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

Nanoelectronics Group

@ NanoElectronics Group, CEA Saclay

Maelle Kapfer Preden Roulleau

• D. C. G.
• P. Jacques
• D. Ritchie,
• I. Farrer ,

@ Cambridge UK OPEN POSITION for 18-24 months Post-doct.

• D. Christian Glattli

PERIODIC DRIVING of a mesoscopic conductor

• Brings new information on electron time-scales:

quantum inductance ~ (h/e2)τ, quantum capacitance ~ (e2/h)τ , charge relaxation (or Büttiker’s) resistance h/2e2

• Can be simply described by the photo-absorption (l>0) or emission (l<0) Floquet probability Pl=|pl|2

For voltage pulses on a contact V(t)=Vdc +Vac(t):

• An interesting regime is when the A.C. voltage amplitude is small, typically eVac ~ hν

→ single-electron transport

) (t V

Energy



) (t I 

F

E

   h    h 2 

F

E

(D)

eVdc= hν V(t) 𝜁 − ℎ𝑤 𝜁 − 2ℎ𝑤

Current Heat current

2

) ( ) ( t v t I

F 

        ) ( * ) ( Im ) ( t dt t d v t I

F Q

  

• M. Moskalets, G. Haack - physica status solidi (b), (2017)

M.F. Ludovico, J. S. Lim, M. Moskalets, L. Arrachea, D. Sanchez (2014)

• Can provide interesting comparison between quantum systems and cycle-operated thermodynamic engines

.... ) / 2 ( ) / ( ) / ( ) ( ) (

. . 2 . . 1 . . 1 . . .

        



q h V O P q h V O P q h V O P V O P V O

dc C D dc C D dc C D dc C D dc A P

   O : current, heat, current noise, heat noise, ….

t l i t i T l

e e dt T p

  2 ) (

1







EF



t

• G. Fève, A. Mahé, J.-M. Berroir, T. Kontos, B. Plaçais, and D. C. Glattli,
• B. Etienne, Y. Jin, , Science 316, 1169 (2007)

quantum capacitor electron source

t

e t

 

 ) ( 

2 2 2

) ( 1 ) (        

t iw t t   1 ) ( 

• J. Dubois, T. Jullien,P. Roulleau, F. Portier,P. Roche, Y. Jin,
• W. Wegscheider, and D.C. Glattli , NATURE 502, 659 (2013)

) (t V

EF

w

e

. 2

) (



 







voltage pulse source:

• simple: voltage pulse on a contact
• Lorentzian pulses create minimal excitation states (levitons)
• time resolved (no quantum jitter)
• long lifetime
• Minimal heat production:

leviton

Levitov, Lee,Lesovik, J. Math. Phys.(1996) Keeling, Klich and Levitov PRL 97, 116403 (2006) Dashti, Misiorny, Kheradsoud, Samuelsson, and Splettstoesser, PRB 100, 035405 (2019)

Energy



) (t I 

F

E

   h    h 2 

F

E

) ( ~  f

(D)

eVdc= hν electrons (no) hole V(t)

 h eVdc 

time 1e- … 1e- (periodic Levitons) Sine Pulses

electrons

) ( ~  f

holes

Energy



) (t I 

F

E

   h    h 2 

F

E

(D)

eVdc= hν V(t) 𝜁 − ℎ𝑤 𝜁 − 2ℎ𝑤

Lorentzian Pulses

) ( ) ( t V V t V

ac dc 



 h eVdc   h eVac 

time 1e- 1e- …

) (t V

Periodic electron injection is well described by Photo-Assisted process

• nly

 l and   l l

t l i t i T l

e e dt T p

  2 ) (

1











t ac

dt t V h e t ' ) ' ( ) ( 

0,00 0,25 0,50 0,75 1,00 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Time delay (period unit) /T SI

coll./SI

Lorentzian Wave =2 ( 2e

• )

Lorentzian Wave =1 ( 1e

• )

VG

V(t)

VG V(t+τ) 2 1 1

( ) ( 1 ) 1 (      e S

HOM I

Electronic Hong Ou Mandel correlation:

2 2 2 2 1 1

( ) ( ( ) ( 2 ) 2 (          e S

HOM I

(many particle HOM experiments open a new field of quantum investigations)

 

• D. C. Glattli, P. Roulleau (2017)

DOI 10.1002/pssb.201600650

• J. Dubois et al., Nature 502,

659–663 (2013)

• E. Bocquillon et al., Science 339,

1054–1057 (2013)

• 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))

THEORY

WDF WDF

EXPERIMENT

Wigner function

(quasi-probability) NEGATIVE PARTS (blue) reflect Quantum Coherence)

Wigner Function of (periodic) levitons

• 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) )

) , ( ) , ( a b e b a

S

i

 





(Leynaas+Mirrheim 1977, Wilczek 1982 ) expected for Fractional Quantum Hall effect (FQHE) quasiparticles (Arovas, Schrieffer, Wilczek 1984) 

C

l e 

2

 

• e / 3

a

z

b

z

B

) , ( 3 exp ) , (

2 2 a b holes b a holes

z z i z z

 

         

Berry phase

Example: for filling factor ν = 1/3 ( = 1 electron/3 quantum states) a quasi-hole particle has a charge e*=-e/3 (Laughlin 1983)

abelian

ν = 1/3;1/5,2/5,3/7, …

) , ( ) , (

, ,

a b U b a

a b b a

   

   

non-abelian

ν = 5/2, 7/2, … Majorana a b c a c b

time BRAIDING

a

z

b

z e*=e/4 a b

To date: no convincing experimental observation of anyons

) , ( ) , ( a b e b a

S

i

 





ν = 5/2 (Moore, Read 1991, Wen 1991) Non-abelian anyons should allow topological quantum computation (Freedman, Kitaev 2002)

Photon 1 Photon 2 Anyon 1 Anyon 2 1 2(1) in

• ut

P(1,2) = (1 – cosθs)/2

X coincidence recording 1(2) 2 Bosons: θS =0 Fermions: θS =π beam- splitter

) 2 , 1 ( ) ( ) 1 , 2 ( ) ( ) 2 , 1 ( ) 2 , 1 (

2 2

a e R T a ir a t b

S

i

   

         ir t t ir S

S B . .

Hong, Ou, & Mandel (1987)

Photon 1 Photon 2 Anyon 1 Anyon 2 X coincidence recording Bosons: θS =0 Fermions: θS =π beam- splitter

) 2 , 1 ( ) ( ) 1 , 2 ( ) ( ) 2 , 1 ( ) 2 , 1 (

2 2

a e R T a ir a t b

S

i

   

1 2(1) in

• ut

1(2) 1

τ

P(1,2) = (1 – g2(τ)cosθS)/2

         ir t t ir S

S B . .

Photon 1 Photon 2 Anyon 1 Anyon 2 X coincidence recording Bosons: θS =0 Fermions: θS =π beam- splitter

) 2 , 1 ( ) ( ) 1 , 2 ( ) ( ) 2 , 1 ( ) 2 , 1 (

2 2

a e R T a ir a t b

S

i

   

         ir t t ir S

S B . .

1 2(1) in

• ut

1(2) 1

τ

P(1,2) = (1 – g2(τ)cosθS)/2

Statistical measurements: Current noise SI = 2 (e*)2 ν (1 – P(1,2))= (e*)2 ν (1 + g2(τ)cos θS )

• 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

III-V semi-conductor heterojunction GaAs/GaAlAs 2D electrons density of state

h eB n 



c

  2 1

c

  2 3

c

  2 5 energy

m B e

c 



 

m A e p m H 2 2 1

2 2

      

eB y Y eB x X

x y

     

) , ( y x

) , ( Y X

eB /  

cyclotron motion

 

 

eB i Y X eB i

y x

      , ,          2 1 n E

c n

  e h Y X B    .

z B ˆ

cyclotron motion is frozen  1D dynamics

Integer Quantum Hall Effect (IQHE) Rhall=(h/e2)1/ =1,2,3, …

) ( 1

2

k e h n e B R

s Hall

   

density of state

h eB n 



c

  2 1

c

  2 3

c

  2 5 energy

m B e

c 



III-V semi-conductor heterojunction GaAs/GaAlAs 2D electrons

z B ˆ

cyclotron motion drift  chiral 1D EDGE CHANNELS

( no current in the bulk ) ( edge current )

x

c

  2 1

c

  2 3

c

  2 5

energy



n=0 n=1 n=2

z B E v

conf drift

ˆ

.

