Surface plasmon polaritons (SPP): an alternative to cavity QED

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. Martin-MorenoI i d Ci i d M i l d AInstituto 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


Outline• Introduction to surface plasmon polaritons (SPP)• Co pling of 1 q ant m emitter (QE) to SPP• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• Conclusion

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!


Roadmap for plasmonicsRoadmap for plasmonics


•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

Main 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 coneWhile SPP are outside the light cone

One problem: coupling light to SPPs


Outline• Introduction to surface plasmon polaritons (SPP)

• Coupling of 1 quantum emitter (QE) to SPP• Coupling of 1 quantum emitter (QE) to SPP• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• Conclusion

Quantization of plasmons (without losses)Q p ( )

Q i i f l i fi ld ( · )(( ) ( )) i k k zkE Quantization of an electric field ( · )

0

(( ) ( ))2

( , )

zk

i k k zk k

kE r u z e aA

r z

( , )r z1 | |ˆ ˆ( )( )

( )zk zzkk

ku z e u uikL k

( ) zikL k

( ) ( )( ) ( ) 1( )m d mdL

with

( ) ( )( ) ( ) 12 ( ) | ( ) |

( )m d mm d

d m

Ldk

†( )H k a aeffective length to normalize the energy of each mode,

†( )EM k k kH k a a

Quantization of modes in a medium with dissipation

“Quantum Optics “ Vogel & Welsch (Wiley 2006)

Noise Polarization associated with absorption ,NP r t

0( , ) Im , ( , )NP r i r f r

2

2( , ) ' Im , , ', ( , )E r i d r r G r rc

f r

†

0

0( , ) ( , )

c

d rH f r f rd

Hereafter, everything is similar to the non-dissipation case ith b i fi ld l i th l f( )f with are bosonic fields playing the role of ( , )f r ka

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 3

0 0 03 /c

02 ( ) kk kzkL 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 3

0 0 03 /c

02 ( ) kk kzkL k

I RWA † · † ·( ; ) ( )i k r ik rg k zH

decay rate

of bare QEIn RWA † †( ) ( )i k r ik r

int 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 TOTTOT TOTi H

t

is complicated and unnecessaryt

Scheme of the quantum dynamics of an open systemIsolated system Open systemIsolated system Open system

γ0

gm g

γSPP=vg/LSPP

gm gm

γ g SPP


Solving

is complicated and unnecessary

Tracing out the reservoir’s

TOT TOTt

Tracing out the reservoir s

degrees of freedom

Scheme of the quantum dynamics of an open systemIsolated system Open systemIsolated system Open system

γ0

gmgm


Solving

is complicated and unnecessaryWhen dissipation at the metal is very high,

TOT TOTt

When dissipation at the metal is very high, tracing out ONLY the vacuum’s degrees of

freedom

Dynamics of the QE population: Weisskopf-Wignert

0 0

0

[ ( )] ( )2

0

( ; ) ( ) ( ) / 2

( ; ) | ( ; ) | ( ; )

( (0) 1) ;

ci k i

K t z c d c t

K z g k z e d J z e

cc t

0

2

0 022

( ; ) | ( ; ) | ( ; )

1 ˆ( ; ) Im[ ( , , )] ( ) ( )[ ]k

K z g k z e d J z e

J z G r rc

g

0 c

•For a single QE with w 0 around the cut-off, spectral density (J) has a non lorentzian shape Þ different dynamicsp ythat a pseudomode (cavity QED) !•Non-markovian noise•Height/width ratio of J determines/allowsHeight/width ratio of J determines/allowssome (fast & local) reversibility !

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 SPPp g q ( )

• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP• Conclusion

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)

sNN H a aH

& quantum regression th.

0 , ,1

0 (k) i jpl k ki ji k

H a aH

· ·† †( ; )( )N

Ns ik r ik rj i iHg k z

a e a e

, ,1

( )Nint

i iH i j i jk kik

a e a eA

( ) | |ˆ( ; ) ·2 ( )

( )z jk zj zk

k kg k z e u ukL 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

& quantum regression th.

† †(k)sN

N H a aH

· ·† †( ; )( )N

Ns ik r ik rj i iHg k z

a e a e

0 , ,1

0 (k) i jpl k ki ji k

H a aH

, ,1

( )Nint

i iH i j i jk kik

a e a eA

Holstein-Primakoff transf.(low excitation Þ no saturation) ( ) | |ˆ ˆ( ; ) ·

2 ( )( )z jk z

j zk

k kg k z e u ukL k

† † †1 ·i j i j i j i j i jb b b b

& Collective bosonic modeß

0

( )2 ( )

( )j zkz

gkL k

No fitting !!!

