Strong back-action of a linear circuit on a single electronic quantum channel F. PIERRE

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1 Strong back-action of a linear circuit on a single electronic quantum channel F. PIERRE F. Parmentier, A. Anthore, S. Jézouin, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly Laboratory for Photonics & Nanostructures (LPN) CNRS/Univ Paris Diderot, Marcoussis, France ν=4 ϕ Nano Team

2 Problematic : quantum laws of electricity? e.g. impedance composition with distinct coherent conductors G I V < L φ V / I = 1/G G=G I V >L φ Z S 200nm V / I = 1/G + Z S

3 Circuit back-action Poisson Fano S I =2eIF hν I Z S (ν) Granularity of charge transfers S I Excitation of the circuit s EM modes Reduction of the conductance G (dynamical Coulomb blockade)

4 Tunnel junction in a very resistive circuit The static Coulomb blockade limit G C V R Charge dynamics ignored if: E C = e 2 /2C >> E h/rc R,1/G >> R K = h/e kΩ G(V) G (k B T<< e 2 /2C) E C = e 2 /2C has to be paid for each tunnel event e 2 /2C ev G(V< e/2c)=0

5 Tunnel junction in an arbitrary linear circuit V G Quantum description of Z S n 1, n j, > Z S (ν) Caldeira & Leggett G(V) k B T G Z S =R//C<<h/e 2 h/rc ev ε= n i hν i (T=0K) P(ε)=Σ <n 1, n j, T e 0,,0> 2 P(ε) 2 θ(ε) Z S <<h/e 2 Re Z S (ε/h) εr K See Ingold & Nazarov in "Single Charge Tunneling" (Ed. Grabert & Devoret, 1992)

6 Scattering matrix description of a coherent conductor coherent conductor set of independent conduction channels t r t r V I Landauer, Büttiker, Martin V I MESOSCOPIC CODE: {τ i } G 2 N = 2e i h i= 1 Landauer formulae τ N τ i= i (1 τ 1 i ) i= S I = 2eI τ N 1 Fano factor i FCS tunnel junction (τ i <<1): S I 2eI e single channel: G = τ, h 2 2 S = 2eI (1 τ ) I

=1except 0<τ N <1for the last channel G h/2e 2 1 0 τ 1 =1 τ 2

7 Quantum Point Contact in a 2DEG A test-bed for coherent conductors V I 2 V QPC 2e 2 2e G= h τ 2 i = h (N-1+τ N ) τ i =1except 0<τ N <1for the last channel G h/2e τ 1 =1 τ 2 = G h/2e 2-1 τ 1 =G h/2e 2 τ 2 = V QPC [V]

8 Principle of the experiment QPC SW 1) SW closed : no back-action Tune & measure intrinsic {τ i } G Z S SW open I 2) : back-action G QPC Measure back-action signal at V 0 for the same intrinsic {τ i } V DS back-action signal relative conductance reduction:

9 Coherent conductor in a linear circuit Theoretical challenge: coherent conductor small perturbation Milestone: weak back-action in a low impedance circuit A. Levy Yeyati, A. Martin-Rodero, D. Esteve & C. Urbina, PRL 87, (2001) D.S. Golubev & A.D. Zaikin, PRL 86, 4887 (2001) Z S << R K = h/e kω t r t r weak back-action (small corrections) short coherent conductor V Z S (ν) I Same energy dependence as for tunnel junctions BUT Renormalized in amplitude by the same Fano factor as shot noise

10 Experimental implementation 200nm 2DEG in GaAs/Ga(Al)As, n S = m -2, µ=55m 2 V -1 s -1 Altimiras, Gennser, Cavanna, Mailly & Pierre, PRL 99, (2007)

Exp tal test of the weak-back action predictions δg/g 0.00-0.01-0.02 {τ 1 =R K G /2,τ 2 =0} R=1.2kΩ T=40mK B=0.2T (1-τ 1 ) {τ 1 =1,τ 2 =R K G /2-1} -0.03 0.0 0.5 1.0 1.5 2.

