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

L

ti

ti

R

L

a

a

tS

tBetA

etAtB

b

b

)()(

)()()(

)(

t

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

000)( )( tif etG

( )i t t ie

t i

Choose with :

20

000)

2)( or,(

22

t

tV

Example: ;01

100

S

0

02

2

1

i

i

e

eS 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)(ˆˆ

11 QitiHQitiH

n

nin eeeeeP

00)( 10 tiHti eetG

Fermi-Edge Singularity:

ReT

eTRS

ti

ti

)(

)(~

iiti

itii

LiLi

eABeABe

eABeeAB

eSSeR

22)(

)(22

1

1

RS ~

Let

2

12 RB

2

12 RA

, with

00

01L

Example: Constant Phase Shift (I)

);( ft

otherwise ,0

0 ,)( 0 fttt

T=1 t T=1/2 t

)( ftG

22220

fttttV 0withe.g. Choose Lorentzian Pulses:

FES FCS

)()(0 fttttV dt

dtV

)( )()(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(4

1)(

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

02

1 RRLLRLLRasym bababiabia00 asym

( ) Imt i

A 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