·† †

, , ,

,1

, ,

1( )

1s

i

Ni

i j i j i

q

j i j i j

Rj i j

is

D q b eN

b b b b

† * †

1,

( ; )( )· ( ) ( )· ( )

1

( )sNj s

int i j i jk kik q s

N

g k z nH S k q ra D q S k q ra D q

N

† †,

·,

1iiq R

i j j qqs

b D 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).

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 ( ) | ( , ) |

| ( ) |k

s ND 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 k

D 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 k

D 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

2 2 22isog g g

•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 ;[ ( )] ( [ ( )]) / 4N

DNg 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

Strong coupling between SPP & excitons:comparison with experimentscomparison with experiments

Our theoryExperimentHakala et al. PRL, 103, 053602 (09)

8 350 ; 1 2 10 ; 1W nm n m Ve 0

40 , 18

50 ; 1.2 10 ; 1

0 , 100isoth th thR

W nm n m

meV R meV R me

V

V

e

exp 115R meV

Quantum effects? : (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

2 2 203e e from microsocpic info.

emetal

e1 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

Quantum effects: 1st & 2nd order coherences1 detector

Young’s interfer. exp.1 detector

Light source Two-slit

(1) ( ) ( )( ,0) ( ) ( ) ( )g t E t E t I tTwo slit

Amplitude interference pattern: the Fouriertransform is the spectrum

g(1) does not distinguish between classical & quantum light

Hanbury-Brown Twiss exp. 2 detectors

(2) ( ) ( ) ( ) ( )( , ) ( ) ( ) ( ) ( )( ) ( )

g t E t E t E t E tI t I t ( ) ( )

Intensity-intensity correlations

Quantum effects: 1st & 2nd order coherences

(1) ( ) ( )( , ) ( ) ( )g t E t E t ( ) ( )

0

1, , , iS r d E r t E r t e

Spectrum (g(1)) Amplitude interf. 1 measurementIt does not distinguish between classical & quantum waves

(2) ( ) ( ) ( ) ( )( , ) ( ) ( ) ( ) ( ) ( ) ( )g t E t E t E t E t I t I t

I i i l i ( (2)) 2Intensity-int. correlations (g(2)) 2 measurements

Classical: First detection does not affect second detection

Quantum: First detection affects second detection

classical fields (1< g (2) <)g (2) can distinguish0

1 classical fields (1< g ( ) <)

quantum fields (0<g (2)<)

g (2) can distinguishbetween q ( g )

(Better to measure Bell’s inequalities)

Quantum effects: g(2) (0) in different regimes

BOSONS FERMIONS2Thermal (gaussian)

mixture

P i di ib i

mixture/2Laser threshold

1Poisson distribution (Coherent sts.)

Poisson distribution (Coherent sts.)

Sub-poissonian distr. 0

non-classical sts. Chaotic

For a number (N) stFor a number (N) st., g(2) (=0) = 1-1/N

Quantum effects in the coupling between SPP & excitons

Quantum effects appear whennon-linear effects are important:Holstein-Primakoff up to 2nd order Mandel paramp

† †, , , , , , ,1 · (1 / 2)i j i j i j i j i j i j i jb b b b b b

0

Mandel param.

† † †0 0 0

s sL LN NN NN

i j i j i j i j i j i jH b b b b b b

Free energy part of the Hamilt. Coherent pumping of plasmons & 0

0 0 , , 0 , , , ,1 1 1 1

i j i j i j i j i j i jj i j i

H b b b b b b

In the quasi-2D limit & usingTh ll ti t

(2) (0) 1g (2) (0) 1g

† †

, ,nl k q k q k k

k k qDH D D D DU

The collective operators:

† †( )( )( ) ( )D t D D t D t

0DU

N

† †(2)

† 2

( )( )( ) ( )( ) lim

( )k k k k

tk k

D t D D t D tg

D D t


Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPPp g q ( )• Collective mode (excitons) strongly coupled to SPP

