Controlling one and two photon transports in onedimension


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1 Sept.,2010 Controlling one and two photon transports in onedimension ChangPu Sun Institute of Theoretical Physics Chinese Academy of Sciences
2 Outline Background and motivations Single photon transport with a controller Two photon transport in waveguide Towards active manipulation for photons 1. L. Zhou, Z. R. Gong, Y.X., Liu, CPS, F. Nori, Phys. Rev. Lett 101, (2008) 2. T. Shi, CPS, Phys. Rev. B 79, (2009) 3. T. Shi, S.H. Fan, CPS, arxiv:
3 Relevant papers Controlling Controlling Quasibound Quasibound States States in in 1D 1D Continuum Continuum Through Through Electromagnetic Electromagnetic Induced Induced Transparency Transparency Mechanism Mechanism Z. Z. R. R. Gong, Gong, H. H. Ian, Ian, Lan Lan Zhou, Zhou, CPS, CPS, Phys. Phys. Rev. Rev. A 78, 78, (2008) (2008) Intrinsic Intrinsic Cavity Cavity QED QED and and Emergent Emergent QuasiNormal QuasiNormal Modes Modes for for Single Single Photon Photon H. H. Dong, Dong, Z. Z. R. R. Gong, Gong, H. H. Ian, Ian, L. L. Zhou, Zhou, CPS, CPS, Phys. Phys. Rev. Rev. A 79, 79, (2009) (2009) Quantum Quantum supercavity supercavity with with atomic atomic mirrors mirrors Lan Lan Zhou, Zhou, H. H. Dong, Dong, Yuxi Yuxi Liu, Liu, CPS, CPS, F.Nori. F.Nori. Phys. Phys. Rev. Rev. A 78, 78, (2008) (2008) LehmannSymanzikZimmermann LehmannSymanzikZimmermann Reduction Reduction Approach Approach to to MultiPhoton MultiPhoton Scattering Scattering in in CoupledResonator CoupledResonator Arrays Arrays T. T. Shi, Shi, CPS, CPS, Phys. Phys. Rev. Rev. B 79, 79, (2009) (2009) 5.Quantum 5.Quantum switch switch for for singlephoton singlephoton transport transport in in a a coupled coupled superconducting superconducting transmissionlineresonator transmissionlineresonator array array J.Q. J.Q. Liao, Liao, J.F. J.F. Huang, Huang, Y. Y. Liu, Liu, L.M. L.M. Kuang, Kuang, CPS, CPS, Phys. Phys. Rev. Rev. A 80, 80, (2009) (2009) 6.Observable 6.Observable Topological Topological Effects Effects of of Mobius Mobius Molecular Molecular Devices Devices Nan Nan Zhao, Zhao, H. H. Dong, Dong, Shuo Shuo Yang, Yang, CPS, CPS, Phys. Phys. Rev. Rev. B 79, 79, (2009) (2009) Möbius Möbius graphene graphene strip strip as as a a topological topological insulator insulator Z. Z. L. L. Guo, Guo, Z. Z. R. R. Gong, Gong, H. H. Dong, Dong, CPS, CPS, Phys. Phys. Rev. Rev. B 80, 80, (2009) (2009)
4 Quantum information and future quantum devices Quantum information Quantum coherent devices Based on whole wave function rather than state density only: Phase effect dominated Emergent quantum phenomena in artificial structures and metamaterials
5 From electronic to single electron transistor (SET) based on current and voltage from the density of electrons rather than phases of the states Controlling quantum state at the level of single electron
6 Optical switch to single photon transistor (SPT) All optical device in quantum level: Controlling one photon by one photon
7 Why controlling photon by photon is difficult? No direct interphoton interaction and direct coupling to external E.M field according to QED Photon self interaction must be mediated by some massive particles in higher order processes
8 Single photon based devices Singlephoton source Photoniccrystal cavity An ideal triggered source of single photon emits one and only one photon in each pulse Our proposal based on superconducting artificial atoms ( PRB 75, ) distributedbraggreflector (DBR) cavity Singlephoton detection Toshiba setup single photon detector
9 Signature of single photon by its statistics g (2) ( τ ) = : I( t) I( t + τ ) : I( t) Δ n> n, g (0)>1, superpoissonian, classical 2 2. Δ n= n, g ( )=1, Poissonian, classical τ Δ n< n, g (0)<g ( )<1, subpoissonian, quantum 1. τ 0 τ A regulated sequence of optical pulses that contain oneandonlyone photon
