an alternative to cavity QED

Transcription

1 Surface plasmon polaritons (SPP): an alternative to cavity QED Strong coupling to excitons & Intermediary for quantum entanglement A. Gonzalez-Tudela, D. Martin-Cano, P. A. Huidobro, E. Moreno, F.J. Garcia-Vidal, C.Tejedor Universidad Autonoma de Madrid L. Martin-Moreno Instituto de Ciencia i de Materiales de Aragon CSIC

2 Strong coupling to excitons SPP & Intermediary for quantum entanglement 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 Conclusion

3 Strong coupling to excitons SPP & Intermediary for quantum entanglement 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 Conclusion

4 Intro: Surface plasmon polaritons Dielectric response of a metal is governed by free electron plasma: 2 p p : plasma frequency i : damping factor Below its plasma frequency () is negative... ß wavevector : c purely imaginary photonic insulator What is a surface plasmon polariton?

5 Intro: Surface plasmon polaritons Dielectric response of a metal is governed by free electron plasma: 2 p p : plasma frequency i : damping factor Below its plasma frequency () is negative... ß wavevector : c purely imaginary photonic insulator What is a surface plasmon polariton?

6 Intro: Surface plasmon polaritons Electromagnetic radiation in dielectric Å Localized Plasmons in a metal surface ß SURFACE PLASMON POLARITONS 200nm 50nm

7 Intro: Surface plasmon polaritons SPP Length Scales span photonics and nano Penetration ti depth into metal Penetration depth into dielectric SPP wavelength SPP propagation length Non-local effects δ m δ d λ SPP δ SPP SPP LRSPP 1 nm 10 nm 100 nm 1 μm 10 μm 100 μm 1 mm 1 cm 10 cm Length scales span 7 orders of magnitude!

8 Intro: Surface plasmon polaritons Roadmap for plasmonics

9 Intro: Surface plasmon polaritons Interesting features of SPPs for photonic circuits: Propagation pg length: mm (Ag or Au) Û lifetime 1 ps Two-dimensional character of EM-fields Optical and electrical signals carried without interference

10 Intro: Surface plasmon polaritons Interesting features of SPPs for photonic circuits: Propagation pg length: mm (Ag or Au) Û lifetime 1 ps Two-dimensional character of EM-fields Optical and electrical signals carried without interference m cbeyond the diffraction limit 1 SPP ( ) d / p 1 d

11 Intro: Surface plasmon polaritons Interesting features of SPPs for photonic circuits: Propagation pg length: mm (Ag or Au) Û lifetime 1 ps Two-dimensional character of EM-fields Optical and electrical signals carried without interference m cbeyond the diffraction limit 1 SPP ( ) d / p 1 d Main problem: coupling in and out to SPPs 3D d c Photons can onlymaeexcitations ti insideid the light cone While SPP are outside the light cone

12 One problem: coupling light to SPPs

13 Strong coupling to excitons SPP & Intermediary for quantum entanglement 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 Conclusion

14 Quantization of plasmons (without losses) Quantization of an electric field with ( ) i ( z z ) E( r) u ( z) e a 20 A r (, z ) 1 ( ) ( z ˆ ˆ ) z u z e u u z L ( ) iz m ( ) d d ( ) ( ) ( ) ( m 1 ) L m d 2 ( ) ( ) d d m effective length to normalize the energy of each mode, H EM ( ) a a

15 Quantization of modes in a medium with dissipation Quantum Optics Vogel & Welsch (Wiley 2006) P N rt, Noise Polarization associated with absorption (, ) 0 Im PN r i r, f ( r, ) 2 (, ) ' Im, Er i dr r Grr, ', f( r, ) 2 0 c H dr d f ( r, ) f( r, ) 0 Hereafter, everything is similar to the non-dissipation case with are bosonic fields playing therole of f( ( r, ) a

16 Interaction of 1 quantum emitter (QE) with SPP HQE 0 z Interaction with a dipole U dr ( r ) E ( r ) g ( ; z) i( r ( ) t) i( r ( ) t) i 0t i 0t Hint () t ( a e a e )( e e ) A ( ) z g ( ; z) E u ( z) e z 0c 0 / 0 ( uˆ uˆ z ) 2 0L ( ) z decay rate of bare QE

