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CLASSICAL OPTICS – QUANTUM PHYSICS ANALOGIES Daniela Dragoman Univ. Bucharest

CLASSICAL OPTICS – QUANTUM PHYSICS ANALOGIES Daniela Dragoman Univ. Bucharest. Classical optics-ballistic electrons analogies Classical-quantum optics. Maxwell. Schr ödinger. Helmholtz. Y A. Electron microscope, 1933 Z. Phys. 87 , 580 (1934). Fresnel electron lenses.

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CLASSICAL OPTICS – QUANTUM PHYSICS ANALOGIES Daniela Dragoman Univ. Bucharest

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  1. CLASSICAL OPTICS – QUANTUM PHYSICS ANALOGIES Daniela Dragoman Univ. Bucharest Classical optics-ballistic electrons analogies Classical-quantum optics

  2. Maxwell Schrödinger Helmholtz Y A Electron microscope, 1933Z. Phys. 87, 580 (1934) Fresnel electron lenses f = 0.25 mm f = 1 mm Y. Ito, A.L. Bleloch, L.M. Brown, Nature 394, 49 (1998) Classical optics-electrons analogies Electron optics Ernst Ruska, Nobel Prize 1986

  3. Talbot effect t x M. Berry, I. Marzoli, W. Schleich, Physics World, June 2001 Photonic crystals W.D. Rau et al., Phys. Rev. Lett. 82, 2614 (1999) A.C. Twitchett et al., Phys. Rev. Lett. 88, 238302 (2002) S. Matthias, F. Müller, U. Gösele, J. Appl. Phys. 98, 023524 (2005) Classical optics-electrons analogies Electron holography Other analogies: D. Dragoman, M. Dragoman,Quantum-Classical Analogies, Springer (2004)

  4. Schrödinger visible light Dirac K.S. Novoselov et al., Science 306, 666 (2004) Ballistic electrons mesoscopic structures

  5. electron Mach-Zehnder Y A Y. Ji et al., Nature 422, 415 (2003) GRIN waveguide/FRFT electron prism magnetic lens D. Dragoman, M. Dragoman, J. Appl. Phys. 94, 4131 (2003) electrostatic lens J. Spector et al., Appl. Phys. Lett. 56, 1290 (1990) J.H. Smet et al., Phys. Rev. Lett. 77, 2272 (1996) J. Spector et al., Appl. Phys. Lett. 56, 2433 (1990) Classical optics-Schrödingerelectrons analogies Helmholtz Schrödinger

  6. AY group velocities boundary conditions quantum: Y, TE: A, TM: , wavefunctions quantum: TE: TM: Classical optics-Schrödinger electrons analogies D. Dragoman, M. Dragoman, Prog. Quantum Electron. 23, 131 (1999)

  7. matrix method • TM excitation • normally incident electrons in addition: condition of same phase across one layer Classical optics-quantum well analogy GaAs/AlAs Lb =25 Å, Lw = 45 Å, mw = 0.067m0,mb = 0.15m0, Vb = 1 eV, Vw = 0 nb =1, nw=1.5 l0=2pc/w0 = 1 mm  Lbo = 0.44 mm Lwo = 0.16 mm D. Dragoman, M. Dragoman, Opt. Commun. 133, 129 (1997) Classical optics-Schrödinger electrons analogies same field, same T, same t

  8. nw = 3.6 (GaAs), nb = 3.47 (Al0.2Ga0.8As) 2pc/wmax = 1 mm GaAs/AlAs L = 100 Å, Lw = Lb = 100 Å Classical optics-quantum wire analogy p = 1 Lwo= 1.02 mm, Lbo=1.34 mm p = 2 Lwo = 0.94 mm, Lbo = 1.25 mm Lo – not determined directly; chosen to match the optical beam width D. Dragoman, M. Dragoman, IEEE J.Quantum Electron. 33, 375 (1997)

  9. GaAs/AlAs l = d = 100 Å, L = 200 Å nw = 3.6 (GaAs) nb = 3.47 (Al0.2Ga0.8As) Classical optics-quantum wire analogy lo = 5 mm Lo = 10 mm, do = 5 mm, independent of p D. Dragoman, M. Dragoman, IEEE J.Quantum Electron. 33, 375 (1997)

