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Two-photon Interference Effects in Quantum Imaging Architectures

Two-photon Interference Effects in Quantum Imaging Architectures

Presented at the 2007 Summer Research Institute, EPFL, Switzerland


Dr. Neil Gunther

July 20, 2007


  1. Two-photon Interference Effects in Two-photon Interference Effects in Quantum Imaging

    Architectures Quantum Imaging Architectures Dr. Neil Gunther Performance Dynamics Company Castro Valley, California, USA www.perfdynamics.com Summer Research Institute 2007 La faculté Informatique et Communications École Polytechnique Fédérale de Lausanne, CH
  2. (c) 2007 Performance Dynamics 2 Collaboration Collaboration  Professor Edoardo

    Charbon, EPFL  Dr. Dmitri Boiko, EPFL  Dr. Giordano Beretta, HP Labs
  3. Splitting the Photon Splitting the Photon

  4. (c) 2007 Performance Dynamics 4 Interference: Wave Picture Interference: Wave

    Picture  Light is a wave – Continuous distribution – Crest and troughs – Classical EM waves  Interference – Wave goes through both slits, no question – Crest and troughs add or cancel at the image plane to produce bands or fringes
  5. (c) 2007 Performance Dynamics 5 Interference: Particle Picture Interference: Particle

    Picture  Quantum theory – All quanta are particles: electron, positron, photon, quark, etc.  Photon is light quantum – Also produces fringes – Random at low counts – Structure appears after 10,000 events or so – Wave theory requires wavefunction to collapse! – Wave theory is the time- averaged effect of single photon events
  6. (c) 2007 Performance Dynamics 6 Photons in Imaging Photons in

    Imaging  Modern optics has its roots in the diffraction theory of imaging in microscopes (Abbe 1873)  Double diffraction for coherent light led to the phase-contrast microscope (Zernike c.1935)  Helmholtz applied Maxwell’s EM wave theory to optics c.1880 – Need to distinguish intensities from amplitudes  Generalized in 1940’s using Fourier transform theory for both analytic and numerical solutions to optical designs – Standard texts: Goodman, Steward, Gaskill – Programs: ImageJ numerical simulator from NIH  What does the quantum theory look like? – Quantum of light is the photon – I will use the Quantum Path Integral (QPI) of Feynman (c.1948) – See how a single photon can be in two places at once: bifurcation – Two-photon effects fail to be explained correctly by wave theory
  7. (c) 2007 Performance Dynamics 7 Diffraction Theory of Imaging Diffraction

    Theory of Imaging  Diffraction plane is Fourier decomposition of object  Image is Fourier recomposition of diffraction plane  Gaussian lens equation is satisfied f i o 1 1 1 = + Focal plane Image plane Object plane Biconvex lens OA Apodizer f o i Diffraction grating Diffraction pattern Image spots SPAD array
  8. (c) 2007 Performance Dynamics 8 Amplitudes vs. Intensities Amplitudes vs.

    Intensities 2 2 ˆ A O I inc ! = 2 ˆ A O I coh ! =  Incoherent light – Convolution of intensity distributions – Image intensity is product of squares  Coherent light – Convolution of amplitudes, not intensities – Image intensity is square of the product  Quantum theory – A false dichotomy – Always use amplitudes
  9. Quantum Path Integral Quantum Path Integral QPI Applied to the

  10. (c) 2007 Performance Dynamics 10 Quantum Design Rules Quantum Design

    Rules  The photon can be represented as an infinite set of paths between source and detector – Need to include all possible paths – Not classical paths but quantum paths or quantum amplitudes (like Fourier optical amplitudes) – Each path belongs to a point particle – Particle-like but distributed in space and time – Provides a connection with Newtonian mechanics  All quantum measurement is an experiment – Photons paths cannot be measured somehow independently of the experiment
  11. (c) 2007 Performance Dynamics 11 Quantum Design Rules Quantum Design

    Rules  Photons do not interact with photons – They only interact with electrons – Matter contains electrons  Introducing any material (matter) into a quantum experiment, can alter the entire experiment  A photon interaction with an electron (momentum exchange) is an event (E)  At this level we don’t need to understand the details of the photon-electron interaction (QED)  You’ll see the details shortly
  12. (c) 2007 Performance Dynamics 12 Quantum Design Rules Quantum Design

    Rules  Path OR-ing Rule – All paths between source and detector must be added  Path AND-ing Rule – Path undergoes an interaction event (E) – Each new segment must be multiplied
  13. (c) 2007 Performance Dynamics 13 Quantum Design Rules Quantum Design

    Rules  Convolution Rule – Generalized case – Superposition of products  This is for 1-photon – We shall return to this rule later for the 2-photon case ( ) ( ) b a b a 2 2 1 1 AND OR AND ! ! ! !
  14. (c) 2007 Performance Dynamics 14 Feynman Feynman’ ’s Photon Model

    s Photon Model  Stopwatch associated with each QPI path  Stopwatch hand (arrow) rotates about 16,000 times/cm for 500 nm red light  When QPI path reaches a detector watch -hand stops  All stopped clock-arrows are added vectorially  Probability (or classical intensity) corresponds to mod-square of vector sum  Like classical phasor model but includes all possible paths, not just geometrical- optics paths
  15. (c) 2007 Performance Dynamics 15 Imaging by the Eye Imaging

    by the Eye  Light does not travel in straight lines (despite Fermat)  Need many QPI paths between source and detector (eye) to provide enough intensity to actually see  Longer QPI paths take a longer time and arrive out of phase  Vector addition of stopwatch arrows head-to-tail  Summation produces the Cornu spiral  Intensity is the square of the resultant vector
  16. (c) 2007 Performance Dynamics 16 Imaging with a Lens Imaging

    with a Lens  Convergent lens forces a subset of QPI paths to take the same time to reach the detector (isochronous paths)  QPI paths through center of lens take longer because glass is thicker and stopwatch rotates more slowly in denser glass  Since lens paths are isochronous arrow rotations are identical  When added head-to-tail they form a larger resultant arrow  Squaring the resultant produces greater intensity than without a lens
  17. What About the Double Slit? What About the Double Slit?

