Particle-particle correlations produced by dynamical scatterer

M. Moskalets Dpt. of Metal and Semiconductor Physics,

NTU "Kharkiv Polytechnical Institute", Ukraine

Keszthely, 2006

Pump is a source of entangled particles

P.Samuelsson and M.Bűttiker,Phys.Rev.B 71, 245317 (2005)

C.W.J.Beenakker, M.Titov, B.TrauzettelPhys.Rev.Lett. 94, 186804 (2005)

A weak amplitude pump

The current noise (CN) is a measure of non-classical correlations:- the BI’s can be formulated in terms of a CN;- the CN produced by the pump violates BI’s.

An arbitrary amplitude pump(a projected state with exactly a single excited electron-hole pair)

The entanglement entropy (of spins) for e-h pairs: relates to CN at weak pumping

while it is unrelated to CN at strong pumping

Thus a current noise (possibly) gives incomplete information about correlations produced by the pump

Our objectives

1. To explore correlations produced by the dynamical scatterer (a pump) at arbitrary in strength but slow driving

2. To establish a relation between the current noise and the particle-particle correlations at strong driving

pump

The set-up

T = 0

=

= 1, 2,…, Nr = 1

= 2

= Nr

= 3 = 4

S(t) = S(t+)

= 2/

All reservoirs are uncorrelated incoming particles are uncorrelated

In a given set up

stationary scatterer does not produce correlationswhile

dynamical one (a pump) can produce

(to illustrate it we analyze an outgoing state)

Outgoing state

1. The stationary case: E(in) = E(out) = E

We will consider particles in states with definite energy E and describe them via the second quantization operators a(E) (for incoming particles) and b(E) (for out-going particles).Perhaps it is better to speak about incoming and out-going modes (single-particle states).However one can speak about the particles belonging to these states.

NrNr

Since all relevant incoming states are either filled (for E ) or empty (for E > ) then due to unitarity of scattering all out-going states are either filled or empty. Thus there are no correlations: it means that the probability for the state to be filled/empty is fixed: 1 or 0 (at zero temperature)

uncorrelated uncorrelated

=1

=Nr

incomingparticles

out-goingstates

The states with different E E’ are statistically independent (therefore we consider the states with the same E)

E

Outgoing state2. The dynamical case: E(in) E(out) = E(in) + nћ, n = 0,±1, ± 2,…, ± nmax

The states with E > + nmaxћ / E < - nmaxћ are fully empty/filled and thus irrelevant

Therefore there are 2nmaxNr relevant state. But only 1/2 incoming states are filled

2nmax

=1 ћ

nmaxNr incoming electrons 2nmaxNr out-going states

The (particles belonging to the) partially filled states can be correlated

E(in)

single-particle occupations are shown

In general,

the dynamical scatterer produces

2-, 3-,…,nmaxNr- particle correlations while

what we see (the order of visible correlations)

depends strongly on how we look at the system

Registered state1. To obtain complete information about outgoing particlesit is necessary to monitor all the relevant 2nmaxNr outgoing stateswhich contain nmaxNr electrons

1

2

3

4

5

6

For instance, one can register the state with exactly a single excited electron-hole pair

3ћ

Such a (registered) state is a multi-electron (3-electron in our case) state.

To characterize it we have to use a multi-electron joint probability which includes multi-particle correlations

single-particle occupations are shown simultaneous occupations are shown

In a presented case it is P(11;15;16) which includes 2-, and 3- electron correlations

Registered state2. If we monitor only several (say 2) outgoing states we get incompletedescription of a whole (multi-particle) outgoing state.However such a description is useful if only these states are in use.For instance, any 2-particle quantities, e.g. a current noise, “monitor” only the states in pairs.

1

4

Other states can be arbitrary occupied.(and contain 1, 2, 2, and 3 electrons, respectively)

Two-particle probabilities P(X1;Y4) include only 2-particle correlations.

P(11;14) P(01;14) P(11;04) P(01;04)

( )1 4

, 0,1P X ; 1

X YY

==å

Reduction of the order of correlations

1. A weak amplitude pump:nmax = 1 the out-going state is an Nr -electron state.

Nr = 2 Nr0 : there are Nr0 orbital channels and 2 spin channels.

2. Spin-independent scattering:

the out-going state is a product of two (spin ,) nmaxNr0 -electron states.

For weak pumping (nmax = 1), spin-independent scattering, and for Nr0= 2 (two single channel leads) the out-going state is effectively a 2-electron state.

Therefore, in this case the current noise represents all the correlations produced by the pump.

Otherwise, the current noise represent only part of correlations.

In a general case there are nmaxNr out-going electrons

To investigate particle-particle correlations we calculate a joint probability

to find several out-going channels occupied and compare it tothe product of occupation probabilities of individual channels

The single-channel occupation probability is a one-particle distribution function.

The joint multi-channel occupation probabilities are multi-particle distribution functions.

