Fermi-Edge Singularityin Tunnel Junctions

Jin Zhang, Y. Sherkunov, N. d’Ambrumenil, B. Muzykantskii

University of Warwick, U.K.

APS March Meeting, Portland, 17th, March, 2010

00)( 10 ff tiHtiHf eetG

Fermi-Edge Singularity

,2

2

4

ff titG

Sudden change of potentialfelt by the Fermi sea:

At zero temperature in single lead

P. W. Anderson, PRL18, 1049 (1967)

10 HH

Orthogonality Catastrophein quantum point contact:

X-Ray Absorption:

Loop ClosedLineOpen

)()()( fff tGtLtS

Spectrum decomposition:

Scattering Matrix Approach

R

Lti

ti

R

L

aa

tS

tBetAetAtB

bb

)()()()(

)(

)(

t

dttVt ')'()(10 HH 10 SS Characterize by

000)( )( tif etG

( )i t t iet i

Choose with :

2000

0)2)( or,( 22

t

tV

Example: ;0110

0

S

00

2

2

1

i

i

ee

S 1ftG ?But how to connect the Hamiltonians?

Keeling Klich Levitov, PRL97, 116403 (2006)

Map onto Full Counting StatisticsFull Counting Statistics:

SfftG ~1det)( fRft 1det)(

00)( ˆˆ11 QitiHQitiH

n

nin eeeeeP

00)( 10 tiHti eetG

Fermi-Edge Singularity:

ReTeTRS

ti

ti

)(

)(~

iiti

itii

LiLi

eABeABeeABeeAB

eSSeR

22)(

)(22

11

RS ~

Let

212 RB

212 RA

, with

0001

L

Example: Constant Phase Shift (I)

);( ft

otherwise ,00 ,

)( 0 fttt

T=1 t T=1/2 t

)( ftG

22220

fttttV 0withe.g. Choose Lorentzian Pulses:

FES FCS

)()(0 fttttV dtdtV

)( )()(0 tttt f

How to represent the delta functionHow to connect and0S 1S

Example: Constant Phase Shift (II)

JZ YS NdA BM, PRB80, 245308 (2009)

When , we have two independent Lorentzian Pulses:

2

0|)1)(1(41)(

ii ee

220 4

2 )( ff titG

)()()( 21 fff tGtGtG

)(1 ftG

2

)(t

ft0

ft

Conclusion

How to connect the two Hamiltonian matters

Map FES problem onto FCS

Thank You

The Effect of Opening & Closing

021 RRLLRLLRasym bababiabia

00 asym

( ) Im t iA tt i

Turn on the scattering as

YS JZ NdA BM, PRB 80, 041313 (R) (2009)

tAT 2

Outcome of the wave function:

Optimal Electronic Entangler

Fermi-edge Singularity in 2DEG

D. Cobden, BM, PRL75, 4274 (1995)

xlc HHHH

aaH c

NNc 1lN

..chbaWH x

',

''* 0|)()(|0)(

baWtaWtbtS fffab

Absorption Spectrum:

bbEH ll

aaH c

1NNc 0lN

',

'',

aaVaaH c

1NNc 0lN