Surface plasmon polaritons (SPP):Surface plasmon polaritons (SPP): an alternative to cavity QED
Strong coupling to excitons & Intermediary for quantum entanglementIntermediary for quantum entanglement
• A Gonzalez Tudela D Martin Cano P A Huidobro• A. Gonzalez-Tudela, D. Martin-Cano, P. A. Huidobro, E. Moreno, F.J. Garcia-Vidal, C.TejedorUniversidad Autonoma de MadridUniversidad Autonoma de Madrid
L M i M• L. Martin-MorenoInstituto de Ciencia de Materiales de AragonCSIC
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• ConclusionConclusion
Intro: Surface plasmon polaritons
p
Dielectric response of a metal is governed by free electron plasma:
i
p
2: : plasma frequency
d i f tBelow its plasma frequency Below its plasma frequency (()) is negativeis negative......
ßß
i : : damping factor
ck :wavevector purely purely imaginary photonic insulatorimaginary photonic insulator
What is a surface plasmon polariton ?p p
Intro: Surface plasmon polaritons
p
Dielectric response of a metal is governed by free electron plasma:Dielectric response of a metal is governed by free electron plasma:
i
p
2: plasma frequency
d i f tBelow its plasma frequency Below its plasma frequency (()) is negativeis negative......
ßß
i : damping factor
ck :wavevector purely purely imaginary photonic insulatorimaginary photonic insulator
What is a surface plasmon polariton ?p p
Intro: Surface plasmon polaritonsElectromagnetic radiation in dielectric Å Localized Plasmons in a metal surface
ßSURFACE PLASMON POLARITONS
200nm 500nm
Intro: Surface plasmon polaritonsSPP Length Scales span photonics and nano
P t ti d thPenetration depth into metal
Penetration depth into dielectric
SPP wavelengthSPP propagation
length
δm δd λSPP δSPPNon-local
effects
SPP LRSPP
10 nm1 nm 100 nm 1 cm1 mm1 μm 100 μm10 μm 10 cm
Length scales span 7 orders of magnitude!
Intro: Surface plasmon polaritons
Intro: Surface plasmon polaritons
•Propagation length: 50-100 mm (Ag or Au) Û lifetime £ 1 psInterestingInteresting featuresfeatures of of SPPsSPPs forfor photonicphotonic circuitscircuits:
p g g m ( g ) p•Two-dimensional character of EM-fields•Optical and electrical signals carried without interference
Intro: Surface plasmon polaritons
•Propagation length: 50-100 mm (Ag or Au) Û lifetime £ 1 psInterestingInteresting featuresfeatures of of SPPsSPPs forfor photonicphotonic circuitscircuits:
p g g m ( g ) p•Two-dimensional character of EM-fields•Optical and electrical signals carried without interference
1( 10 100 ) /SPP dk m c Beyond the diffraction limit
p 1 d
Intro: Surface plasmon polaritons
•Propagation length: 50-100 mm (Ag or Au) Û lifetime £ 1 psInterestingInteresting featuresfeatures of of SPPsSPPs forfor photonicphotonic circuitscircuits:
p g g m ( g ) p•Two-dimensional character of EM-fields•Optical and electrical signals carried without interference
1( 10 100 ) /SPP dk m c Beyond the diffraction limit
p
One problem: coupling in and out to SPPs
1 d
coupling in and out to SPPs
k k Ph t l k it ti i id th li ht3D dk k c Photons can only make excitations inside the light cone
While SPP are outside the light cone
One problem: coupling light to SPPs
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• ConclusionConclusion
