the presence of the AC perturbation mimics a finitefrequency noise measurement. At zero temperature, the shot noise exhibit divergencesat each integer value of the ratio ω0/ω[13]. These divergences are not physical sincethey appear in a range of frequencies where the perturbative calculation turn out to beno more valid. For this reason, we have performed finite temperature calculations whichprevent divergences in the backscattering current and shot noise. At finite temperature,the shot noise reads:S(0,0) =(e∗)2Γ202pi2a2Γ(2ν)(avF)2ν (2piβ)2ν−1×+∞∑n=−∞J2n(ω1ω)cosh((ω0 +nω)β2)∣∣∣∣Γ(ν + i(ω0 +nω)β2pi)∣∣∣∣2, (7)where Γ is the Gamma function and β = 1/kBT .DISCUSSIONWe have first test the validity of our result by setting ν = 1, the value which correspondsto non-interacting system (normal metal). The derivative of the shot noise according tothe bias voltage exhibits staircase behavior as shown on Fig. 2. Steps occur every timeω0 is an integer multiple of the ac frequency. This is in complete agreement with the−4 −3 −2 −1 0 1 2 3 4−8−6−4−202468ω0/ωdS(0,0)/dω0FIGURE 3. Noise derivative in the fractional quantum Hall regime at a filling factor ν = 1/3 forkBT/h¯ω = 0.05 (solid line) and kBT/h¯ω = 0.15 (dotted line). We take ω1/ω = νeV1/h¯ω = 3/2.results obtained be Lesovik and Levitov for a Fermi liquid[14]. When the temperatureincreases, the steps are rounded.For non-integer value of the filling factor (ν = 1/3, for example), the shot noisederivative exhibits evenly spaced singularities, which are reminiscent of the tunnelingdensity of states singularities for Laughlin quasiparticles. The spacing is determinedby the quasiparticle charge νe and the ratio of the bias voltage with respect to the acfrequency, and the amplitude is governed by temperature (see Fig. 3). Photo-assistedtransport can thus be considered as a probe for effective charges at such filling factors,and could be used in the study of more complicated fractions of the quantum Hall effect.REFERENCES1. M. Reznikov, M. Heiblum, H. Shtrikman, and D. Mahalu, Phys. Rev. Lett. 75, 3340 (1995).2. A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, ibid. 76, 2778 (1996).3. G. B. Lesovik, JETP Lett. 70, 208 (1999).4. M. Büttiker, Phys. Rev. Lett. 65, 2901 (1990); Phys. Rev. B 45, 3807 (1992).5. C.W.J. Beenakker and H. van Houten, Phys. Rev. B 43, 12066 (1991).6. T. Martin and R. Landauer, Phys. Rev. B 45, 1742 (1992).7. C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. 72, 724 (1994).8. C. de C. Chamon, D. E. Freed, and X. G. Wen, Phys. Rev. B 51, 2363 (1995).9. P. Fendley, A. W. W. Ludwig, and H. Saleur, Phys. Rev. Lett. 75, 2196 (1995).10. L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Lett. 79, 2526 (1997); R. de-Picciotto,M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, Nature 389, 162 (1997).11. A. Crépieux, R. Guyon, P. Devillard, and T. Martin, Phys. Rev. B 67, 205408 (2003).12. M. Büttiker and R. Landauer, Phys. Rev. Lett. 49, 1739 (1982).13. A. Crépieux, P. Devillard, and T. Martin, Phys. Rev. B 69, 205302 (2004).14. G.B. Lesovik and L.S. Levitov, Phys. Rev. Lett. 72, 538 (1994).

Photo--assisted current and shot noise in the fractional quantum Hall effect

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