Introduction Model and theory Optical analogs in quantum Hall regime current and noise

Visibility of current and shot noise in electrical MVisibility of current and shot noise in electrical Mach-Zehnder and ach-Zehnder and

Hanbury Brown Twiss interferometersHanbury Brown Twiss interferometers V. S.-W. Chung(鐘淑維 )1,2, P. Samuelsson3 ,and M. Büttiker1

1 Départment de Physique Théorique, Université de Genève, Genève 4, CH-1211 Swizterland2 Department of Electonics and Engineering, Chiao Tung University, HsinChu 30010, Taiwan3 Division of Solid State Theory, Lund University, Sölvegatan 14 A, S-223 62 Lund, Sweden

Ref.: P.R.B 72, 125320 (2005)

My supervisors: C.S. Chu(EP,NCTU) and C.Y. Chang(EE,NCTU)

Introduction Model and theory

Optical analogs in quantum Hall regime current and noise Scattering approach to current and shot noise Dephasing probe model

Mach-Zehnder(MZ) interferometers A fully coherent condition Effect of dephasing

Hanbury Brown Twiss(HBT) interferometers A fully coherent condition Effect of dephasing

Conclusion

IntroductionIntroduction

With the advent of mesoscopic physics, it has become possible to experimentally investigate quantum phase coherent properties of electrons in solid state conductors in a controllable way and opens up the possibility of investigating electrical analogs of various optical phenomena.

For photons and conduction electrons, both the wave-nature of the particles as well as their quantum statistics are displayed in a clearcut fashion in interferometer structures.

4

Various with electronic interferometers ballistic transport of the electrons have been investigated experimentally last decades, as e.g.

Ref.: H. Brown et al., Nature 177, 27(1956) http: //mpej.unige.ch/~buttiker/

Two-particle Optical HBT interferometers

One-particle Optical HBT interferometers

One-particle electronic HBT interferometers

W.D. Oliver et al.,Science 284, 299(1999)

Only very recently two of the authours and Sukhorukov proposed a direct electronic analog of the optical HBT interferometer, which permits demonstrate two-particle interference in an unambiguous way.

In this work, we investigate and compare in detail the current and zero frequency noise in single particle MZ and two-particle HBT interferometers. Dephasing is studied with the help of the dephasing probe model.



Mach-Zehnder interferometers A fully coherent condition Effect of dephasing

Hanbury Brown Twiss interferometers A fully coherent condition Effect of dephasing

Conclusion

Model and Theory:Model and Theory:Optical analogs in quantum Hall regimeOptical analogs in quantum Hall regime

B The transport takes

place along edge states, realizing the beams of electrons.

The QPC’s work as the electronic beam splitters with controllable transparency.

Model and Theory: Model and Theory: current and noisecurrent and noise

t

( )I t

I

Model and Theory:Model and Theory:Scattering approach to current and shot noiseScattering approach to current and shot noise

System

1

23

4

5.........N

1a1b

2a

3a

4a

5aˆNa

2b

3b

4b

5b

ˆNb

1

ˆ ˆ N

b E s E a E


† †' ˆ ˆˆ ˆ ˆ'exp ' ' i E E te

I t dEdE b E b E a E a Eh

12

ˆ ˆ ˆ ˆ(0) ( ) ( ) (0) ,

ˆ ˆ ˆwhere ( ) ( ) ( ) .

S dt I I t I t I

I t I t I t

Ref. : Ya. Blanter and M. Büttiker, Phys. Rep. 336,1(2000)

22

, , 1 .e

S dE A E E A E E f E f Eh


21

; , . e

I dE G E f E G E A E Ee h

Here , ' ' .A E E s E s E

Model and Theory: Model and Theory: Dephasing probe modelDephasing probe model

Zero currents into lead γ, achieved by the distribution fun. inside the probe, which conserves (i) total currents; (ii) currents at each energy at the probe.

Phase broken but no energy dissipated.

A phenomenological model.

System

12

3

4

5.........N


, ,f E t f E f E t

, , ;

1, , ,

j E t j E j E t

j E t j E t G E f E te

System

12

3

4

.........N

.........


Condition I: 0 .

G Ej E f E f E

G E

Condition II: , 0 , , , .

G Ej E t j E t j E t j E t

G E

2

.dp G E G E G E G ES E S E S E S E S E

G E G E G E





Conclusion

1

2

3

4

2 1

A

B j j

j

j j

i R TS

T i R

Mach-Zenhder interferometers:Mach-Zenhder interferometers:A fully coherent conditionA fully coherent condition

A

B

1

2

3

4

1

2


2

4

0 0

3 4

The current in lead 4 is

4 csch sin cos ;2 2

where = 2 ; ; .

Current conservation gives .

D

A B B A A B A B

BB

c c c

vhe LF c

eh

eI T R T R eV T T R R

h

k T eV eVk T

E E E

E E

I V I


max min

max min

Visibility is quantified as

amp.I

II I

I I I

,

4csch sin

2A B A B B B

I MZA B B A c c

T T R R k T k T eV

T R T R eV E E


0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0 I,MZ

kBT/E

c

eV/Ec=0.1

eV/Ec=1

eV/Ec=3

eV/Ec=5

As .A BT T

Ref. :Y. Ji et al., Nature 422, 415 (2003)I. Neder et al., P.R.L. 96, 16804 (2006)

2

34 0 0 2 2

0

2

0

2cos cos 2 ;

2 2

with

6 ,

2 , 2 ,

coth 2 ,2

2 csch coth2

c c

A A B B A A B B

A A A A A B A B A B A B

BB

Bj B

c

e eV eVS c S c S c S

h E E

c T R T R T R T R

c T R T R T T R R c T T R R

eVS eV k T

k T

j k T eVS k T

E

sin cos , ( 1,2).