Quantization of plasmons (without losses)Q p ( )
Q ti ti f l t i fi ld( · )(( ) ( ))
2z
ki k k z
k k
kE r u z e aA
Quantization of an electric field 0( ) ( )
2( , )
kk kAr z
1 | |ˆ ˆ( )( )
( )zk z zkz
k
ku z e u uikL k
i h( )
( ) ( )( ) ( ) 1( )m d mdL
with
( ) ( ) 12 ( ) | ( ) |
( )m dd m
Ldk
ff ti l th t li th f h d
†( )EMH k a a effective length to normalize the energy of each mode,
( )EM k k kH k a a
Interaction of 1 quantum emitter (QE) with SPP†
0QEH
z( )· ( )U dr r E r
Interaction with a dipole
( )· ( )U dr r E r
0 0( · ( ) ) † ( · ( ) ) †( ; )
( ) ( )( )k
i t i ti k r k t i k r k tint k k
g k zH t a e a e e e
A
A
( ) | |ˆ ˆ( ; ) · ( ) ·( )2 ( )
zk zzkk k
k kg k z E u z e u ukL k
2 3 30 0 03 /c
02 ( )zkk k
zkL k decay rate
of bare QE
Interaction of 1 quantum emitter (QE) with SPP†
0QEH
z( )· ( )U dr r E r
Interaction with a dipole
( )· ( )U dr r E r
0 0( · ( ) ) † ( · ( ) ) †( ; )
( ) ( )( )k
i t i ti k r k t i k r k tint k k
g k zH t a e a e e e
A
A
( ) | |ˆ ˆ( ; ) · ( ) ·( )2 ( )
zk zzkk k
k kg k z E u z e u ukL k
2 3 30 0 03 /c
02 ( )zkk k
zkL k
I RWA † · † ·( ; ) ( )ik r ik rg k zH decay rateof bare QE
In RWA † †( ; ) ( )ik r ik rint k k k
gH a e a e
A
Interaction of 1 quantum emitter (QE) with SPP† , 0 One QE with w0 only couples to a
bright SPP º symmetric linear
z
bright SPP º symmetric linear comb. (J0 Bessel funct.) of all themodes with
0; SPPk k
The higher coupling does notcoincide with the higher b-factor
plradiation to plasmonstotal radiation
Scheme of the quantum dynamics of an open system
gm
Solving TOT TOT TOTi Ht
is complicated and unnecessaryt
Scheme of the quantum dynamics of an open system
γQEgm gm
γSPP=vg/LSPP
Solving TOT TOT TOTi Ht
is complicated and unnecessary
Tracing out the reservoir’s
t
degrees of freedom
Dynamics of the QE population: Weisskopf-Wigner
[ ( )] ( )2
0
( ; ) ( ) ( ) / 2( )
ci k i
t
K t z c d c tc t
With dissipation0 0[ ( )] ( )2
0
2
( ; ) | ( ; ) | ( ; )
1 ˆ( ; ) Im[ ( )][ ]
ci k iSPP
k
K z g k z e d J z e
J z G r r
With dissipationWithout dissipation
SPP 0 020
0 0
( ; ) Im[ ( , , )]
ˆ( ; ) ... ( , , )...
[ ]SPPT
J z G r rc
J z G r r
0 0 0( ) 2 [ ( ; ) (PT
SPJz J z 0 0; )]z
Dynamics of the QE population: Weisskopf-Wigner
[ ( )] ( )2
0
( ; ) ( ) ( ) / 2( )
ci k i
t
K t z c d c tc t
With dissipation0 0[ ( )] ( )2
0
2
( ; ) | ( ; ) | ( ; )
1 ˆ( ; ) Im[ ( )][ ]
ci k iSPP
k
K z g k z e d J z e
J z G r r
With dissipationWithout dissipation
SPP 0 020
0 0
( ; ) Im[ ( , , )]
ˆ( ; ) ... ( , , )...
[ ]SPPT
J z G r rc
J z G r r
0 0 0( ) 2 [ ( ; ) (PT
SPJz J z 0 0; )]z
•For a single QE, the SPP spectral densityJSPP governs the dynamics !•Non lorentzian shape of JSPPÞ differentp SPP dynamics that a pseudomode (cavity QED) !•Height/width ratio of JSPP determines thepossible reversibility !possible reversibility ! •g0
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglementgg p gp g yy qq gg
Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• ConclusionConclusion
Exciton collective mode of emitters in a plane
More complicated system:
†,ij ij
emitters in a plane Dynamics described bymaster eq. for density matrix
& quantum regression th
zj0
,ij ij
† †(k)sN
H a aH
& quantum regression th.