11 Exp tal test of the weak-back action predictions δg/g {τ 1 =R K G /2,τ 2 =0} R=1.2kΩ T=40mK B=0.2T (1-τ 1 ) {τ 1 =1,τ 2 =R K G /2-1} : δg G R K G /2 N i= 1 τ 2 (1-τ 2 ) (1+τ 2 ) τ i (1 τ i ) N τ i= 1 i S I /2eI shot noise across a coherent conductor R K G /2 Kumar, Saminadayar, Glattli, Jin & Etienne, PRL 76, 2778 (1996) Quantitative agreement data/thy: back-action signal intrinsic Fano factor Altimiras, Gennser, Cavanna, Mailly & Pierre, PRL 99, (2007) For the reduction of the back-action signal at τ~1, see also: Cron et al., XXXVIth Moriond proceedings (2001)

12 Strong back-action of a linear circuit: theory V QPC Problem unsolved in g al BUT important advances for Z S (ν)=r Z S (ν) V DS Z S (ν)=r, 1 channel, T=0 [1] δg/g << 1 DCB corrections linkedto noise in presence of back-action Z S (ν)=r << R K, T=0 [1-3] Z S (ν)=r K, 1 channel, T=0 [1] 1 [1] Safi & Saleur, PRL 93, (2004) [2] Kindermann & Nazarov, PRL 91, (2003) [3] Golubev, Galaktionov & Zaikin, PRB 72, (2005)

13 Experimental implementation SW strong back-action regime 2DEG dans GaAs/Ga(Al)As, n S = m -2, µ=55m 2 V -1 s -1

14 Experimental implementation SW strong back-action regime 2DEG dans GaAs/Ga(Al)As, n S = m -2, µ=55m 2 V -1 s -1

15 Back-action signal in the known tunnel regime thy with R K G =0.19 thy with R K G =0.18 R=26 kω ev<k V=0 B T Strong back-action (~ 90% red.) Expected EM environment Agreement between methods

16 Further check of the environment switch

17 Back-action signal vs intrinsic transmission (1-R K G ) 0.00 Altimiras et al., 2007 δg/g δ G G ( 1 R G ) K R K G

18 Back-action signal vs intrinsic transmission (1-R K G ) 0.00 Altimiras et al., 2007 V=0 δg/g δ G G ( 1 R G ) K R K G

19 Back-action signal vs reduced transmission (1-R K G QPC ) V=0

20 Back-action signal vs reduced transmission Exp tal finding:,, 0 V=0,, 0 1

21 Proposed generalized expression For a single electronic channel in an arbitrary linear environment Hyp: Exp tal finding valid for all Z S, T & V,, 1 Back-action signal for a tunnel junction (can be calculated from Ingold & Nazarov, 1992 ),, 1,, 1,, Finite bias test: DATA PREDICTION R=26 kω T=25 mk

22 Proposed expression vs recent predictions Z S (ν)=r << R K, T=0 [1-3] Z S (ν)=r K, 1 channel, T=0 [1] 1 ) [1] Safi & Saleur, PRL 93, (2004) [2] Kindermann & Nazarov, PRL 91, (2003) [3] Golubev, Galaktionov & Zaikin, PRB 72, (2005) = = = 1 Proposed expr.: For a R//C environment at T=0: =

23 Experimental data SUMMARY suggest a generalized expression single channel in an arbitrary linear circuit,, = 1,, 1,, in good agreement with theoretical predictions in simplified frameworks! Parmentier, Anthore, Jezouin, le Sueur, Gennser, Mailly & Pierre, Nature Phys. (advanced online publication)

24 François Parmentier Anne Anthore Sébastien Jézouin Hélène le Sueur (now: CSNSM, univ Paris-Sud) Ulf Gennser Antonella Cavanna ϕ Nano Team Dominique Carles Mailly Altimiras (now: SPEC, CEA) Thanks: D. Estève, P. Joyez, F. Portier, H. Pothier, C. Urbina I. Safi Y. Nazarov F. Lafont ϕ Nano Team Founding agencies:

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