Controlling one- and two photon transports in one-dimension

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1 Sept.,2010 Controlling one- and two photon transports in one-dimension Chang-Pu 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 Quasi-Normal Quasi-Normal 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 super-cavity super-cavity with with atomic atomic mirrors mirrors Lan Lan Zhou, Zhou, H. H. Dong, Dong, Yu-xi Yu-xi Liu, Liu, CPS, CPS, F.Nori. F.Nori. Phys. Phys. Rev. Rev. A 78, 78, (2008) (2008) Lehmann-Symanzik-Zimmermann Lehmann-Symanzik-Zimmermann Reduction Reduction Approach Approach to to Multi-Photon Multi-Photon Scattering Scattering in in Coupled-Resonator Coupled-Resonator 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 single-photon single-photon transport transport in in a a coupled coupled superconducting superconducting transmission-line-resonator transmission-line-resonator 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 meta-materials

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 inter-photon 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 Single-photon source Photonic-crystal 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, ) distributed-braggreflector (DBR) cavity Single-photon 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 one-and-only-one 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 two-level 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: Lee-Fano-Aderson model Quasi-Normal Mode Quasi-Bound State Feshbach Resonance Physical Implementation Circuit QED with Superconducting qubit Photonic Crystal Defect cavity Coupled Nanomechanical resonators

14 Tight-binding boson model H ( + a a ) + 1 h c =... ξ +.. c j j j Non-Linear 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 eigen-state + E = u ( j) a 0g + u 0e H k k j ke j Ω = E Ω k k k Two channel scattering equation Single-photon 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 Breit-Wigner 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 Super-cavity: analog of super-lattice Super-cavity: e g e g Zhou, Dong, Liu, Sun, Nori Phys. Rev. A 78, (2008)

23 Wide-Band 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 Anti-Bunching g (2) ( τ ) = : I( t) I( t + τ ) : I( t) τ

29 Coulomb (electron) blockade Coulomb interaction prevents electron from tunneling to Island 1. Non-linear 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 Anti-bouncing 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 Nano-mechanical 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 Side-coupling case Direct-coupling case Reflection of photons in the side-coupling case = Transmission of photons in the direct-coupling 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 Lehmann-Symanzik-Zimmermann 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 anti-bunching=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 anti-bouncing 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,Y-x 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 H-type 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 time-domain 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 non-trivial 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, single-valued Gauge transf. Ψ ϕ / 2 ( ϕ) = e i ψ ( ϕ) How about a more complicated topologically non-trivial boundary?? 2 ϕ 2 ψ ( ϕ ) = E ψ ( ϕ ) ( ϕ) = ψ ( ϕ + π ) ψ 2 anti-periodic, multi-valued

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 c-ring d-ring

55 Physical Realization of Mobius systems Boson: heating a bundle of photonic crystal fibers been Fermion: synthesizing aromatic hydrocarbons with twisted Pi-electrons J. Am. Chem. Soc. (1982) Tetrahedron Lett. (1964) Nature (2002) Chemical Reviews (2006)

56 Non-Abelian induced gauge field in continuous limit φ = 0 ( + 1) 4 = σ z In the pseudo-spin 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 ), Shan-Hui Fan (Univ. Stanford ) Lan Zhou (HNNU), Yu-xi Liu (Tsinghua Univ) [ my previous Post docs] Students: Hui Dong, Tao Shi, Dazi Xu, Yue Chang, Jin-Feng Huan Post Doc Dr. Qing Ai + some regular visitors

59

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