† †, ,0
10 (k) N i jpl k ki j
i k
H a aH
· ·† †( ; )( )NNs ik r ik rj i iH
g k za e a e
† †, ,
1( )Nint i iH i j i jk k
ika e a e
A
( ) | |ˆ( ; ) ·2 ( )
( )z jk zj zkk kg k z e u u
kL k
0
( )2 ( )
( )j zkz
gkL k
No fitting !!!
Exciton collective mode of emitters in a plane
More complicated system:
†,ij ij
emitters in a plane Dynamics described bymaster eq. for density matrix
& quantum regression th
zj0
,ij ij
† †(k)sN
H a aH
& quantum regression th.
† †, ,0
10 (k) N i jpl k ki j
i k
H a aH
· ·† †( ; )( )NNs ik r ik rj i iH
g k za e a e
† †, ,
1( )Nint i iH i j i jk k
ika e a e
A
Collective mode ( ) | |ˆ( ; ) ·2 ( )
( )z jk zj zkk kg k z e u u
kL k
Holstein-Primakoff transf.(low excitation Þ no saturation)
ß
0
( )2 ( )
( )j zkz
gkL k
No fitting !!!
·† †,
1
1( )
1
si
Niq r
j i jis
D q eN
† * †
1,
( ; )( )· ( ) ( )· ( )
1
( )sN
j sint i j i jk k
ik q s
N
g k z nH S k q ra D q S k q ra D q
N
·† †,
1 ( ) iiq ri j jqs
D q eN
·1
1( )s
i
Nik r
is
S k eN
Structure factor
Experimental evidence of t li fstrong coupling of SPP & excitonsQE are not just in a plane
Ensembles of organic molecules• J. Bellessa, et al, Phys. Rev. Lett. 93, 036404 (2004).• J. Dintinger, et al, Phys. Rev. B 71, 035424 (2005).
k l l h ( )• T. K. Hakala, et al, Phys. Rev. Lett. 103, 053602 (2009).• P. Vasa, et al, Nano Lett. 12, 7559 (2010).• T. Schwartz, et al, Phys. Rev. Lett. 106, 196405 (2011).
Semicond. nanocrystals & Quantum wells • P. Vasa, et al., Phys. Rev. Lett. 101, 116801 (2008).
ll l h 8 (2008)D=110meV
• J. Bellessa, et al, Phys. Rev. B 78 (2008).• D. E. Gomez, et al, Nano Lett. 10, 274 (2010).• M. Geiser, et al, Phys. Rev. Lett. 108, 106402 (2012).
(Semi)-classical description
20
2 20
1 2 / 3/ ;1 1/ 3
Nf e mi N
Polarizability of 1 emitter
Eff. dielectricfunct of emitters 0 1 1/ 3i N 1 emitter funct. of emitters
e5 =1T@0 3
20 00 02 2 2
2 23m cmf
Oscillator strength
e t l
e(w)e3 =1
SPP
Absorbances
W
d
0 02 2 203e e from microsocpic info.