   

drift

v 

. conf

E 

ν = 2

Integer Quantum Hall Effect (IQHE) Fractional Quantum Hall Effect (FQHE) Rhall=(h/e2)1/ =1,2,3, … Rhall=(h/e2)1/ =1/3,2/5,3/7, …2/3, 3/5, 4/7, … 1/3 2/5

a

z

b

z

e*=e/3

B

2D-electrons Anyons Ѱ(a,b)=eiθ Ѱ(b,a) =π/3

Tunneling through a ν=2/5 Jain FQHE state

FQHE → C-F. IQHE ν = 1/3 → ν=1 ν = 2/5 → ν=2 ν = 3/7 → ν=3 …. → …. e*=e/5 e*=e/3 B: WB in 2/5 Plateau 1/3 A: WB in 1/3 while 2/5 reflected

A B

νB=2/5 νB=2/5 νB=2/5

• J. K. Jain Composite-fermion approach for the fractional quantum Hall effect. Phys. Rev. Lett. 63, 199-202 (1989)

ν = 1/3

• 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

μR μL e*Vdc Vdc e* μR –μL = e*Vdc IB(Vdc) SI

DC =2e* |IB| (Schottky)

Vdc SI

DC Shot Noise DC Transport

DC SHOT NOISE (weak coupling)

e* e*

Vdc

IB

               

dc B B dc B DC I

V e T k T k V e I e S * 2 2 * coth * 2

weak coupling limit

μR μL e*Vdc V(t)

Photon-Assisted Shot Noise (PASN)

e* μL+hf μL-hf

Vdc SI

P0 SI

DC

P-1 SI

DC(Vdc-hf/e*)

P1 SI

DC(Vdc+hf/e*)

DC Shot Noise

• hf/e*

hf/e* Pl : probability to absorb (emit) l >0 (<0) Photons V(t)=Vdc + Vaccos(2πft)

Lesovik and Levitov (1994)

• C. de C. Chamon, D. E. Freed, X. G. Wen (1995)
• A. Crépieux, P. Devillard, T. Martin (2004)

SI

PASN = P0 SI DC(Vdc) + P1 SI DC(Vdc+hf/e*) + P-1 SI DC(Vdc-hf/e*)+…

μR μL e*Vdc V(t)

Photon-Assisted Shot Noise (PASN)

e*

Vdc

V(t)=Vdc + Vaccos(2πft) μL+hf μL-hf

• hf/e*

hf/e* PASN DC Shot noise

SI

PASN = P0 SI DC(Vdc) + P1 SI DC(Vdc+hf/e*) + P-1 SI DC(Vdc-hf/e*)+…

Pl : probability to absorb (emit) l >0 (<0) Photons V(t)=Vdc + Vaccos(2πft)

SI

Lesovik and Levitov (1994)

• C. de C. Chamon, D. E. Freed, X. G. Wen (1995)
• A. Crépieux, P. Devillard, T. Martin (2004)

2/5 1/3 A B e/5 e/3 case A: case B: =2/5 =2/5 =2/5 =2/5 1/3

Experimental Set-up and samples

V It IB Samples: ns =1.07 1011 cm-2 μ=3 106 cm2V-1s-1 (from I. Farrer, D. Ritchie, Cambridge UK) S.E.M. view Nanolithography at SPEC (M. Kapfer) =Vdc+Vac(t)

Experimental Set-up and samples

14 Tesla Dry Magnet 13mK base temperature 2.2 MHz Δf=~150kHz Home-made Cryo-amp. (0.22nV)2/Hz CROSS-SPECTRUM 0-26GHz It IB V1 V2

DC Shot noise for the 1/3-FQHE state

e/3 B=2/5 B=2/5

=

1/3

2/5 1/3 A B

               

dc B B dc B DC I

V e T k T k V e I e S * 2 2 * coth * 2

e*= e/3 ! confirms ’97-’98 experiments

(Saclay PRL 97, Weizmann Nat. 97 and 99)

e/3 Vdc It+It(t) IB+IB(t)