17 Interaction of 1 quantum emitter (QE) with SPP HQE 0 z Interaction with a dipole U dr ( r ) E ( r ) g ( ; z) i( r ( ) t) i( r ( ) t) i 0t i 0t Hint () t ( a e a e )( e e ) A ( ) z g ( ; z) E u ( z) e z 0c 0 / 0 ( uˆ uˆ z ) 2 0L ( ) z decay rate of bare QE In RWA g ; z ( ) ( i r i r ) int H a e a e A

18 Interaction of 1 quantum emitter (QE) with SPP, 0 z One QE with w 0 only couples to a bright SPP º symmetric linear comb. (J 0 Bessel funct.) of all the modes with ; 0 SPP The higher coupling does not coincide with the higher b-factor radiation to plasmons total radiation pl

19 Scheme of the quantum dynamics of an open system g m Solving TOT i H TOT TOT tt is complicated and unnecessary

20 Scheme of the quantum dynamics of an open system Isolated system Open system g m g m γ 0 γ SPP =v g/l SPP Solving TOT i H TOT TOT t is complicated and unnecessary Tracing out the reservoir s s degrees of freedom

21 Scheme of the quantum dynamics of an open system Isolated system Open system γ 0 g m Solving TOT i H TOT TOT t is complicated and unnecessary When dissipation at the metal is very high, tracing out ONLY the vacuum s degrees of freedom

22 Dynamics of the QE population: Weissopf-Wigner t c ( t) K ( t ; z) c( ) d 0c( t)/2 ; c (0) i[ 0( )] c i( 0) K (;) z g (;) z e d J ( ;) z e ˆ 2 ( ; ) [ Im[ ( 2 0, 0, )] ] g ( ) ( ) 0 c J z G r r For a single QE with w 0 around the cutoff, spectral density (J) has a non lorentzian shape Þ different dynamics that a pseudomode (cavity QED)! Non-marovian noise Height/width ratio of J determines/allows some (fast & local) reversibility!

23 SPP Strong coupling to excitons & Intermediary for quantum entanglement 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 Conclusion

24 Exciton collective mode of emitters in a plane ij, 0 ij z j More complicated system: Dynamics described by master eq. for density matrix & quantum regression th. N s H H 0 () a a N 0 pl i, j i, j i1 N s g ( ; z ) i r i r j N H int ( i i a ij, e a ij, e ) i1 A ( ) z z j g ( ; z ) ˆ e u j u z 2 L ( ) 0 No fitting!!! ( ) z

25 Exciton collective mode of emitters in a plane 1 b b b b i, j i, j i, j i, j i, j ij, 0 ij z j More complicated system: Dynamics described by master eq. for density matrix & quantum regression th. N s H H 0 () a a N 0 pl i, j i, j i1 N s g ( ; z ) i r i r j N H int ( i i a ij, e a ij, e ) i1 A Holstein-Primaoff transf. (low excitation Þ no saturation) ( ) z z j ( ; ) ˆ & Collective bosonic mode 0 z ß 1 D q b e Ns j( ) i, j Ns i1 iq R 1 iq Ri i, j j, q N s q b D e i g ˆ z j e ( u u ) z 2 L ( ) No fitting!!! g ( ; z ) n H S q ra D q S q ra D q N s j s * int ( ( ) i j ( ) ( ) i j ( )) q, i1 Ns Ns 1 i r i S ( ) e N s i 1 Structure factor

26 Experimental evidence of strong coupling of SPP & excitons QE are not just in a plane Ensembles of organic molecules J. Bellessa, et al, Phys. Rev. Lett. 93, (2004). J. Dintinger, et al, Phys. Rev. B 71, (2005). T. K. Haala, l et al, Phys. Rev. Lett. 103, (2009). P. Vasa, et al, Nano Lett. 12, 7559 (2010). T. Schwartz, et al, Phys. Rev. Lett. 106, (2011). Semicond. nanocrystals & Quantum wells P. Vasa, et al., Phys. Rev. Lett. 101, (2008). 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, (2012). D=110meV

