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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
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
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
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
•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
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
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
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 TOTTOT TOTi H
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 TOTTOT TOTi H
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
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 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)
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 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
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
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
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 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
SPPSPPStrongStrong couplingcoupling toto excitonsexcitons & & IntermediaryIntermediary forfor quantum quantum entanglemententanglement
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