Photon-Assisted Shot Noise for the 1/3-FQHE state

e/3 B=2/5 B=2/5

=

1/3

2/5 1/3 A B

) 2 cos( ) ( ft V V t V

ac dc

  

Vac  100 μV for -67dBm Vdc+Vaccos(2πft) f=22GHz

e/3 B=2/5 B=2/5

=

1/3

2/5 1/3 A B

) 2 cos( ) ( ft V V t V

ac dc

  

Vac  200 μV for -61dBm f=22GHz

Photon-Assisted Shot Noise for the 1/3-FQHE state

Vdc+Vaccos(2πft)

e/3 B=2/5 B=2/5

=

1/3

2/5 1/3 A B

) 2 cos( ) ( ft V V t V

ac dc

  

Vac  200 μV for -61dBm

 

*) / ( *) / ( ) ( ? ? ) (

1

e hf V S e hf V S P V S P V S

dc DC I dc DC I dc DC I dc PASN I

    

2 1 1 1 2

* *                   



hf V e J P P hf V e J P

ac ac

Photon-Assisted Shot Noise for the 1/3-FQHE state

f=22GHz

Excess PASN for the 1/3-FQHE state

 

*) / ( *) / ( ) ( ) (

1

e hf V S e hf V S P V S P V S S

dc DC I dc DC I dc DC I dc PASN I I

      

Killing the non photon-assisted part !

Excess PASN: Finding a flat variation for the low |Vdc| range provides a determination of P0

Excess PASN for the 1/3-FQHE state

WHY a FLAT VARIATION?

Vdc SI

P0 SI

DC

P-1 SI

DC(Vdc-hf/e*)

P1 SI

DC(Vdc+hf/e*)

DC Shot Noise

• hf/e*

hf/e*

• hf/e*

hf/e* Excess PASN

Vdc

SI = SI

PASN (Vdc) –

P0 SI

DC (Vdc)

= P1 [SI

DC (Vdc – hf / e*) + SI DC (Vdc + hf / e*) ]

Finding a flat variation for the low |Vdc| range provides a determination of |p0|2 as: P0 +2 P1 1 , this gives P1

Killing the non photon-assisted part !

Excess PASN for the 1/3-FQHE state

 

*) / ( *) / ( ) ( ) (

1

e hf V S e hf V S P V S P V S S

dc DC I dc DC I dc DC I dc PASN I I

      

Excess PASN:

Finding a flat variation for the low |Vdc| range provides a determination of |p0|2 as: P0 +2 P1 1 , this gives P1 comparison using fJosephson=e*Vdc/h with e*=e/3

Killing the non photon-assisted part !

Excess PASN for the 1/3-FQHE state

 

*) / ( *) / ( ) ( ) (

1

e hf V S e hf V S P V S P V S S

dc DC I dc DC I dc DC I dc PASN I I

      

Excess PASN:

New Measurement of e* for the 1/3-FQHE State

MEASURING e* from Excess PASN: threshold voltage : VJ=hf/e* scales with frequency! Best fit of data with e* free parameter e*=1/(3.07±0.05)

 

*) / ( *) / ( ) ( ) (

1

e hf V S e hf V S P V S P V S S

dc DC I dc DC I dc DC I dc PASN I I

      

• M. Kapfer et al. SCIENCE, Vol. 363 pp. 846-849 (2019)

DC Shot noise for the 2/5-FQHE state

2/5 1/3 A B

               

dc B B dc B DC I

V e T k T k V e I e S * 2 2 * coth * 2

e*= e/5 ! Vdc It+It(t) IB+IB(t)  - <IBIt> e/5 B=2/5 B=2/5 e*=e/5 confirms Weizmann results (Reznikov 1999) on 2/5

New Measurement of e* for the 2/5-FQHE State

MEASURING e* from Excess PASN: threshold voltage : VJ=hf/e* scales with frequency! Best fit of data with e* free parameter

f=fJ

(e/5)Vdc/h ( GHz )

e*=e/(5.17±0.31)

 

