experiments with entangled photons

Experiments With Entangled Photons

Paulo Henrique Souto RibeiroInstituto de Física - UFRJ

Summer School of OpticsConcépcion January/2010

Quantum Optics Group at IF/UFRJ

Group membersExperiments:Prof. Paulo Henrique Souto Ribeiro Prof. Stephen Patrick Walborn

Theory:Prof. Luiz Davidovich Prof. Nicim ZaguryProf. Ruynet Matos FilhoProf. Fabricio Toscano

Msc and PhD students: Adriana Auyuanet Larrieu, Adriano H. de Oliveira Aragão, Bruno de Moura Escher , Bruno Taketani, Daniel Schneider Tasca, Gabriel Horacio Aguilar, Osvaldo Jimenez farias, Gabriela Barreto Lemos, Rafael Chaves.

UFRJ

UFMG

USP-SÃO PAULO

UFAL

UFF

Outline:

Part I-Simultaneity in parametric down-conversion-Violation of a classical inequality-Consequences of simultaneity: i)localized one-photon state; ii)the Hong-Ou-Mandel interferometer iii) measurement of the tunneling time

Part II-Polarization entanglement-Bell’s inequalities -Entanglement measurement

Part III-Entanglement dynamics-Kraus operators-Entanglement sudden death-Process tomography-Evolution of entanglement

Part VI-Spatial correlations-The transfer of the angular spectrum-Continuous variables etanglement- EPR paradox-Non-gaussian entanglement-Non-local optical vortex

Part I

- Simultaneity in parametric down-conversion

- Violation of a classical inequality

- Consequences of simultaneity:

i) localized one-photon state;

ii) the Hong-Ou-Mandel interferometer

iii) measurement of the tunneling time

Parametric Down-conversion

Espontaneous emission

Stimulated emission

TwinPhotons

p i s

p i sk k k

Parametric Down-conversion

Observation of simultaneity

Observation of simultaneity

Parametric down-conversion: quantum state

Time evolution

Time evolution operator

Time integral

Simultaneity in parametric down-conversion

Quantum state for weak interaction

Simultaneity in parametric down-conversion

Quantum state including some approximations

Simultaneity in parametric down-conversion

, , ,ˆ ˆ( ) ( )s i s i s iI t t E t E t t

ˆ ˆ ˆ ˆ, ( ) ( )i s s s i i i i s sC t t t E t E t E t E t t

Calculation of expectation values

.1ˆ,

i k r t

k kk

E r t l a e

Electric field operator

Intensity

Coincidence

Simultaneity in parametric down-conversion:very simple view

Simultaneity in parametric down-conversion:very simple view

0

1 2( ) 1 1i si t ti s i s i st c vac c d d v e

i tE t c d a e

2

, ( ) ( )

( )

i s s s i i i i s s

i i s s

C t t t E t E t E t E t t

E t E t t

Quantum state: simple version

Electric field operator: plane wave, almost monochromatic

Coincidence

Simultaneity in parametric down-conversion:very simple view

0 02

,

1 1i i s s

i s

i t t t i t t ti s i s i s

C t t

d d v e e

2

, i sii s i sC t t d e

1 2

1 2

0

2

1 2,

1 1

i s

i s

i t i t

i s i t ti s i s i s

d a e d a eC t t

d d v e

Plane wave pumping field 0 i s i sv

Coincidence detection

Coincidence detection

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 = 370ps

even

ts (

norm

aliz

ed)

time delay (ns)

Measurement of time delays

=168ps

=185ps

Simultaneity in parametric down-conversion:very simple view + detection filters

0 02

( ) ( 1)

,

1

i i s s

i s

i s

i t t t i t t ti s i s i s

C t t

d d ef efv

22 2

( ), i sis si iC t t d ef F

1 2

1 2

0

1 2

2

1 2 )

