Nanoelectronics Group Post-doct. @ Cambridge UK PERIODIC DRIVING - PowerPoint PPT Presentation

D. Christian Glattli Maelle Kapfer Preden Roulleau D. C. G. P. Jacques @ NanoElectronics Group, CEA Saclay D. Ritchie, OPEN POSITION I. Farrer , for 18-24 months Nanoelectronics Group Post-doct. @ Cambridge UK

PERIODIC DRIVING of a mesoscopic conductor V ( t ) • Brings new information on electron time-scales: quantum inductance ~ (h/e 2 ) τ , quantum capacitance ~ (e 2 /h) τ , charge relaxation (or Büttiker’s ) resistance h/2e 2 • Can provide interesting comparison between quantum systems and cycle-operated thermodynamic engines • Can be simply described by the photo-absorption (l>0) or emission (l<0) Floquet probability P l =|p l | 2 (D) T 1      i ( t ) i l 2 t p dt e e For voltage pulses on a contact V(t)=V dc +V ac (t): Energy    2 h l T eV dc = h ν    h 0 E  E  F F 𝜁 − ℎ𝑤         P . A . O ( V ) P O ( V ) P O ( V h / q ) P O ( V h / q )  𝜁 − 2ℎ𝑤 dc 0 D . C dc 1 D . C . dc 1 D . C . dc     P O ( V 2 h / q ) .... 2 D . C . dc  I ( t ) V(t) O : current, heat, current noise, heat noise, …. • An interesting regime is when the A.C. voltage amplitude is small, typically eV ac ~ h ν → single -electron transport  F  2 M. Moskalets, G. Haack - physica status solidi (b), (2017) I ( t ) v ( t ) Current M.F. Ludovico, J. S. Lim, M. Moskalets, L. Arrachea, D. Sanchez (2014)    d ( t ) *   Heat current Q  I ( t ) v Im ( t )   F   dt

   1   t   2  ( t ) e quantum capacitor ( )      2 2 ( ) electron source 0 t E F G. Fève, A. Mahé, J.-M. Berroir, T. Kontos, B. Plaçais, and D. C. Glattli, B. Etienne, Y. Jin, , Science 316, 1169 (2007)  voltage pulse source:   2    . w ( ) e 1   ( t )  t iw E F V ( t ) t leviton J. Dubois, T. Jullien,P. Roulleau, F. Portier,P. Roche, Y. Jin, W. Wegscheider, and D.C. Glattli , NATURE 502, 659 (2013) • simple: voltage pulse on a contact • Lorentzian pulses create minimal excitation states ( levitons ) Levitov, Lee,Lesovik, J. Math. Phys.(1996) Keeling, Klich and Levitov PRL 97, 116403 (2006) • time resolved (no quantum jitter) • long lifetime • Minimal heat production: Dashti, Misiorny, Kheradsoud, Samuelsson, and Splettstoesser, PRB 100, 035405 (2019)

 dc  V ( t ) V V ( t ) (D) ac Energy    2 h eV dc = h ν electrons    h  E E  F F eV dc   (no) hol e Lorentzian Pulses h ~   0 I ( t ) f ( ) 1e - …  time 1e - V(t) l 0 only (periodic Levitons ) T t 1 e         i ( t ) i l 2 t ( t ) V ( t ' ) dt ' p dt e e ac l h T V ( t ) 0 (D) Energy    2 h eV dc = h ν electron s eV ac      h h  E E Sine Pulses  F F eV dc   𝜁 − ℎ𝑤 h holes 𝜁 − 2ℎ𝑤 time  0 and l 0 ~  1e - … 1e -   I ( t ) f ( ) l 0 V(t) Periodic electron injection is well described by Photo-Assisted process

Electronic Hong Ou Mandel correlation: 1,6 - ) Lorentzian Wave  =2 ( 2e 1,4 - ) Lorentzian Wave  =1 ( 1e 1,2 1,0 0 V G coll. /S I V(t+ τ ) 0,8 S I 0,6 V G 0,4 0,2 V(t ) 0,0 0,00 0,25 0,50 0,75 1,00 Time delay (period unit)  /T  2      HOM D. C. Glattli, P. Roulleau (2017) S ( 1 e ) 1 ( 0 ) ( I 1 1 DOI 10.1002/pssb.201600650 J. Dubois et al., Nature 502,  659 – 663 (2013) 2 2          HOM S ( 2 e ) 2 ( 0 ) ( ( 0 ) ( E. Bocquillon et al., Science 339, I 1 1 2 2 1054 – 1057 (2013) (many particle HOM experiments open a new field of quantum investigations)

Wigner Function of (periodic) levitons THEORY EXPERIMENT WDF WDF Wigner function (quasi-probability) NEGATIVE PARTS (blue) reflect Quantum Coherence) T. Jullien, P. Roulleau, B. Roche and D. C. Glattli Nature , 514 603-607 (2014) C. Grenier et al., New J. Phys. 13, 093007 (2011))