27 Excitonic collective mode in the volume of width W Coupling depends on distance z j g ( ; z j ) More complicated collective mode For many QE with disorder S ( q),0 ( ) N L H g z n a D a D int j1 ( ; ) j s j ( ) j ( ) 1 D ( ) g ( ; z ) D ( ) NL N g j j g ( ) j1 g g z n z g z ; [ D i ( ), D j ( )] ij NL s N 2 2 d ( ) (, j ) (, ) s j1 H D ( ) D( ) a ( ) a( ) g ( ) ( a ( ) D( ) a( ) D ( )) N N 0 Decay of the collective mode D N s n s N d z ( z) g (, z) 2 g ( ) Average of random orientations momentum is conserved 2 No fitting!

28 Dynamics under coherent pumping of a SPP with -vector Average of random orientations Coherent pumping (classical field) Reservoirs

29 Dynamics under coherent pumping of a SPP with -vector H D ( ) D( ) a ( ) a( ) g ( )( a( ) D ( ) a ( ) D( )) N N 0 H t ae ae L i L t i L t () t ( ) D a N L i [, H H ] D a DD Exciton Plasmon Pure dephasing decay decay (vibro-rotation) rotation) c c c c c c c (2 )

30 Dynamics under coherent pumping of a SPP with -vector H D ( ) D( ) a ( ) a( ) g ( )( a( ) D ( ) a ( ) D( )) N N 0 H t ae ae L i L t i L t () t ( ) D a N L i [, H H ] D a DD Exciton Plasmon Pure dephasing decay decay (vibro-rotation) rotation) At the crossing ( 0 ) between exciton and SPP, Rabi splitting is analytical 0 0 c c c c c c c (2 ) N 2 2 N 2 R [ g ( 0 )] ( D a ) / 4 with [ g ( 0 )] n R ih

31 Strong coupling between SPP & excitons 0 0.1meV giso g 2g g N dependsd onw with saturation due to SPP z-decay gg N practically independ. on s g s,iso (s=1) g i g N n eV m 6 3 g spp

32 Strong coupling between SPP & excitons s 1 nm ; W 500nm ev ; 0.1g N (40 mev ) N 0.1 mev ; 0.1 g ( RT ) Polariton populations µ absorption spect. n 10 m 6 3

33 Strong coupling between SPP & excitons s 1 nm ; W 500nm ev ; 0.1g N (40 mev ) N 0.1 mev ; 0.1 g ( RT ) Polariton populations µ absorption spect. n 10 m 6 3 Rabi splitting (at 0 ) N 2 2 N 2 R [ g ( 0 )] ( ) /4;[ g ( 0 )] n D a 0 0 Wea Strong coupling coupling At RT, the incoherent processes (g f ) determine a critical density for observing strong coupling

34 Strong coupling between SPP & excitons: comparison with experiments Experiment Haala et al. PRL, 103, (09) Our theory W 50 nm; n1.210 m ; 1eV 8 3 W 50 nm; n1 210 m ; 0 1eV R th th th 40 mev, R 180 mev, R iso 100meV R exp 115 mev

35 Quantum effects? : (Semi)-classical description N N 2 Polarizability of fe 0 / m 1 2/3 ; 2 2 Eff. dielectric 1 emitter 0 i 1 1/3 funct. of emitters W s d e 5 =1 e(w) e 3 =1 e e metal e 1 1 R T@0 3 2m0 2 2 m 0c Oscillator strength f from microsocpic info. 3e e SPP 0 Absorbance A=1-R-T 0 0 e 1 =3; d=50nm; s=1nm; W=500nm ω 0 =2eV; N=10 6 μm -3 γ =γ 0 +γ deph ; γ 0 =0.1meV; γ deph =40meV

36 Quantum effects: 1st & 2nd order coherences Young s interfer. exp. 1 detector (1) ( ) ( ) g t E t E t I t (,0) () () () Light source Two-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 () t ItIt ()( ) Intensity-intensity correlations