  10. GaAs/AlAs Lx = Ly = 100 Å, Lw = 100 Å, Lb = 20Å • nw = 3.6 (GaAs) • nb = 3.41 (Al0.3Ga0.7As) • Lxo = Lyo = 0.5 mm • Lwo = 2.16 mm, Lbo = 0.68 mm 2pc/wmin = 1.27 mm • 2pc/wmax = 1.2 mm • D. Dragoman, M. Dragoman, Opt. Commun. 150, 331 (1998) Classical optics-quantum dot analogy

  11. n1=3.41 (Al0.3Ga0.7As) n2=3.6 (GaAs) lxo= 0.5 mm lyo= 0.5 mm Lxo = Lyo = 1 mm do = 0.5 mm D. Dragoman, M. Dragoman, Opt. Commun. 150, 331 (1998) Al0.12Ga0.88As/GaAs m1 = 0.076m0 m2 = 0.067m0 V1 = 0.1 eV, V2 = 0 lx = ly = 100 Å Lx = Ly = 200 Å d = 100 Å Classical optics-quantum dot analogy

  12. quantum optics: coupled modes conditions AlGaAs: d1 = 6 mm, d2 = 0.5 mm, D = 4 mm ncl = 3, nco2 = 3.1, ncl1 = 3  Lwo = 2 mm, Lbo = 0.32 mm InAs/AlSb: Lb = 10 Å, Lw = 15 Å D. Dragoman, J. Appl. Phys. 88, 1 (2000) Classical optics-ballistic electrons analogy: type II and III heterostructures

  13. quantum states M. Dragoman et al., J. Appl. Phys. 106, 044312 (2009) polarization states orthogonality evolution law in gyrotropic and electro-optic media (AgGaSe2) Stokes parameters refraction at an interface Snell law D. Dragoman, J. Opt. Soc. Am. B, in press Classical optics-Dirac electrons analogy

  14. D. Dragoman, M. Dragoman, J. Appl. Phys. 101, 104316  (2007) m1 = m3 = 0.4m0, m2 = –0.02m0, V1 = V3 = 0, V2 = 0.5 eV d = 30 nm d = 30 nm slab homogeneous m1 homogeneous m2 d = 34 nm Metamaterials for ballistic electrons harmonic plane waves ballistic electrons GaN, AlN, In0.53Ga0.47As, InAs, InP (left-handed) metamaterial  barrier in a semiconductor with negative effective mass

  15. Classical-quantum optics in phase space coherent quantum state Wmix Wint pure quantum state coherent optical field

  16. mimicking quantum decoherence in phase space optical incoherent field in phase space S. Deléglise et al., Nature 455, 510 (2008) D. Dragoman, M. Dragoman, Opt. Quantum Electron. 33, 239 (2001) Classical-quantum optics in phase space

  17. Classical-quantum optics in phase space x z n = 0 n = 1 n = 5 valid also for superpositions of Fock states D. Dragoman, Optik 111, 393 (2000) squeezed states fractional Fourier transform of order a D. Dragoman, Optik 112, 497 (2001) Fock states Review on classical optics-quantum phase space analogies: D. Dragoman, Prog. Opt. 42, 433 (2002)

  18. classic quantum D. Dragoman, J. Opt. Soc. Am. A 26, 274 (2009) Classical-quantum optics analogies: the fractional Fourier transform

  19. Classical-quantum optics analogies: computation NOT C-NOT D. Dragoman, Optik 113, 425 (2002) logic gates: image forming devices + phase shifters 2nstates of an n-qubit system can only be realized by 2n distinct optical paths OR NOT ?

  20. classic quantum • Conclusions • Analogies between classical and quantum states exist, although they refer to completely different realities (fermions versus bosons) or theoretical approaches (operators versus algebraic functions) • The quantum-classical analogies offer a means to generate, measure, and design optical structures similar to mesoscopic quantum structures but much easier to fabricate, control and characterize • These analogies can be used to study phase space distribution functions of quantum optical states without worrying about decoherence and the impossiblity of measuring non-commutative variables • The analogies between classical optics and quantum physics reveal, if used properly, the (sometimes subtle) differences between these two realms and help understanding the essence of quantum behavior • For all these reasons it is my belief that quantum-classical analogies are worth pursuing

  21. Thank you for your attention!

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