*) / ( *) / ( ) ( ) (

1

e hf V S e hf V S P V S P V S S

dc DC I dc DC I dc DC I dc PASN I I

      

• M. Kapfer et al. SCIENCE, Vol. 363 pp. 846-849 (2019)

 h eVdc   h eVac 

time 1e- 1e- …

) (t V

• WE HAVE SHOWN the MICROWAVE CONTROL of ANYONS is possible
• NEXT STEP : sine-wave to Lorentzian pulses.
• → TIME RESOLVED STOCHASTIC SOURCE OF ANYONS

See: theory prediction for photo-assisted current noise at ν=1/3

• J. Rech, D. Ferraro, T. Jonckheere, L. Vannucci, M. Sassetti,

and T. Martin, . Phys. Rev. Lett. 118, 076801 (2017) Also heat noise at at ν=1/3

• L. Vanucci;F. Ronetti, J. Rech, D. Ferraro, T. Jonckheere, T. Martin, M. Sassetti, PRB 95, 245415 (2017)



V(t) V(t+)

e/3 (e) (e) e/3

Id (t) Iu (t) =1/3 FQHE

state

=1/3 FQHE

state H.O.M. Beam Splitter ANYON SOURCE ANYON SOURCE

ANYON COLLIDER ) cos( ) ( 1

. 2 stat d u

g I I       

A DC version can be find in: ``Current Correlations from a Mesoscopic Anyon Collider ‘’ B. Rosenow, I. P. Levkivskyi, B. I. Halperin, (2016)

PERSPECTIVE : ANYON BRAIDING INTERFERENCE

PASN Josephson Relation (photon quantum )

h f = e* V

SCHOTTKY (charge granularity )

SI = 2 e* IB

SHOT NOISE combined with Microwave PHOTONS → 2 ways to determine carrier charge

weak signal but accurate good signal but lack of accuracy, IB : model dependent very accurate good accuracy

OLD METHOD NEW METHOD e/3 and e/5: M. Kapfer et al. SCIENCE, Vol. 363 pp. 846-849 (2019) e/3 finite frequency noise, R. Bisognin et al. Nature Communications (2019)

The Josephson Frequency of fractionally charge anyons

• M. Kapf

pfer er, P. Roulleau, I. Farrer, D. A. Ritchie, and D. C. Glattli,

SCIENCE, Vol. 363 pp. 846-849 (2019)

Levitons :

• J. Dubois et al, Natur

ure e 502, 659 (2013)

• T. Jullien et al., Nature

re 514, 603 (2014)

• X. Waintal
• H. Saleur

I Safi

• Th. Martin, J. Rech, T. Jonkheere
• M. Freedman

All members of Nanoelectronics Group at Saclay ANR FullyQuantum AAP CE30 UltraFastNano FET Open H2020.

Nanoelectronics Group

SPEC CEA-Saclay

ACKNOWLEDGEMENTS

OPEN POSITION for 18-24 months Post-doct. soon coming !

 h eVdc   h eVac 

time 1e- 1e- …

) (t V

• WE HAVE SHOWN the MICROWAVE CONTROL of ANYONS is possible
• NEXT STEP : sine-wave to Lorentzian pulses.
• → TIME RESOLVED STOCHASTIC SOURCE OF ANYONS

See: also: Jérome Rech’s Talk, Wednesday Session 3-C, and

• J. Rech, D. Ferraro, T. Jonckheere, L. Vannucci, M. Sassetti,

and T. Martin, . Phys. Rev. Lett. 118, 076801 (2017)

 

     e dt t I t V h e e t I dt t V e ) ( ) ( * ) ( 2 ) ( * 1   

Why only charge e levitons in FQHE.

e e

strong barrier :

 = 1  = 1

V

) (t I h V e I /

2 0 

h V e I /

2 0 

) (t I B I I 

1 2   D eI SI 1 2   D eI S

B I

) 1 ( 2 D D I e S I  

B

I I I h V e I   

2

/

transmitted (D) reflected (1-D) (rarely transmitted electrons) (incoming electrons)

e e

(rarely transmitted holes) weak barrier :

Poisson’s statistics

• G. B. Lesovik,

JETP Letters49, 594 (1989) Schottky (1918)

h/eV

e e/3

strong barrier :