37 Quantum effects: 1st & 2nd order coherences 1 (1) ( ) ( ) ( ) ( ) i g (, t) E () t E ( t) Sr, d E r, te r, t e 0 Spectrum (g (1) ) Amplitude interf. 1 measurement It does not distinguish between classical & quantum waves (2) ( ) ( ) ( ) ( ) g (, t) E () t E ( t) E ( t) E () t I() t I( t) Intensity-int. i correlations (g (2) ) 2 measurements Classical: First detection does not affect second detection Quantum: First detection affects second detection g (2) can distinguish between 0 1 classical fields (1< g (2) <) quantum fields (0<g (2) <) (Better to measure Bell s inequalities)

38 Quantum effects: g (2) (0) in different regimes BOSONS Thermal (gaussian) mixture Laser threshold Poisson distribution ib i (Coherent sts.) Sub-poissonian distr. non-classical sts. For a number (N) st For a number (N) st., g (2) (=0) = 1-1/N 2 /2 1 0 FERMIONS Poisson distribution (Coherent sts.) Chaotic

39 Quantum effects in the coupling between SPP & excitons Quantum effects appear when non-linear effects are important: Holstein-Primaoff up to 2nd order Mandel param. i, j 1 bi, jbi, j bi, j(1 bi, jbi, j/2) bi, j Free energy part of the Hamilt. Coherent pumping of plasmons & 0 NL Ns NL Ns N 0 0 i, j i, j 0 i, j i, j i, j i, j j1 i1 j1 i1 H b b b b b b g In the quasi-2d limit & using Thecollective operators: H U D D D D U D (2) nl D q q,, q 0 N ( ) lim t D t D D t D t ()( )( ) () DD () t 2 g (2) (0) 1 g (2) (0) 1

40 Strong coupling to excitons SPP & Intermediary for quantum entanglement 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 Conclusion

41 QE-QE Q coupling mediated by plasmonic waveguides Cilindrical wire V-Channel Waveguide Wvegude to reinforce QE-QE effects Fields are stronger in the channel than in the cilinder

42 1QE: b and Purcell factors Metallic nanostructures increase the emission from a QE (Purcell) but, Is italways a coherent emission of SPP s??? (b) total radiation radiation to plasmons pl Purcell factor ; factor QE radiation i to vacuum total radiation i 0 The channel is more convenient than the cilinder

43 b and Purcell factors h=20nm b- factor is very stable in a broad range of dipole orientations while Purcell factor decreases significantly ifi whenthedipoleisnot properly oriented h=150nm

44 Dispersion & b factor for V-channel h=140nm V angle= 20 degrees b-factor

45 Two QE s dynamics All the degrees of freedom (SPP, dissipation, radiation) can be traced out producing effective coherent & incoherent interactions between the two QE s that can be computed from the classical Green s function: ˆ (, ) (, ) [ ˆ ˆ (, )..] c 3 H d r d a r a r ω ˆ ˆ 0ii d i i h 0 der 0 i12, i12, 2 d c Er ˆ (, ) i d r (, ) (,, ) (, i r Grr ω a r ) ˆ i ˆ ˆ 1 ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆˆ t [ Hs HL, ] ij ( σσ i j σσ i j 2 σ j σi ) 2 i, j L 2 ˆ H ( ) ˆ ˆ g ˆ ˆ ˆ s 0 i i i ij i j i i j g 1 i Lt H L ˆ e hc * 2 d Re G ( r, r, 0 ) d 2 0 c ij i i j j * ij d (,, 0 ) 2 i ImG ri rj d j 0c i i i

46 Scheme of levels 1 3 ee g 1 g 2 ( ge 1 2 eg 1 2 ) 2 QE e g Modulation of g 12 would allow to swicth on/off red and blue paths

47 Two QE s dynamics It is possible to identify the effects of SPP & dissipation SPP Green s function t t t t i E ( r ) E ( r ) i ( zz) ß GSP P (, r r) e t *t 2 0 dsuz( E H ) S Effective interaction ( ) J ( ) hc.. Coherent ij ij J 1,2 1, J ( 0) 2 iq( ) x x e Incoherent d /2 g g,pl e sin( r d ) 2 d /2 e cos( d) ij ij,pl p/2 shift allows switching on/off Coherent versus incoherent interactions Control of different decay paths r

