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)
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