Surface plasmon polaritons (SPP): an alternative to cavity QEDSPP d Beyond the diffraction limit p...

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


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!


•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




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

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


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


† † † †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)


† † † †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

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


12 cos 2

pl

d

12 sin 2pl

dg


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

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

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)

Surface plasmon polaritons (SPP): an alternative to cavity QEDSPP d Beyond the diffraction limit p...

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Transcript of Surface plasmon polaritons (SPP): an alternative to cavity QEDSPP d Beyond the diffraction limit p...