48 Two QE s dynamics It is possible to identify the effects of SPP & dissipation SPP Green s function t t t t i E ( r ) E ( r ) i ( zz) ß GSP P (, r r) e t *t 2 0 dsuz( E H ) S Effective interaction ( ) J ( ) hc.. Coherent ij ij J 1,2 1, J ( 0) 2 iq( ) x x e Incoherent d /2 g g,pl e sin( r d ) 2 d /2 e cos( d) ij ij,pl r ˆ i [ Hˆ ˆ, ˆ s HL ] t 1 ( ˆˆ ˆ ˆ ˆ ˆ 2 ˆ ˆˆ ij σσ i j σσ i j σ jσi) 2 i, j Hˆ ( ) ˆ ˆ g ˆ ˆ s 0 i i i ij i j i i j ˆ 1 ilt H ˆ L i i e hc.. 2 i p/2 shift allows switching on/off Coherent versus incoherent interactions Control of different decay paths For inequivalent dipoles e d/2 ij,pl d ii jj i j r cos( )

49 Coherent (gij) & incoherent (gij) effective couplings bt between QE s Incoherent coupling much more important than the coherent one because it switchs on/off each decay path with respect to the other 12

50 Entanglement measure Concurrence Complex definition Wooters, PRL 80, (1988) What we need to now: for pure states Separable states (e.g. ) => Entangled states (e.g. ) =>

51 Entanglement measure Concurrence Complex definition Wooters, PRL 80, (1988) What we need to now: for pure states Separable states (e.g. ) => Entangled states (e.g. ) => For mixed states: Concurrence - Linear entropy diagram SL Tr 3 Ûmaximally entangled mixed states

52 Spontaneous decay of a single excitation 1 ( t 0) 1 e1g2 ( ) Concurrence becomes: t I pl /(2 () [ () ()] 4 [ ()] sinh[ ) ] C t t t Im t e e t C C

53 Stationary entanglement (in the previous viewgraph) Spontaneous decay mediated by plasmons produces finite-time entanglement starting from an unentangled state 1 ( t 0) 1 e1g2 ( ) 2 But one wants both to obtain and manipulate stationary entanglement. This can be done by means of lasers: In the coherent part of the In the coherent part of the master equation

54 Stationary entanglement d 12 cos 2 pl d g12 sin 2 pl

55 Stationary entanglement d 12 cos 2 pl d g12 sin 2 pl

56 Stationary entanglement d 12 cos 2 pl d g12 sin 2 pl

57 How is stationary entanglement generated? d 12 cos 2 pl

58 How is stationary entanglement generated? d 12 cos 2 pl

59 How is stationary entanglement generated? d 12 cos 2 pl

60 Stationary state concurrence Stationary density matrix 0.15, Stationary state tomography

61 How to measure stationary concurrence: QE-QE correlation Second order cross-coherence between the two QE s (2) g b=1 V channel b= , Cylinder b=0.6

62 Effect of pure dephasing ˆ ˆ ˆ ˆ i i i i 2 i [ ] ˆ ˆ deph[ ] [, ], Pure dephasing reduces, but not critically, both correlations & concurrence

63 Purity Concurrence - Linear entropy diagram 2 S L Tr Û maximally entangled mixed states Still room for improvement! b=0.94 L=2mm d/l SPP= W 1 =0.15g ; W 2 =0 W 1 =W 2 =0.1g W 1 =-WW 2 =0.1g

64 Strong coupling to excitons SPP & Intermediary for quantum entanglement 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 Conclusion

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

66 Conclusion: Plasmon-polaritons share many properties & capabilities with cavity exciton-polaritons (Cav-QED). Strong coupling to excitons SPP & Intermediary for quantum entanglement A. Gonzalez Tudela et al,, A. Gonzalez Tudela et al,, Phys. Rev. Lett. 110, Phys. Rev. Lett. 106, (2013) (2011) D. Martin Cano, et al, Phys. Rev. B 84, (2011)

67 Conclusion: Plasmon-polaritons share many properties & capabilities with cavity exciton-polaritons (Cav-QED). Strong coupling to excitons SPP & Intermediary for quantum entanglement A. Gonzalez Tudela et al,, A. Gonzalez Tudela et al,, Phys. Rev. Lett. 110, Phys. Rev. Lett. 106, (2013) (2011) D. Martin Cano, et al, Phys. Rev. B 84, (2011) Thans for your attention