1

(

1

),

(

i s

i s

i t i t

i s i t ti s i s i s

f fd a e d a eC t t

d d v e

Plane wave pumping field i s i sv

Simultaneity in parametric down-conversion:very simple view + detection filters

( )f

0 5 10 15 20 25 30

0,0

0,2

0,4

0,6

0,8

1,0

= 3.8 x 1013 Hz

tran

smit

ance

(%)

frequency (Hz)

Interference filter: typical = 10nm, = 3.8 x 1013 Hz, t = 82 fs << 100ps

0 100 200 300 400 500 600

0,0

0,2

0,4

0,6

0,8

1,0

= 82 x 10-15 s

amplit

ude

time(fs)

( )f ( )tF

Simultaneity in parametric down-conversion:very simple view + timing resolution

Localized one photon state

Localized one photon state

Violation of a classical inequality

Violation of a classical inequality

Hong, Ou and Mandel Interferometer

Hong, Ou and Mandel Interferometer:single mode approach

Beam splitter Input-output relations

122

211

ariatb

ariatb

21

221

22211

122121

aaraataairtaairt

ariatariatbb

trFor

221121 aaaairtbb

22

211

21221

212111

aaraataairtaairt

ariatariatbb

11

222

21221

121222

aaraataairtaairt

ariatariatbb

Hong, Ou and Mandel Interferometer:single mode approach

Beam splitter Two-photon input state

Coincidence probability

2111 aa

011

11),(

2

2211

2

21

22121

21

21

aa

aa

aaaairt

bbbbbbC

011

11),(

2

222

112

1221

2

112

1111

21

21

aa

aa

aaraataairtaairt

bbbbbbC

011

11),(

21

21

112

222

1221

222222

aa

aa

aaraataairtaairt

bbbbbbC

Hong, Ou and Mandel Interferometer

2.1

i s

C e

( )f

.c

( )f

2c

Single-photon tunneling time

Part II

- Polarization entanglement

- Bell’s inequalities

- Entanglement measurement

Polarization entanglement:generation

Kwiat et al. PRL 75, 4337 (1995)

H V1

2HV ie

HH1

2VVie

Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)

Polarization entanglement:generation

V V1

2H Hie

Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)

Polarization entanglement:generation

12 1 2 1 2

1

2 H H V V

Mixed state

12 1 2 1 2

1

2 H H V V

Pure entangled state

Mixed states and entangled states

Detection of entanglement:violation of the Bell inequality

Bell-CHSH inequality

1 1 2 2 2 1 1 2, , , , 2 S E E E E

, , , ,,

, , , ,

C C C CE

C C C C

Bell inequality and Bell states

1,2 1 2 1 2

1

2H V V H 1,2 1 2 1 2

1

2H H V V

Bell states for the photon polarization

Coincidence rate for +: 2

, i sC E E

Bell inequality and Bell states

1,2 1 2 1 2

1

2 H H V V

Bell states for the photon polarization

2

2

2

2

2

,

cos cos cos cos

cos cos sin sin

cos

i s

i s i s

C E E

a a H H V V

H H V V

H H H H

Bell inequality and Bell states

Coincidence rate for +:

0 0 0 01 1 2 2

0 0 0 01 1 2 2

0 , 22,5 , 45 , 67,5

90 , 112,5 , 135 , 157,5

2

1 1 1 1

2

1 1 1 1

, , cos 22.5 0.854

, , cos 67.5 0.146

C C

C C

2

1 2 1 2

2

1 2 1 2

0.146

0.8

, , cos 67.5

, , cos 22 5 4. 5

C C

C C

Maximal violation

Bell inequality and Bell states

2

2 1 2 1

2

2 1 2 1

, , cos 22.5 0.854

, , cos 67.5 0.146

C C

C C

2

2 2 2 2

2

2 2 2 2

, , cos 22.5 0.854

, , cos 67.5 0.146

C C

C C

Maximal violation

Bell inequality and Bell states

0 0 0 01 1 2 2

0 0 0 01 1 2 2

0 , 22,5 , 45 , 67,5

90 , 112,5 , 135 , 157,5

1 2

0.146 0.146 0.854 0.854,

0.146 0.146 0.854 0 854

2

. 2

E

1 1

0.854 0.854 0.146 0.146 2,

0.854 0.854 0.146 0.146 2

E

2 1

0.854 0.854 0.146 0.146 2,

0.854 0.854 0.146 0.146 2

E

2 2

0.854 0.854 0.146 0.146 2,

0.854 0.854 0.146 0.146 2

E

1 1 2 2 2 1 1 2, , , 2, 2 2 .83 E E E ES

Maximal violation 0 0 0 01 1 2 2

0 0 0 01 1 2 2

0 , 22,5 , 45 , 67,5

90 , 112,5 , 135 , 157,5

Bell inequality and Bell states

Violation of a Bell inequality

- Detects but does not quantify the entanglement properly - Some entangled states do not violate the Bell inequality- Valid for dichotomic or dichotomized systems

Bell inequality and entanglement

Take a set of measurements :

(H,H); (H,V); (V,H); (V,V); (H,D); (H,L); (D,H); (R,H);

(D,D); (R,D); (R,L); (D,R); (D,V); (R,V); (V,D); (V,L)

C C C C C C C C

C C C C C C C C

Reconstruction of the density matrix

Quantum state tomography

Quantum state tomography

12

V V VV

V V V V V VV V

V

HH HH H HH H HH HH

HH H H H H H H

HH H H H H H H

HH H H

V V V V VV V

VV V VV V VV VV VV

Quantum state tomography

12

V V VV

V V V V V VV V

V

HH HH H HH H HH HH

HH H H H H H H

HH H H H H H H

HH H H

V V V V VV V

VV V VV V VV VV VV

Quantum state tomography

With one can compute all quantities related to the system

Concurrency:

0,

0

y y y

iC

i

Direct measurement of entanglement

Mintert, Kus, and Buchleitner, Phys. Rev. Lett. 95 260502 (2005).

12 01 10

2C P

Direct measurement of entanglement using copies of states

1 2

1 11 1 1 11 1

10;

20C

1

1 1

I / 2

1I / 4 ( )

41

14

P C

Direct measurement of entanglement:pure states

Pure state

Two copies

Maximally entangled state

Two copies

Experiment with entangled photons

1 21 2H2

VH1

Vie

Two copies of a state in a single photon

Polarization state

11 22

1

2ia be ba

Linear momentum state

Two copies of a state in a single photon

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 1;

2 212

i iMOM POL

i i

a a e b b H H e V V

a a e b b H H e V V

Simultaneous entanglement in polarization and linear momentum

Two copies of a state in a single photon

1 1

2 2aV bH aH bV

1

21

2

CNOT H V b b

CNOT H V a a

Bell state projection

Bell states combining momentum and polarization

aH bH aV aV

bH aH bV bV

C-NOT with a SAGNAC interferometer

Spatial rotations with cilyndrical lenses

Spatial rotations with cilyndrical lenses

Direct measurement of entangled with two copies

S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich,

F. Mintert, A. Buchleitner, Nature 440 1022 (2006)

Direct measurement of entangled with two copies

S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich,

F. Mintert, A. Buchleitner, Nature 440 1022 (2006)

Direct measurement of entangled with two copies

Part III

-Entanglement dynamics

-Kraus operators

-Entanglement sudden death

-Process tomography

-Evolution of entanglement

0,0

0,2

0,4

0,6

0,8

1,0

t

P1(e) e P

2(e)

0,0

0,2

0,4

0,6

0,8

1,0

?

t

Concurrency

Entanglement dynamics

1,2 1 2 1 2

1

2e g g e

1,2 1 2 1 2

in the computational basis

11 0 0 1

2

T. Yu, J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004). T. Yu, J. H. Eberly, Phys. Rev. Lett. 97, 140403 (2006).