 = 1/3  = 1/3

V

) (t I

h V e I 3 /

2 0 

h V e I 3 /

2 0 

) (t I B I I 

1 2   D eI SI

1 3 2   D I e S

B I B

I I I h V e I   

2

3 /

transmitted (D) reflected (1-D) (rarely transmitted electrons) (incoming electrons)

e e

(rarely transmitted holes) weak barrier :

e e e/3 e/3

derived from chiral-Luttinger liquid approach ( X.G. Wen 1995, C. Kane + M. Fisher 1994; Fendley, Ludwig + Saleur (1995)) First observation: CEA Saclay 1997 Weizmann 1997

h/3eV =1/3 =1/3

JOSEPHSON RELATION

q V = h f

qV (L) conductor (R) conductor q micro-wave photon hf

• D. C. GLATTLI

CEA Saclay

2e JOSEPHSON RELATION

q V = h f

qV

S S

I

q = 2e Inverse AC Josephson Effect

V I

• S. Shapiro (1963)

energy

2e JOSEPHSON RELATION

q V = h f

qV

S S

I

q = 2e Inverse AC Josephson Effect

V I hf/q

• hf/q
• S. Shapiro (1963)

hf

e JOSEPHSON RELATION

q V = h f

qV

N N

I

HERE: no Josephson Effect BUT: current Shot Noise

V SI Lesovik + Levitov (1994)

e JOSEPHSON RELATION

q V = h f

qV

N N

I

V SI hf/q

• hf/q

Lesovik + Levitov (1994)

HERE: no Josephson Effect BUT: Photo-Assisted Shot Noise

hf micro-wave photon

e* JOSEPHSON RELATION

e*V = h f

qV

HERE: Fractional QH Effect and Photo-Assisted Shot Noise

e* V =1/m =1/m q=e* = e/m hf Lesovik + Levitov (1994) X.G. Wen (1995) V SI hf/e*

• hf/e*

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

       ``single side band spectrum’’

Lorentzian pulse

) ( / t i t i x k i

e e e

    

EF ) ( ~  f

 ) (t V el. hole

) ( ) (         p p

LEVITONS : a wave property + Fermi statistics

requires :

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

       ``single side band spectrum’’

Lorentzian pulse

) ( / t i t i x k i

e e e

    

EF ) ( ~  f

 ) (t V el. hole

w i t e

t i

 



) ... (

) ( 

LEVITONS : a wave property + Fermi statistics

t ' t i iw    

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

       ``single side band spectrum’’

Lorentzian pulse

) ( / t i t i x k i

e e e

    

EF ) ( ~  f

 ) (t V el. hole

w i t w i t e

t i

  

 ) ( 

) /( 2 /

2 2

w t w dt d   

LEVITONS : a wave property + Fermi statistics

t ' t i iw    

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

       ``single side band spectrum’’

Lorentzian pulse

) ( / t i t i x k i

e e e

    

EF ) ( ~  f

 ) (t V el. hole

2 ) (

          

 

w i t w i t e

t i

) /( 4 /

2 2

w t w dt d   

LEVITONS : a wave property + Fermi statistics

t ' t i iw    

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

``double side band spectrum’’ for non-integer charge

Lorentzian pulse

) ( / t i t i x k i

e e e

    

) (t V hole

e q t i

w i t w i t e

/ ) (

          

 

) /( 2 /

2 2

w t w e q dt d   

EF ) ( ~  f

 el.

      

fractional q

LEVITONS : a wave property + Fermi statistics

« Dynamical orthogonality catastrophe » Levitov 1995

Lorentzian pulse

) (t V

fractional q

LEVITONS : a wave property + Fermi statistics

 energy

F

E





t ac

dt t V e t ' ) ' ( ) (  



    

 

 / ) (

) (

t i t i

e e dt p

 



) (t Vac

``double side band spectrum’’ for non-integer charge

) ( / t i t i x k i

e e e

    

hole

e q t i

w i t w i t e

/ ) (

          

 

) /( 2 /

2 2

w t w e q dt d   

EF ) ( ~  f

 el.

      

minimal excitation states (levitons) for integer charge only