10 Single photon transistor (SPT) proposal D. E. Chang et al, Nature Physics, 3,807(2007) With electromagnetically induced transparency (EIT) mechanism Model : Linear waveguide coupled to a local twolevel system The setup was based on the theory by a series papers in S.H. Fan, et. al (Stanford), e.g., J. T. Shen and S. Fan, Phys. Rev. Lett. 95, (2005); 98, (2007); ibid. 98, (2007);Opt. Lett. 30, 2001 (2005)
11 Our questions about this SPT Setup One shot control : one photon by one photon? No, nly strong light controls the EIT Wide band or narrow band? Narrow one due to the single resonate point Localize photon for quantum memory? No, this localization need bound state of Photon! The linear dispersion that could not trap photon Dirac Type particle,klein paradox
12 Our questions about this SPT Setup Evanesce wave coupling for Photonic crystal defect cavity
13 Controlling photons with local atoms e g Bethe Ansatz Discrete Coordinate Scattering Equation Quantum Field Theory Quantum Devices Photon transistor\switch Quantum storage Photonic logic device Physics: LeeFanoAderson model QuasiNormal Mode QuasiBound State Feshbach Resonance Physical Implementation Circuit QED with Superconducting qubit Photonic Crystal Defect cavity Coupled Nanomechanical resonators
14 Tightbinding boson model H ( + a a ) + 1 h c =... ξ +.. c j j j NonLinear dispersion sin k k Higher E Ωk = ω 2ξ 2ξcosk k k π / π / π Low E Simulating waveguide in high energy limit cos k 1 k 2 / 2
15 CRA Based single photon transistor (SPT) L. Zhou, Z. R. Gong, Y.X. Liu, C. P. Sun, F. Nori, Phys. Rev. Lett 101, (2008) Local controller Circuit QED setup e g Фx H c j a j a j a j a j 1 j h. c. H I e e J a 0 g e e g a 0,
16 Discrete coordinate scattering equation Stationary eigenstate + E = u ( j) a 0g + u 0e H k k j ke j Ω = E Ω k k k Two channel scattering equation Singlephoton amplitude Vacuum state of the cavity field Excited state amplitude ( Ek ω) uk ( j) = ξ[ uk ( j + 1) + uk ( j 1)] + Jukeδ j0 ( E Ω) u = Ju (0) k ke k
17 Resonate potential in effective scattering equation ( Ω ω V( E )) u ( j) = ξ u ( j+ 1) + u ( j 1) k k k k k Resonance Potential V( E ) k = J E k 2 δ j0 Ω Energy dependent
18 Working mechanism of SPT E k < Ω E k > Ω E k = Ω
19 Solution 1: 2 bound photon states e g ikx Ae, x > 0 u() j = ikx Ae, x < 0 E 2 ik g ω + 2Je = 0 E Ω E = ω 2Jcosk 2J E B1 ω + 2J E E 2J g 2 E 2 4J 2 E E < ω 2J g 2 E 2 4J 2 ω 2J E B2 ω 2J
20 Solution 2: single photon scattering For j<0 u Lk ikj ikj ( j) = e + re For j>0 Rk ikj ( j) se u = The boundary condition at j=0 r = 2 J 2iξ sin k J 2 ( ω Ω 2ξ cos k)
21 BreitWigner and Fano line shape high energy limit Phase Diagram of reflection Low energy limit R( Δ) = J [ 4ξ ( ω Ω Δ) ] Δ + J Δ = ω Ω 2ξ cos k 4
22 Supercavity: analog of superlattice Supercavity: e g e g Zhou, Dong, Liu, Sun, Nori Phys. Rev. A 78, (2008)
23 WideBand Scattering of Single Photon Yue Chang, Z. R. Gong, C. P. Sun. arxiv:
24 Two photon transport in CRA waveguide Two photon effect: The very quantum nature of light T. Shi and C. P. Sun, Phys. Rev. B 79, (2009); arxiv:
25 Tow photons in one dimension Anti bunching single photon case two photon case Photon blockade T. Shi, CPS, arxiv: (2009)
26 Signature of photon blockade via statistics g (2) ( τ ) 1 0 τ 2 1.g (0)>1, No Blockade 2 2.g ( )=1, No Blocade τ g ( )<g (0)<1, Blockade τ Photon bunching Photon antibunching Photon antibunching A two photon interference effect, tends to enhance the single photon effect for single photon counting or source
27 Photon Bunching
28 Photon AntiBunching g (2) ( τ ) = : I( t) I( t + τ ) : I( t) τ
29 Coulomb (electron) blockade Coulomb interaction prevents electron from tunneling to Island 1. Nonlinear potential 2. For certain gate voltage H = 2 Q 2C H = ( Q e) 2C Δ E = = ( Q e) Q e( Q e/2) 2C 2C C Δ E < 0 ( tunneling) Δ E > 0 ( no tunneling)
30 Photonic analog of Coulomb blockade effect Strong repulsive interaction of photons is induced by nonlinear medium effectively the excitation of medium by a first photon can block the transport of a second photon. H= ξaa+ kaa ( ) 2 nonlinear medium Imamoglu, A.,et al. Rev. Lett. 79, 1467 (1997).