emetale1 1 R
AbsorbanceA=1-R-T
d
k
e1=3; d=50nm; s=1nm; W=500nm
ω0=2eV; N=106 μm-3
γ =γ0 +γdeph ; γ 0 =0.1meV; γdeph=40meV
Excitonic collective modein the volume of width W
( ; )jg k z
Coupling depends on distance zj
More complicated collective mode
Average of randomorientations
For many QE with disorder momentum is conserved,0( ) kS k q
† †( ; ) ( ) ( )( )LN
H g k z n a D k a D k
int1
†† †
( ; ) ( ) ( )
; [ ( ), ( )]1( ) ( ; ) ( )
( )L
j s j jk kjk
i j ij
N
j jND k g k z D k
H g k z n a D k a D k
D k D k
2 2
1; [ ( ), ( )]
( ) | ( , ) | d | ( , ) |
( ) ( ; ) ( )( )
L
i j ij
N sNj
j jNj
g
g k g k z n z g k z
g k
No fitting !1
† †0
( ) | ( , ) | | ( , ) |
( ) ( ) ( ) ( ) ( )
j sj
N Nk k
g g g
H D k D k a k a k g k
† †( ( ) ( ) ( ) ( ))a k D k a k D k
g
Decay of the collective mode 22 d ( ) | ( , ) || ( ) |ks N
D N s
n z z g k zg k
Dynamics under coherenti f S ipumping of a SPP with k-vector
Average of random orientations
CoherentCoherentpumping
(classical field) Reservoirs
Dynamics under coherenti f S ipumping of a SPP with k-vector
† † † †0
†
( ) ( ) ( ) ( ) ( )( ( ) ( ) ( ) ( ))
( ) ( )L L
N Nk k
i t i tL
H D k D k a k a k g k a k D k a k D k
H t
†
[ ]
( ) ( )
k
L
k
L
D aN L
i t i tLk k k kH t a e a e
i H H
†[ , ] 2 2 2k kk kD ak k k k D Di H H
Excitondecay
Plasmondecay
Pure dephasing(vibro-rotation)
† † †(2 )c c c c c c c decay decay (vibro rotation)
Dynamics under coherenti f S ipumping of a SPP with k-vector
† † † †0
†
( ) ( ) ( ) ( ) ( )( ( ) ( ) ( ) ( ))
( ) ( )L L
N Nk k
i t i tL
H D k D k a k a k g k a k D k a k D k
H t
†
[ ]
( ) ( )
k
L
k
L
D aN L
i t i tLk k k kH t a e a e
i H H
†[ , ] 2 2 2k kk kD ak k k k D Di H H
Excitondecay
Plasmondecay
Pure dephasing(vibro-rotation)
At the crossing (k0) between exciton and SPP,† † †(2 )c c c c c c c
decay decay (vibro rotation)
Rabi splitting is analytical
22 2[ ( )] ( ) / [ ( )]4N NR k i h k
0 0
220 0
2[ ( )] ( ) / [ ( )]4k k
ND a
NR g k with g k n
Strong coupling betweenSPP & excitons
0 0.1meV
N d d W•gN depends on W with saturationdue to SPP z-decay
•gN practically independ. on sg p y p
gs i (s=1)
0 1 Ng
gs,iso(s=1)
gspp
6 3
0.1
10k g
n m
0 2eV
Strong coupling betweenSPP & i
1 ; 500N
s nm W nm SPP & excitons
0
0
(40 )2 ; 0.1
0.1 ; 0.1 ( )
Nk
N
meVeV g
meV g RT
Polariton populations µ absorption spect.
6 310 6 310n m
Strong coupling betweenSPP & i
1 ; 500N
s nm W nm SPP & excitons
0
0
(40 )2 ; 0.1
0.1 ; 0.1 ( )
Nk
N
meVeV g
meV g RT
Rabi splitting (at k0)
220
20 ;[ ( )] ( [ ( )]) / 4
ND
Ng kR k ng
Polariton populations µ absorption spect.0 0
00 ;[ ( )] ( [ ( )]) / 4k kD a g kR k ng
Weak Strong6 310 coupling coupling6 310n m
At RT, the incoherent processes (gf ) determine a critical density for observing strong coupling
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• ConclusionConclusion
QE-QE coupling mediated by plasmonic waveguidesQ Q p g y p g
Cilindrical wire WaveguideV-Channel W vegu deto reinforce
QE-QE effects
Fields are stronger in the channelthan in the cilinderthan in the cilinder
1QE: b and Purcell factors
Metallic nanostructures increase the emission from a QE (Purcell) but,
I i l h i i f SPP’ ??? (b)
; pltotal radiation radiation to plasmonsPurcell factor factorQ d l d
Is it always a coherent emission of SPP’s??? (b)
0
;f fQE radiation to vacuum total radiation
The channel is more convenient than the cilinder
b and Purcell factorsh=20nm
b- factor is very stable in a broad range of dipoleorientations while Purcellf t d i ifi tl
h=150nmfactor decreases significantlywhen the dipole is notproperly orientedproperly oriented