Single-particle distribution function

S(t)

a(Em)

b(En) En = E + nћ , < E < + ћ

adiabatic driving: SF(En,Em) = Sn-m()

p

ppnn ESE ab ,

1,1

0,00

†)(

p

pEfEEEf pppp

in aa

1,1

0,0

1

2

,†)(

n

nSEEEf

ppnnnn

out

bb

(it is a sum of squared single-particle scattering amplitudes)

incoming particles:

out-going particles:

Two-particle distribution function

B,(En,Em)

mnmnmnout EEEEEEf ,,, ,

†,

)(, BB

a two-particle operator: (the order is irrelevant)

a two-particle distribution function (a joint probability):

an electron-electron correlation function (a covariance):

,

2

( ), ,

1

, 0outn m n p m p

p

f E E S S

mnnm EEEE bbbb ††

while incoming electrons are not correlated: ,0,)(, mninf

Electrical noise and two-particle correlations

The zero-frequency current noise power produced by the pump at arbitrary driving amplitude can be expressed in terms of electron-electron (2-particle) correlations:

tItItItIddt ˆˆˆˆ

2

1

0

P

The factor /2 counts all the statistically independent sets of states within the interval 0 < E - < ћ.

Multi-particle correlations

NjiS

N

EEfNEEfP

jiji

N N

NNNN

pnN

ijp p

N

nnout

nnout

N

,,1,,,det!

1

,...,1,...,1:;...1

,)(

1 1

2)(

)(†)(1

1 1

1,...,11

,...,1

MM

BB

i) a multi-particle (N-particle) operator:

ii) a multi-particle distribution function in terms of NN Slater determinants:

1

1

††1)(

11111,...,1 1,...,

N

jnn

Pnn

Nnn

out

jnjjjn

N

nNNnNN EEEEEEf bbbb

PN = (n1,n2,…,nN) is a permutation of integers from 1 to N. The cyclic permutations are excluded.

NNN bbbbbbbbbb 14332211 †††††

iii) a multi-particle correlation function:(it is a sum of squared multi-particle scattering amplitudes)

Multi-particle correlations

0

)( ln1

,...,1

ΛKITrf

N

N

Nout

A generating function:

1,,

†,p

pmpnnmmn SSK bb

Here: A pair correlator:

The unit matrix: A diagonal matrix:

1

2

0

0

Λ

1 0

0 1

Multi-particle correlationsA three-particle distribution function:

,,,,,,, fffffffffff

1,,

1,,

1,,Re2,,,,

rrnrl

qqlqm

ppmpnlmn SSSSSSEEEf

The sign of correlations:stationary driven

2-particle

3-particle

- , < 0

2, , 0><

Higher order current cumulants and multi-particle correlations

Nth-order current correlation function (symmetrized in lead indices):

(the sum runs over the set of all the permutations PN=(r1,…rN) of integers from 1 to N; 1=0)

The zero frequency current cross-correlator (different leads) can be expressed in terms of the N-particle correlation functions for outgoing particles:

( ) ( ){ }

( ) ( )

( ) ( ) ( )

( )

1

1

1 1

, 11

, 1 1

, , 1

1 ˆ, ,!

, , ,

ˆ ˆ ˆ ,

0, , 0 .

N jjN

N

j j jj j j

N N

N

N rrP j

N N

r r rr r r

N

IN

I I I

a a a

a a

a a a

a a a a

w w w

w w d w w

w w w

w w

=

¢ = D

= + + +

D = -

º = =

å ÕP

P

P P

L

L

L L

K

K K K

K

Higher order current cumulants and multi-particle correlations

The multi-particle correlation functions ( i.e., irreducible parts of multi-particle probabilities )

are the quantities which are directly related to the higher order current cumulants

( ) ( )

( ) ( )

1

2

~cos ,

~cos

V t t

V t t

d w

d w j+

IL IR

V1(t) V2(t)

Example: a resonant transmission pump

-2 -1 0 1 2 3

0.00

0.05

0.10

0.15

0.80

0.85

0.90

0.95

1.00

Single-particle distribution function

weak pumping, a single-particle distribution function

fL

n = 0

n = -1

ћ

E-, ћ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

weak pumping, a single-particle distribution function - the dependence on

, 2

fL(n=0)

f(h)L(n=-1)

-15 -10 -5 0 5 10 15

0.00

0.05

0.10

0.15

0.80

0.85

0.90

0.95

1.00

strong pumping, a single-particle distribution function

E-, ћ

fL

IL 1e/cycle = /2

no dc current = 0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

0.05

0.10

0.15

0.20

strong pumping, a single-particle distribution function - the dependence on

fL(n=0)

f(h)L(n=-1)

, 2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

0.01

0.02

0.03

Two-particle correlations

weak pumping, the dependence on at transmission resonance

, 2

fL,n=0 f(h)R,m=-1

f (1L,n=0 ;0R,m=-1)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.00

0.01

0.02

0.03

strong pumping, the dependence on at transmission resonance

, 2

fL,n=0 f(h)R,m=-1

f (1L,n=0;0R,m=-1)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

-0.0015

-0.0010

-0.0005

0.0000

0.0005

Three-particle correlations strong pumping, the dependence on , at transmission resonance

, 2

f

(L,0; L,+1; R,-1)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

strong pumping, the dependence on , at transmission resonance

, 2

fL, 0 fL, +1 f(h)R, -1

f (1L,0;1L,+1;0R, -1)

+ 2-particle correlations

Conclusion

• The current noise generated by the pump can be expressed in terms of two-particle correlations at arbitrary strength of a drive

• The N-particle distribution functions depends on multi- particle correlations up to Nth order

•The multi-particle correlations can be experimentally probed via the Nth-order cross-correlator of currents flowing into the leads attached to the pump