0,0

0,2

0,4

0,6

0,8

1,0

t

P(e)

Amplitude decay channel

0 0 0 0

1 0 1 1 0 0 1

E ES S

E E ES S Sp p

Quantum channel and Kraus map

0,0

0,2

0,4

0,6

0,8

1,0

t

P(e)

Operadores de Kraus para o canal de amplitude

1 2 3 4

1 0 0 ˆ, , 00 1 0 0

pK K K K

p

†$( ) K K

Quantum channel and Kraus operators

0 0 0 0

1 0 1 1 0 0

0 0

0 1 0 1

1

E ES S

E E ES S

E ES S

E E

S

ES S S

H H

V p

p

V p H

p

Amplitude decay channel for one photon polarization

HH

V1 pp V H

Environment

Environment

Amplitude decay channel for one photon polarization

0 0 0 0

1 0 1 1 0 0

0 0

0 1 0 1

1

E ES S

E E ES S

E ES S

E E

S

ES S S

H H

V p

p

V p H

p

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

Amplitude decay channel for one photon polarization

V V1

2H Hie

Kwiat et al. PRA 60, R773 (1999)White et al. PRL 83, 3103 (1999)

Polarization entangled state

M. P. Almeida et al., Science 316, 579 (2007)

Experimental observation of theentanglement sudden death

M. P. Almeida et al., Science 316, 579 (2007)

HH VV3

ie

HH 3 VVie

Experimental observation of theentanglement sudden death

/ 2

/ 2

H

V

H V

R H i V

$ ,with ,

and , , ,

j j j j

j H V R

1 2 3 4

$ Kraus operators

, , e

j

K K K K

Process tomography

[( $) ]C I

Reconstruction of the Kraus operators

( )C

$

[( $) ]C I

'

[( $) ] [( $) ] ( )

For pure states

C I C I C

T. Konrad et al., Nature Physics 4, 99 (2008).

A dynamical law for the entanglement

[( $) ] [( $) ] ( )

For mixed states

C I C I C

$

'

( )C [( $) ]C I

[( $) ] [( $) ] ( ) C I C I C

A dynamical law for the entanglement

$

'

( )C [( $) ]C I

[( $) ] [( $) ] ( ) C I C I C

A dynamical law for the entanglement

$

'

$

[( $) ]C I

( )C [( $) ]C I

[( $) ] [( $) ] ( )C I C I C

A dynamical law for the entanglement

A dynamical law for the entanglement

O. Farias et al., Science 324, 1414 (2009)

A dynamical law for the entanglement

O. Farias et al., Science 324, 1414 (2009)

[( $) ] ( )C I C

[( $) ]C I

A dynamical law for the entanglement:experimental test

( ) mixed state C [( $) ]C I

Inequality

[( $) ] [( $) ] ( ) C I C I C

$

A dynamical law for the entanglement:generalization for mixed states

T. Konrad et al., Nature Physics 4, 99 (2008).

$'I

( )C [( $$') ]C I

[( $) ]C I

[( $) ] [( $$') ] ( )C I C I C

$' $

( )C $' $

[( $) ]C I

A dynamical law for the entanglement:generalization for mixed states

( )C [( $) ]C I

[( $) ] [( $$') ] ( )C I C I C

$' $

$

[( $ '$) ]C I $'

A dynamical law for the entanglement:generalization for mixed states

[( $') ]I

A. Jamiołkowski, Rep. Math. Phys. 3, 275 (1972)

$'

1 2tr 1 i i ii

r e e i i ii

r e f

How to find $'

1/ 2$' i j i j a i j babf f r r f e e f

A dynamical law for the entanglement:generalization for mixed states

O. Farias et al., Science 324, 1414 (2009)

[( $ '$) ] ( )C I C

[( $) ]C I

A dynamical law for the entanglement:generalization for mixed states

experimental test

Part VI

-Spatial correlations

-The transfer of the angular spectrum

-Continuous variables etanglement- EPR paradox

-Non-gaussian entanglement

-Non-local optical vortex

Spatial correlations in the far field

Spatial correlations in the far field

Spatial correlations in the far field

Spatial correlations in the far field

Spatial anti-bunching:non-classical behavior

Cauchy-Swartz inequality

Homogeneity and stationarity

0δ 2,22,2

t,,,, 222,2 ρIρIρρ 11

2ρρδ 1

For 0

222,2 ,, ρρρρ 11 C

-15 -10 -5 0 5 10 15

0,0

0,2

0,4

0,6

0,8

1,0

C(

)