31 Mechanism of photon blockade K. M. Birnbaum et al., Nature (London) 436, 87 (2005)) (0) λ 2 + 2g (0) λ2 1 λ± ( n) =Ω+ ( n ) ωc ± ( Ω ωc) + 4ng ω c g (0) λ 1 + (0) λ1 λ () n = α () n n, e + β ( n n+ 1, g ± ± ± ω c ω g c 0 c g Spectrum of JC model Δ E = λ (2) λ (1) = ω ( Ω ω) + 16 g + ( Ω ω) + 4g c + = ω 2 g ( resonance) c
32 Antibouncing means photon blockade? K. M. Birnbaum et al., Nature (London) 436, 87 (2005)) (0) λ 2 + 2g (0) λ2 ω c ω c ω c ω c g c ΔE ω g (0) λ 1 + (0) λ1 0 c g
33 Mechanism and Experiment of photon blockade K. M. Birnbaum et al., Nature (London) 436, 87 (2005)) D 1 e U g PBS B S D 2
34 Photon blockade due to anharmonicity of energy levels Transmission line coupled to nonlinear Nanomechanical resonator via quantum transducer setup [CPS, L. F. Wei, Y Liu, F. Nori Phys. Rev. A 73, (2006)] Y.D. Wang, CPS C. Bruder, in preparation, 2010
35 Theoretical approaches for two photon 1.Quantum trajectory approach: L. Tian and H. J. Carmichael, Phys. Rev. A 46, 6801 (1992). 2. Numeircal Master equation approach e.g., R. J. Brecha et al., Phys. Rev. A 59, 2392 (1999). 3.Mean field approach: K. Srinivasan and O. Painter, Phys. Rev. A 75, (2007). 4. Exact solution with Bethe Ansatz and QFT J. T. Shen, S. Fan, Phys. Rev. Lett. 98, (2007); L. Zhou et al.,phys. Rev. Lett. 101, (2008); H. Dong et al., Phys. Rev. A 76, (2009); T. Shi and C. P. Sun, Phys. Rev. B 79, (2009); arxiv:
36 Duality of two configurations for two photon e g e g Sidecoupling case Directcoupling case Reflection of photons in the sidecoupling case = Transmission of photons in the directcoupling case H W k k a k a k H I V k a k a H. c. / L H JC c a a e e g a g e a e g, J. T. Shen and S. Fan, Phys. Rev. A 79, (2009).
37 LehmannSymanzikZimmermann Reduction in QFT Two photon effect T. Shi, CPS, Phys. Rev. B 79, (2009) S = it + pp; kk pp; k, k S S + S S pk p k p k pk T ppkk = ( E α Ω) δp 1+ p2, E V g [( E 2 Ω)( E 2 α) 4 g ]. π ( E λ ) ( k λ )( p λ ) 2s i 1s i 1s s=± s=± i= 1,2 S pk = t, kδ kp
38 QFT Calculations 1 X = S X = S k, k out in E = k + k X out t out r out rt out t out dx 1 dx 2 t 2 x 1, x 2 a R x 1 a R x 2 0 g r out dx 1 dx 2 r 2 x 1, x 2 a L x 1 a L x 2 0 g rt out dx 1 dx 2 rt 2 x 1, x 2 a L x 1 a R x 2 0 g T. Shi, Sanhui Fan, C. P. Sun, Phys. arxiv (2010).
39 QFT Calculations 2 t 2 x 1, x e iex c t k1 t k2 cos Δ k x F, x, 4 4 E V g s=± se ( 2 λ1s)exp[ i( 2 λ1, s)] x F( λ, x) = ; 4( λ λ ) [( E λ ) ( k λ )] 1+ 1 s=± 2s i= 1,2 i 1s G (2) ( τ) = S a + ( x) a + ( x+ τ) a ( x) a ( x+ τ) S out S S S S out For S=L,R, (2) 2 g ( τ) = t2( x, x+ τ) / D T. Shi, Sanhui Fan, C. P. Sun, Phys. arxiv (2010).