• First step to realize a single anyon source • Shows that FQHE abelian anyons with charge e*=e/3 and e/5 can be manipulated with microwave by well-defined Photon-Assisted processes. • Validates the possibility to realize on-demand single anyon sources for time domain anyon braiding. • Based on Photon-Assisted Shot Noise (PASN) measurements • Photon-Assisted process revealed by the anyonic Josephson relation e*V/h=f ( X. G. Wen (1991) )

    i ( a , b ) e ( b , a ) S (Leynaas+Mirrheim 1977, Wilczek 1982 ) expected for Fractional Quantum Hall effect (FQHE) quasiparticles B (Arovas, Schrieffer, Wilczek 1984)  0 Example: for filling factor ν = 1/3 ( = 1 electron/3 quantum states) 2 e    - e / 3 l C a quasi-hole particle has a charge e*=-e/3 (Laughlin 1983)         2 holes   2 holes ( z , z ) exp i ( z , z ) a b   b a 3 z a Berry phase z b

    i ( a , b ) e ( b , a ) S abelian ν = 1/3;1/5,2/5,3/7, … time     ( a , b ) U ( b , a ) non-abelian ν = 5/2, 7/2, …     , , a b b a Majorana (Moore, Read 1991, Wen 1991) b c a ν = 5/2 z a e*=e/4  a  b z b b c a Non-abelian anyons should allow topological quantum computation (Freedman, Kitaev 2002) To date: no convincing experimental observation of anyons BRAIDING

Hong, Ou, & Mandel (1987) Photon 1 Photon 2 1(2) 1 beam- in out splitter coincidence recording X 2(1) Anyon 1 2 Bosons: θ S =0   2 2   b ( 1 , 2 ) t a ( 1 , 2 ) ( ir ) a ( 2 , 1 ) ir t    S   i  B . S .     t ir ( T R e ) a ( 1 , 2 ) S Fermions: θ S = π Anyon 2 P(1,2) = (1 – cos θ s )/2

Photon 1 Photon 2 1(2) 1 beam- τ in out splitter coincidence recording X 2(1) Anyon 1 1 Bosons: θ S =0   2 2   b ( 1 , 2 ) t a ( 1 , 2 ) ( ir ) a ( 2 , 1 ) ir t    S   i  B . S .     t ir ( T R e ) a ( 1 , 2 ) S Fermions: θ S = π Anyon 2 P(1,2) = (1 – g 2 ( τ )cos θ S )/2

Photon 1 Photon 2 1(2) 1 beam- τ in out splitter coincidence recording X 2(1) Anyon 1 1 Bosons: θ S =0   2 2   b ( 1 , 2 ) t a ( 1 , 2 ) ( ir ) a ( 2 , 1 ) ir t    S   i  B . S .     t ir ( T R e ) a ( 1 , 2 ) S Fermions: θ S = π Anyon 2 P(1,2) = (1 – g 2 ( τ )cos θ S )/2 Statistical measurements: Current noise S I = 2 (e*) 2 ν (1 – P(1,2))= (e*) 2 ν (1 + g 2 ( τ )cos θ S )

• EDGE STATE and DC SHOT NOISE in FQHE • PHOTON-ASSISTED TRANSPORT • Photon-assisted processes • A JOSEPHSON Relation for Photon Assisted Shot Noise (PASN) • Experimental Results • e*=e/3 • e*=e/5 • CONCLUSION and PERSPECTIVES

B ˆ energy z 2D electrons 5   c 2 e B  c  m 3   c 2 III-V semi-conductor heterojunction GaAs/GaAlAs 1   c 2 eB n   h density of state      2    1 2        ,    1 H p e A  i eB      E n x y 2 m 2 m n c   2  ( x , y )   y X x eB   h     .     / eB B X Y  X , Y i   e x eB Y y ( , ) X Y eB cyclotron motion is frozen  1D dynamics cyclotron motion

Integer Quantum Hall Effect (IQHE) energy R hall =(h/e 2 )1/   =1,2,3, … 5   c 2 e B  c  m 3   c 2 1   c 2 eB n   h density of state B h 1   R   Hall 2 e n e ( k ) s

energy B ˆ z 2D electrons n=2 5   c 2  3   c 2 n=1 III-V semi-conductor heterojunction GaAs/GaAlAs n=0 1   c 2 x  v drift  E  conf .  E   ν = 2 conf . ˆ v z drift B ( no current in the bulk ) ( edge current ) cyclotron motion drift  chiral 1D EDGE CHANNELS

Fractional Quantum Hall Effect (FQHE) Integer Quantum Hall Effect (IQHE) R hall =(h/e 2 )1/   =1,2,3, … R hall =(h/e 2 )1/   =1/3,2/5,3/7, …2/3, 3/5, 4/7, … 2/5 1/3 B 2D-electrons z a e*=e/3 z Anyons Ѱ (a,b)=e i θ Ѱ (b,a)  = π /3 b

Tunneling through a ν =2/5 Jain FQHE state FQHE → C -F. IQHE ν = 1/3 → ν =1 B ν B =2/5 B : WB in 2/5 A e*=e/5 ν = 2/5 → ν =2 ν = 3/7 → ν =3 …. → …. ν = ν B =2/5 Plateau 1/3 1/3 A : WB in 1/3 ν B =2/5 e*=e/3 while 2/5 reflected J. K. Jain Composite-fermion approach for the fractional quantum Hall effect . Phys. Rev. Lett. 63, 199-202 (1989)

• EDGE STATE and DC SHOT NOISE in FQHE • PHOTON-ASSISTED TRANSPORT • Photon-assisted processes • A JOSEPHSON Relation for Photon Assisted Shot Noise (PASN) • Experimental Results • e*=e/3 • e*=e/5 • CONCLUSION and PERSPECTIVES