2,2 C δ δ

Spatial anti-bunching:non-classical behavior

S. Mancini, V. Giovannetti, D. Vitali, and P. TombesiPhys. Rev. Lett. 88, 120401 (2002).

S. Mancini, V. Giovannetti, D. Vitali, and P. TombesiPhys. Rev. Lett. 88, 120401 (2002).

2 2

2 1 2 1 1x x p p

Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. ZollerPhys. Rev. Lett. 84, 2722 (2000).

Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. ZollerPhys. Rev. Lett. 84, 2722 (2000).

2 2

2 1 2 1 2x x p p

Inseparability

DGCZ criterion

MGVT criterion

Inseparability

2 2 22 1 2 1 2

2 2

2 1 2 1

1

. : 1 2

The state is inseparable if

x x p p aa

ex a x x p p

Inseparability:proof

1 2 1 2

1 1;u a x x v a p p

a a

1 2 ii

p

2 22 2 2 2

2 2 2 2 2 21 2 1 22 2

1 2 1 2

2 2

1 1

2

i i ii

i i i i ii

i ii i i ii i

u v p u v u v

p a x x a p pa a

ap x x p p p

a

u v

Inseparability:proof

2 2

2 2 2 22 21 2 1 22 2

2 22 2

2 2 2 221 1 2 22

2 22 2

1 1

1

iì ì ì ìi

i i i iì i ì ii i i i

iì ì ì ìi

i i i iì i ì ii i i i

u v

p a x x a p pa a

p u p u p v p v

p a x p x pa

p u p u p v p v

Inseparability:proof

2 2

1 1 1 1

2 2

2 2 2 2

1,

, 1

From the Heisenberg uncertainty principle:

ì ì

ì ì

x p x p

x p x p

2 2 22

2 22 2

1 11

ii

i i i iì i ì ii i i i

u v p aa

p u p u p v p v

Inseparability:proof

22

22

1

From the Cauchy-Schwartz inequality:

and

i i i

ii

ìi i i

i i iìi i i

p p u p u

pp p v p v

2 2 22

2 2 2 2

11

i i i ii i i i

i i i i

u v aa

p u p u p v p v

0

Inseparability criterion

2 2 22

1

u v a

a

DGCZ criterion

Inseparability

22 2

2 1 2 1

2 2

,

. : , ,

The state is inseparable if

i i

i i i i

x x p p x p

ex x p i x p

Inseparability:proof

1 2 ii

p

2 2

222 2 2

1 22

222 2 2

1 22

1

1

i i iì ii ii i i

i i iì ii ii i i

u v

p a x x p u p ua

p a p p p v p va

1 2 1 2

1 1;u a x x v a p p

a a

Inseparability:proof

2 2

2 2

2 2 21 22

2 2

2 2 21 22

1

1

i i ii ii ii i i

i i ii ii ii i i

u v

p a x x p u p ua

p a p p p v p va

22

1i i i ii

ìi i i

p p v pp v

Using the Cauchy-Schwartz inequality:

and

Inseparability:proof

2 2

2 2 21 22

2 2 21 22

01

10

i i ii

i i ii

u v

p a x xa

p a p pa

22

1i i i ii

ìi i i

p p v pp v

Using the Cauchy-Schwartz inequality:

and

Inseparability:proof

2 2 2 2 2 21 2 1 22 2i ii i i i

i i

u v p x x p p p

2 2 2

Using the inequality:

Using again the Cauchy-Scwarz inequality:

2 2 2 21 2 1 2

212 2 2 2 4

1 2 1 2

i ii i i ii i

i i i i ii

p x x p p p

p x x p p

21

2 2 2 2 2 2 41 2 1 24 i i i i i

i

u v p x x p p

Inseparability:proof

Using the uncertainty principle:

2 2

1 1 1 12 2 2 21 1 2 2

, ,

4 4and

i i i i

x p x px p x p

21

2 2 2 2 2 2 41 2 1 24 i i i i i

i

u v p x x p p

2 22 22 21 2 1 2 1 1, ,i iu v x p x x p p x p

MGVT criterion

Inseparability

2 2 21 2 1 2x x p p

1 1

2 2 22 0 2 0| | 0.01x px p

Inseparability

Inseparability

2 2 21 2 1 2x x p p

1 1

2 2 22 0 2 0| | 0.01x px p

Inseparability

1

1

2 2

2 0 1 2

2 2

2 0 1 2

|

|

x

p

x x x

p p p

I t is claimed that

Therefore the inequality is violated

1 1

1 1 1

1 1 1

2 0 2 0

22 22 0 2 2 0 2 2 2 0 2

22 22 0 2 2 0 2 2 2 0 2

( | ) ( | )

| ( | ) ( | )

| ( | ) ( | )

and is measured andx p

x x x

p p p

P x P p

x x P x dx x P x dx

p p P p dp p P p dp

1 1

2 2 22 0 2 0| | 0.01x px p

Non-gaussian entanglement

Gaussian states are completely characterized by the secondorder momenta:

22 2 ( ) ( )x x P x dx x P x dx

Then, DGCZ, MGVT and other criteria based on second ordermomenta are non optimal for non-gaussian states.

Higher order criterion

E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005)

To the second order:

† †

† † † † † † †

† †2

† †

† † † † † † †

1 a a b b

a a a a a a b a b

M a aa a a ab ab

b ab a b b b bb

b ab a b b b bb

a and b are annihillation operators for modes a and b.

Higher order criterion

E. Shchukin and W. Vogel Inseparability criteria for continuous bipartite quantum states. Phys Rev Lett. 95, 230502 (2005)

† †

† † † † † † †

† †2

† †

† † † † † † †

1 a a b b

a a a a a a b a b

M a aa a a ab ab

b ab a b b b bb

b ab a b b b bb

The state has a positive partial transpose, if and only if all principal minors are non-negative.

Gaussian and non-gaussian states

Production of a gaussian state with parametric down-conversion

2 2 2 2, exp / 4 exp / 4x x N x s x t

Gaussian and non-gaussian states

Production of a non-gaussian state with parametric down-conversion

2 2 2 2, exp / 4 exp / 4x x N xx s x t

01 modeHG

Higher order criterion

We found a non-gaussian state that does not violate any second order criterion:

2 2 2 2, exp / 4 exp / 4x x N xx s x t

According to R. Simon Phys. Rev. Lett. 84, 2726 (2000), if

0 ;

,

a b a b a b a bx x p p x p p x

x x x

with

is satisfied, no second order criterion is violated.

For 0.57 < s/t < 1.73 satisfies the inequality.

Higher order criterion

†

† † †

1

HO

ab

D

a b a ab b

2 2 2 2, exp / 4 exp / 4x x N xx s x t

However it gives the negative minor below for the higher order criterion

Isomorphism between a multimode singlephoton field and a single mode multiphoton field

† †1 1

2 2anda a

ir x a a p a a

r

4 2 2 2 24

2 2 2 2 22

11

12 2 0

a b a b a b a b

a b a b a b a b a b a b

r x x x p p x p pr

x p p x x x p p r x x p pr

The inequality is violated for r=1/t and 0.68 < s/t < 1.53

Experimental observation of genuine non-gaussian entanglement

Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)

Experimental observation of genuine non-gaussian entanglement

Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)

Experimental observation of genuine non-gaussian entanglement

Quantum entanglement beyond Gaussian criteria R. M. Gomes, A. Salles, F. Toscano, P. H. Souto Ribeiro and S. P. WalbornProc. Nat. Acad. Sci. 106, 21517-21520(2009)