40 Strong coupling regime : g V 2 R=Reflection T=Transmission T 2 ( a ) Δ k p g H2L H0L Δ 1 ( b) E 2 R T Eê2 E λ 2 + = λ1 = 1 ωa = ω = 10 g H2L HτL ( c) 4 R T R T τ antibunching=blockade g H2L HτL 1 8 ( d ) τ g (2) ( τ ) 1 large bunching
41 Weak coupling regime : g V 2 photon blockade effect vanishes T 2 ( a ) Δ k Δ p g H2L H0L Δ ( b ) Eê2 g H2L HτL ( c) τ g H2L HτL ( d ) τ
42 Reflected antibouncing photons reflection 2 nd order coherence T. Shi, CPS, Phys. Rev. B 79, (2009);2010,in Arxive
43 Summary for two photon The two photon transports in waveguide coupled to a cavity embedded a TLS : Exact solution by LSZ reduction. Photon blockade effect in strong coupling regime. Vanishing of Photon blockade effect in weak coupling regime. Analytic results agree with observations in recent experiment
44 Towards active manipulation for photon via Quantum Zeno dynamics Photonic Feshbach Resonance Induced gauge field with Mobius topology
45 Active control via quantum Zeno dynamics High frequency modulation a A t a cos t Band structure and bound states in frequency Domain Ω HI = [ ωa +Ω cos( νt)] e e + G J0 e g + h.c ν L.Zhou,S. Yang,Yx Liu,C. P. Sun, F. Nori PHYSICAL REVIEW A 80,
46 Dynamic Quantum Zeno Effect ( γ ) exp ixsin = J ( x)exp( inγ ) n n Ω HI = Gexp[ i( Δ sin νt] e g + h.c., ν + Ω i( nν Δ) t = G Jn e e g + h.c., n= ν Ω HI G J0 e g + h.c., ν Decoupling at the zeros of some Bessel function!! Ω / ν = , ,...
47 Numerical : Quantum Zeno Switch for SPT Photon Delocalization from bound state due to Zeno effect
48 Photonic analog of Feshbach Resonance Predicted in Nuclear physics experiment with cold atoms both in MIT!
49 Wave Equation of Single Photon in Htype E a u a j J a u a j 1 u a j 1 g au a 0 g b u b 0 E E b u b j J b u b j 1 u b j 1 g au a 0 g b u b 0 E g a j,0 g b j,0 sexp( ikj ), j> 0 ua () j = exp( ik j) + rexp( ik j), j < 0 Bound state u b j B exp ikj, j 0 B exp ikj, j 0 s 1 B g a g b J b J a sin k sin k.
50 Photonic Feshbach Resonance E a1 A scattering state in chain a ω + 2J a a and a bound state chain b ω a E b1 ω 2J a s g g BE Ω a b. E a2 g = + 2 = 0 E Ω 2 b a ik b E ω b J b e ω + 2J b ω 2J b ω b b b S=0, Total Reflection E b2
51 Numerical with FDTD FDTD = Finite difference timedomain Without bound state in another chain With bound state forming in another chain Coupled cavity arrays with defect in photonic crystal
52 How to have more controllable parameters for photon According quantum electrodynamics (QED), no direct interaction exist between two photons, thus magnetic or electric fields could not control the photon straightforwardly. In this sense, photon is very different from electron Motivated by AB effect, we use the nontrivial spatial topology to induced an equivalent field for photon
53 Aharanov Bohm effect in a mesoscopic ring i + ϕ Ψ Ψ ( ϕ ) = E Ψ ( ϕ ) ( ϕ) = Ψ( ϕ + 2π ) φ =1 ϕ 2 Periodic, singlevalued Gauge transf. Ψ ϕ / 2 ( ϕ) = e i ψ ( ϕ) How about a more complicated topologically nontrivial boundary?? 2 ϕ 2 ψ ( ϕ ) = E ψ ( ϕ ) ( ϕ) = ψ ( ϕ + π ) ψ 2 antiperiodic, multivalued
54 Tight binding boson model with Mobius topology Mobius boundary condition:, N j N j j j j j a A b V V ε ε = = M = b a b a N N j j j j c a d b = Cut off of upper band in transmission spectrum cring dring
55 Physical Realization of Mobius systems Boson: heating a bundle of photonic crystal fibers been Fermion: synthesizing aromatic hydrocarbons with twisted Pielectrons J. Am. Chem. Soc. (1982) Tetrahedron Lett. (1964) Nature (2002) Chemical Reviews (2006)
56 NonAbelian induced gauge field in continuous limit φ = 0 ( + 1) 4 = σ z In the pseudospin representation The Mobius boundary condition induce an effective magnetic flux in the conduction band. D. Loss, P. Goldbart, A. V. Balatsky, Phys. Rev. Lett. 65, 1655 (1990)
57 Suppression of conduction band transmission Conclusions also valid to the fermion system
58 Acknowledgements Franco Nori (Riken & Univ. Michigan ), ShanHui Fan (Univ. Stanford ) Lan Zhou (HNNU), Yuxi Liu (Tsinghua Univ) [ my previous Post docs] Students: Hui Dong, Tao Shi, Dazi Xu, Yue Chang, JinFeng Huan Post Doc Dr. Qing Ai + some regular visitors
59
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