• Coupling & entanglement of 2 QE mediated by SPP• Conclusion

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 factorQE di i l di i

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 0

1 2 1 2

†

, ,( , ) ( , ) [ ( , ) . .]i i i i

i iH d d a d ha c

r r r ω d E r

23ˆ ( ) ( ) ( ) ( )i d E r r r G r r ω r 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 i

i 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 L

i ii t

L e h cH 0( )s i i i ij i ji i j

g

2 i ii

L 2

*00

2 ( , , )ij i i j jIm d G r r d2

*00( , , )ij i i j jg Re d G r r d 02

0

( , , )ij i i j jImc

d G r r d02

0

( , , )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 ( )2

e 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 interactionCoherent

( )t *t

0

( ) ( )( , )2 ( )

ik zS

zP

zP

S

i edS

E r E rG r ru E H

/2 sin( )dg g e k d † †( ) ( )J h Coherent

Incoherent

,pl r

/2,pl r

sin( )2

cos( )

ij ij

dij ij

g g e k d

e k d

0 1 2

† †1,2 1 2 2 1

( )01,2

( ) ( ) . .( )2

iq x x

J h cJJ e

p/2 shift allows switching on/off

2

•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 interactionCoherent

( )t *t

0

( ) ( )( , )2 ( )

ik zS

zP

zP

S

i edS

E r E rG r ru E H

/2 sin( )dg g e k d † †( ) ( )J h Coherent

Incoherent

,pl r

/2,pl r

sin( )2

cos( )

ij ij

dij ij

g g e k d

e k d

0 1 2

† †1,2 1 2 2 1

( )01,2

( ) ( ) . .( )2

iq x x

J h cJJ e

p/2 shift allows switching on/off ˆ ˆ ˆ ˆ[ , ]s L

i H Ht

2

•Coherent versus incoherent interactions•Control of different decay paths† † †

,

1 ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ( 2 )2 ij i j i j j i

i j

t

σ σ σ σ σ σ

0† †ˆ ˆ ˆ ˆ ˆ( )s i i i ij i j

i i jH g

† .ˆ .1ˆ Li i

i tL e h cH . .

2 i ii

L e h cH

For inequivalent dipoles/2

,pl rcos( )dij ii jj i j e k d

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 measuregConcurrence

Complex definition

What we need to know: for pure states

Wooters, PRL 80, (1988)

Separable states (e.g. ) =>Entangled states (e.g. ) =>

p

Entangled states (e.g. )

Entanglement measuregConcurrence

Complex definitionWooters, PRL 80, (1988)

What we need to know: for pure statesSeparable states (e.g. ) =>Entangled states (e.g. ) =>

p

Entangled states (e.g. )

Ûmaximally entangledmixed statesFor mixed states:

Concurrence -

24 13LS Tr

Linear entropy diagram

3

Spontaneous decay of a single excitationConcurrence becomes:

pl /(2 )2 2( ) [ ( ) ( )] 4 [ ( )] i h[ ]tC I

1 21( 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 12 sin 2

pl

dg

How is stationary entanglement generated?

y g g

12 cos 2pl

d

12 12 12

12 12


y g g

12 cos 2pl

d

12 12

12 12


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 ii

Pure dephasing reduces, but not critically, bothcorrelations & concurrence

PurityConcurrence - Linear entropy diagram 24 1

3LS Tr

Ûmaximally entangledmixed states

Still room fori !

b=0.94L=2mmd/lSPP=0.5-1.5

improvement! SPPW1=0.15g ; W2=0W1=W2=0.1gW1=-W2=0.1gW1 W2 0.1g


Outline• Introduction to surface plasmon polaritons (SPP)• Coupling of 1 quantum emitter (QE) to SPPp g q ( )• Collective mode (excitons) strongly coupled to SPP• Coupling & entanglement of 2 QE mediated by SPP

C i• Conclusion

ConclusionConclusion:: Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons (Cav QED)& capabilities with cavity exciton-polaritons (Cav-QED).

ConclusionConclusion:: Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons (Cav QED)

SPP

& capabilities with cavity exciton-polaritons (Cav-QED).

Strong coupling to excitons & Intermediary for quantum entanglement

A. Gonzalez‐Tudela et al, A. Gonzalez‐Tudela et al, ,Phys. Rev. Lett. 106 , 020501(2011)

,Phys. Rev. Lett. 110 , 126891 (2013)

D. Martin‐Cano, et al, Phys. Rev. B 84, 235306 (2011)

ConclusionConclusion:: Plasmon-polaritons share many properties& capabilities with cavity exciton polaritons (Cav QED)

SPP

& capabilities with cavity exciton-polaritons (Cav-QED).

Strong coupling to excitons & Intermediary for quantum entanglement

A. Gonzalez‐Tudela et al, A. Gonzalez‐Tudela et al, ,Phys. Rev. Lett. 106 , 020501(2011)

,Phys. Rev. Lett. 110 , 126891 (2013)

D. Martin‐Cano, et al, Phys. Rev. B 84, 235306 (2011)

Thanks for your attention

Surface plasmon polaritons (SPP): an alternative to cavity QED

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