Dispersion & b factor for V-channel
h=140nmV‐angle= 20 degrees
b factorb-factor
Two QE’s dynamicsyAll the degrees of freedom (SPP, dissipation, radiation) can be traced outproducing effective coherent & incoherent interactions between the twoproducing effective coherent & incoherent interactions between the twoQE’s that can be computed from the classical Green’s function:
3 † †ˆ ˆ ˆˆ ˆ( ) ( ) [ ( ) ]H d d d h
d E 3 † 00 01 2 1 2
†
, ,
ˆ ˆ( , ) ( , ) [ ( , ) . .]i i i ii i
H d d a d ha c
r r r ω d E r 2
3ˆ ( ) ( ) ( ) ( )i d E G 3
20
( , ) ( , ) ( , , ) ( , )ii d ac
E r r r G r r ω r
† † †ˆ 1ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ[ ] ( 2 )i H H σ σ σ σ σ σ,
[ , ] ( 2 )2s ij i j i j jL ii j
H Ht
σ σ σ σ σ σ
0† †ˆ ˆ ˆ ˆ ˆ( )s i i i ij i jH g † .ˆ .1ˆ Li i i tL e h cH 0( )s i i i ij i j
i i j
g
2 i iiL 2
*00
2 ( )Im d G r r d2
*00( )g Re
d G r r d 020
( , , )ij i i j jImc
d G r r d
020( , , )ij i i j jg Rec
d G r r d
Scheme of levelse
gQE
1 2 1 2 1 2 1 213 0 ( )e e g g g e e g
g
1 2 1 2 1 2 1 23 0 ( )2e e g g g e e g
Modulation of g12 would allow to swicth on/off red and blue paths
Two QE’s dynamicsIt is possible to identify the effects of SPP & dissipationSPP Green’s function t t t t( ) ( )i E r E r
y
SPP Green s functionß
Effective interactionsCoherent
( )t *t
0
( ) ( )( , )2 ( )
ik zS
zP
zP
S
i edS
E r E rG r ru E H
CoherentIncoherent
/2,pl r
/2
sin( )2
cos( )
dij ij
d
g g e k d
e k d
,pl rcos( )ij ij e k d
p/2 shift allows switching on/off •Coherent versus incoherent interactions•Control of different decay paths
Two QE’s dynamicsIt is possible to identify the effects of SPP & dissipationSPP Green’s function t t t t( ) ( )i E r E r
y
SPP Green s functionß
Effective interactionsCoherent
( )t *t
0
( ) ( )( , )2 ( )
ik zS
zP
zP
S
i edS
E r E rG r ru E H
CoherentIncoherent
ˆ ˆ ˆ ˆ[ ]i H H
/2,pl r
/2
sin( )2
cos( )
dij ij
d
g g e k d
e k d
† † †
[ , ]
1 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( 2 )2
s L
ij i j i j j ii j
H Ht
σ σ σ σ σ σ ,pl r
cos( )ij ij e k d
p/2 shift allows switching on/off ,2 i j
0† †ˆ ˆ ˆ ˆ ˆ( )s i i i ij i j
i i j
H g
•Coherent versus incoherent interactions•Control of different decay paths
† .ˆ .1ˆ2
Li i
i
i tL e h cH
/2,pl rcos( )
dij ii jj i j e k d
For inequivalent dipoles
Coherent (gij) & incoherent (gij) ff ti li b t QE’effective couplings between QE’s
Incoherent coupling much more important than the coherent onepbecause it switchs on/off each decaypath with respect to the other
12
Entanglement measureg
ConcurrenceComplex definitionComplex definition….
What we need to know:
Separable states(i.e ) =>
Entangled states(i. e. ) =>
Spontaneous decay of a single excitationConcurrence becomes:
pl /(2 )2 2( ) [ ( ) ( )] 4 [ ( )] i h[ ]tC I 1 2
1( 0) 1 ( )2
t e g
pl /(2 )2 2( ) [ ( ) ( )] 4 [ ( )] sinh[ ]tC t t t Im t e e t
C
C
Stationary entanglementy g(in the previous viewgraph) Spontaneous decay mediated by plasmonsproduces finite-time entanglement starting from an unentangled stateproduces finite-time entanglement starting from an unentangled state
1 21( 0) 1 ( )2
t e g
But one wants both to obtain and manipulatestationary entanglement.stationary entanglement. This can be done by means of lasers:
In the coherent part of theIn the coherent part of themaster equation
Stationary entanglementy g
2 d 12 cos 2
pl
d 12 sin 2
pl
dg
Stationary entanglementy g
12 cos 2
pl
d
12 sin 2pl
dg
Stationary entanglementy g
12 cos 2pl
d
d p 12 sin 2
pl
dg
How is stationary entanglement generated?