2 2B

B c c c

jk TjeV jeVj

k T E E E


2 22N,MZ N,MZ

0 0 0 0

; .c S c S

c S c S


0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

N,MZ

TB=0.01

TB=0.1

TB=0.3

TB=0.5

(a)

N,MZ

TA

(b)

N,MZ

2N,MZ

In the limit , :

2;

6

2.

6

c B

A A B B A B A B

A A B B A A B B

A B A B

A A B B A A B B

E k T eV

T R T R T T R R

T R T R T R T R

T T R R

T R T R T R T R


0 1 2 3 4

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

N,MZ

eV/Ec()

N,MZ

no

ise

visi

bili

ties

kBT/E

c

2

N,MZ

Consider the situation , :

csch 1 ,

( 1,2).

B c

j B B B

c c c

eV k T E

j k T j k T jk T

E E E

j

N,MZ

In the opposite limit , :

2sin .

2

B c

j c

c

k T eV E

E jeV

jeV E

Mach-Zenhder interferometers:Mach-Zenhder interferometers:Effect of dephasing Effect of dephasing

1

2

3

4

2

15

B

A

1

1

iS

i

Mach-Zenhder interferometers:Mach-Zenhder interferometers:Effect of dephasingEffect of dephasing

4

Then the current in lead 4 is found to be

4 csch sin cos .2 2

The effect of dephasing can thus be simply incorporated

in the visibi

1

li

dpA B B A A B A B

BB

c c c

eI T R T R eV T T R R

h

k T eV eVk T

E E E

, ,

ty as

.1dpI MZ I MZ


2

34 0 0 2 2

The cross correlator between contacts 3 and 4 in the presence of dephasing probe is

2cos cos 2 .

2 2

The visibilities of the two osci

1

l

1dp

c c

e eV eVS c S c S c S

h E E

,dp 2 ,dp 2N,MZ N,MZ N,MZ N,MZ

lations in the presence of dephasing can simply be

written

and1 1 .

Ref.: F. Marquardt et al., P.R.L. 92, 56805(2004) A. A. Clerk et al., P.R.B 69, 245303(2004)


The effect of dephasing, introduced with the voltage probe, both for the current and noise, is for arbitrary dephasing strengnth identical to a phase averagephase average.

2

0

2 20

cos 1 cos ,

with the Lorentzian distribution

1, ln 1 .

2

n

d n n

aa

a

Ref.: S. Pilgram et al., cond-mat/0512276


Multiplicative: (1-ε)→ (1-ε)n.

(1-ε)n =exp(-L/Lφ) with Lφ =-d/ln(1- ε) and L=nd.

(1-ε)1/2→ exp(-L/2Lφ); (1-ε) → exp(-L/Lφ).

1 2

n

1 2

3

4A

B

dephasing terminalsreserviors

d





Conclusion

Hanbury Brown Twiss interferometers: Hanbury Brown Twiss interferometers: A fully coherent conditionA fully coherent condition

A B

C

D

1 2

3 4

5

6

7

8

1 3

4 2

1

2

3

4

6

7

8

A B

C

D2

13

45

Hanbury Brown Twiss interferometers: Hanbury Brown Twiss interferometers: A fuA fully coherent conditionlly coherent condition

1

2

3

4

6

7

8

A B

C

D2

13

4

2

5

2

6

2

7

2

8

;

;

;

.

A C A D

A D A C

B C B D

B D B C

eI V T T R Rh

eI V T T R Rh

eI V T T R Rh

eI V T T R Rh

5

2

58 0,58 0

2

57 0,57 0

0,58

0,57

, , ,

67 58 68 57

2cos ;

2

2cos ,

2

with

,

,

2 .

; .

c

c

A B C C B A D D

A B C C A B D D

j jJ A B C D

C D C D

e eVs c S c S

h E

e eVs c S c S

h E

c T R T R T R T R

c T T T R R R T R

c T R

S S T T S S T T


,N.HBT

0, 0

;

=5,6 and =7,8.

c S

c S


,58 ,67N.HBT N.HBT

,57 ,68N.HBT N.HBT

In the limit ,

as or ,

the visibility are given by

2;

2;

c B

C D C D

A A B B

A B B A

A A B B

A B A B

E k T eV

T T T R

T R T R

T R T R

T R T R

T T R R

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

1.0

TA

N,HBT

N,HBT

TB=0.1

TB=0.3

TB=0.5

TB=0.7

TB=0.9

Hanbury Brown Twiss interferometers: Hanbury Brown Twiss interferometers: EffeEffect of dephasingct of dephasing

1

2

3

4

6

7

8

B

C

D2

13

4

9

2

58 0,58 0

2

57 0,57 0

, , ,, ,

1~4

2cos ;

2

2cos ;

2

.

exp / 2 ; .

1

1

1

1

dp

c

dp

c

dpN HBT N HBT

ii

e eVs c S c S

h E

e eVs c S c S

h E

L L L L

ConclusionConclusion MZ interferometers (amplitude interferometers) exhibit the current

visibility with period h/e and the shot noise visibilities with periods of both h/e and h/2e. In contrast, HBT interferometers (intensity interferometers) exhibit no AB-effect in the current and only exhibit h/e-effect in the shot noise.

Our investigation shows the shot noise visibility of HBT interferometers as a function of temperature, voltage, dephasing rate is qualitatively similar to the h/e component of MZ interferometers. It is contrary to the naive expectation that the visibility of two particle processes of HBT interferometers should be related to the two particle processes, i.e. the h/2e component of MZ interferometers. Instead it is the number of times AB flux enclosed which decides the behavior of the visibility.

Documents

Introduction Model and theory Optical analogs in quantum Hall regime current and noise