y g g
12 cos 2pl
d
12 12 12
12 12
How is stationary entanglement generated?
y g g
12 cos 2pl
d
12 12 12
12 12
How is stationary entanglement generated?
y g g
12 cos 2pl
d
12
12
12 12
Stationary state concurrence
Stationary state tomographyStationary state tomographyStationary
density matrixdensity matrix
1 20.15 , 0
How to measure stationary concurrence: QE QE l tiQE-QE correlation
1 2 0.1
Second order cross-coherence
† †(2) 1 2 2 1g
between the two QE’s
12 † †1 1 2 2
g
1 2 0.1
1 20.15 , 0 1 2,
b=1 V‐channel b=0.9
Cylinder b=0.6
Effect of pure dephasing
[ ] † †
deph ˆ ˆˆ ˆ ˆ ˆ[ ] [ , ],2[ ]i i i i
i
Pure dephasing reduces, but not critically, bothcorrelations & concurrence
PurityConcurrence - Linear entropy diagram 24 13LS Tr
Ûmaximally entangledmixed states
Still room fori !
b=0.94L=2mmd/lSPP=0.5-1.5improvement! SPPW1=0.15g ; W2=0W1=W2=0.1gW1=-W2=0.1gW1 W2 0.1g
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• ConclusionConclusion
ConclusionConclusion:: Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons& capabilities with cavity exciton-polaritons.
ConclusionConclusion: : Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons
SPP
& capabilities with cavity exciton-polaritons.
Strong coupling to excitons & Intermediary for quantum entanglement
A. Gonzalez‐Tudela et al, Phys. Rev. L 106 020501(2011)
A. Gonzalez‐Tudela et al, Xi 1205 3938 Lett. 106 , 020501(2011)
D. Martin‐Cano, et al, Phys. Rev. B 84, 235306 (2011)
arXiv:1205.3938
ConclusionConclusion:: Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons
SPP
& capabilities with cavity exciton-polaritons.
Strong coupling to excitons & Intermediary for quantum entanglement
A. Gonzalez‐Tudela et al, Phys. Rev. L 106 020501(2011)
A. Gonzalez‐Tudela et al, Xi 1205 3938 Lett. 106 , 020501(2011)
D. Martin‐Cano, et al, Phys. Rev. B 84, 235306 (2011)
arXiv:1205.3938
Thanks for your attentionСпасибо за ваше вниманиеСпасибо за ваше внимание
(Semi)-classical description
20
2 20
1 2 / 3/ ;1 1/ 3
Nf e mi N
Polarizability of 1 emitter
Eff. dielectricfunct of emitters 0 1 1/ 3i N 1 emitter funct. of emitters
e5 =1T@0 3
20 00 02 2 2
2 23m cmf
Oscillator strength
e t l
e(w)e3 =1
SPPs
W
d
0 02 2 203e e from microsocpic info.
Absorbanceemetale1 1 R
d
kA=1-R-T
e1=2.25; d=30nm; s=1nm; W=500nm
ω0=2eV; N=106 μm-3 0 μ
γ =γ0 +γdeph ; γ 0 =0.1meV; γdeph=40meV
Red: light line ω = c kRed: light line ω c k
Red thin: light line ω = c k / Ö(ϵ1)
Green: SPP dispersion relation ω = c k/ Ö(ϵ1 ϵm/(ϵ1+ ϵm))
Dispersion relation for condensationp2a d Chain of silver spheres
a=25nm PRB 70 125429 (’04)d=75nm PRB, 70, 125429 ( 04)
Exp : PRB 65 193408 (’02)Exp.: PRB, 65, 193408 ( 02)
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