  18. (c) 2007 Performance Dynamics 18 QPI for Young Interferometer QPI

    for Young Interferometer  Each photon is represented by all possible QPI paths  QPI paths can go through both slits and photon appears to bifurcate  Classical particles e.g., Newton’s corpuscles, can take only one path  This is why Newton never got his Opticks theory to be consistent  Apply the Convolution Rule to get superposition  Hence: “a single photon interferes with itself ” (Dirac c.1930) Laser
  19. Young- Young-Afshar Afshar-Wheeler -Wheeler (YAW) Interferometer (YAW) Interferometer Quantum Path

    Integral Analysis
  20. (c) 2007 Performance Dynamics 20 Beyond the Fringe Beyond the

    Fringe  What happens if the screen in the usual Young interferometer is replaced by a convergent lens?  The lens should form an image of both slits or pinholes in the image plane posterior to the lens
  21. (c) 2007 Performance Dynamics 21 Fourier Imaging Simulation Fourier Imaging

    Simulation  Indeed it does!  Vote: Is Young interference still present? (wave property, V)  Vote : If it is present, is each image spot still produced by its respective diagonal pinhole? (particle/path property, K)
  22. (c) 2007 Performance Dynamics 22 Is Interference Present? Is Interference

    Present?  QM requires V2 + K2 ≤ 1  Afshar claims V=1 and K=1 concurrently so that V2 + K2 = 2 and therefore QM is broken!  A wire comb is carefully positioned so that each wire sits in a dark fringe center  Significant scattering blurs image spot when only one pinhole is open If you voted ‘yes’ for both questions, then QM is fundamentally wrong! QM excludes measuring both “wave” and “particle” properties in the same expt.
  23. (c) 2007 Performance Dynamics 23 The Null Experiment The Null

    Experiment  No significant image bluring occurs if both pinholes are open  This could only be true if interference is present, even though we don’t observe any fringes directly  No detection event
  24. (c) 2007 Performance Dynamics 24 Composition of Image Spots? Composition

    of Image Spots?  Consider a digital camera with lens and SPAD detectors in the image plane. (Only 2 detectors shown)  Geometric and QPI paths show each image spot is produced by paths from its respective source  Photons do not interact with photons (Rule 1)  Each detector sees only one source (diagonal pinhole)
  25. (c) 2007 Performance Dynamics 25 Plethora of QPI Paths Plethora

    of QPI Paths  There are many more possible QPI paths than those defined by classical geometrical optics  QPI paths can go from S1 to D1, for example (non classical)  But different travel times (optical path-lengths) through the lens means stopwatches arrive with their clock-hands out of phase  They tend to scroll up and cancel each other vectorially  They make no contribution to the intensity measured at detector D1
  26. (c) 2007 Performance Dynamics 26 Coherent Bifurcation Coherent Bifurcation 

    How can light from both sources get to the same detector?  Camera uses incoherent light  YAW uses coherent light so photons from S1 are phase- reinforced by photons from S2 (classical path) and act like phase twins  Tweedle-dum and Tweedle-dee do contribute to the image intensity at D1  Light bifurcates and each detector sees both sources  Path information (K) is lost and QM survives. (Whew!)  Can we measure this bifurcation experimentally?
  27. (c) 2007 Performance Dynamics 27 Abbe-Rayleigh Abbe-Rayleigh Wave Theory Wave

    Theory  Imaging with coherent light based on diffraction  Another way to understand how each image point is combination of waves from both sources  Image is formed by interference of waves at the image plane sourced from points in the diffraction plane  Afshar should have known this!
  28. (c) 2007 Performance Dynamics 28 QPI and Scalar Wave Theory

    QPI and Scalar Wave Theory  QPI is mathematically equivalent to a scalar (s = 0) wave theory  Must be relativistic because photon is relativistic  Must be massless because electromagnetic force has infinite range  Real photon is a vector (s = 1) gauge boson  Correct Green’s function solves Maxwell-Lorentz equations  QPI corresponds to a massless Klein-Gordon Green’s function  Sufficient when polarization can be ignored
  29. (c) 2007 Performance Dynamics 29 Complete QED Rules Complete QED

    Rules  QED contains all possible relativistic interactions  Maxwell-Lorentz equation for the photon (spin 1)  Includes polarization of vector photon field (A µ )  Dirac equation for the electron and positron (spin 1/2)  All Feynman diagrams are tri-vertex graphs in QED
  30. (c) 2007 Performance Dynamics 30 Event Rule in Complete QED

    Event Rule in Complete QED  Incoming photon energy is absorbed by an electron in an atom of material in the imaging device e.g., lens  That electron temporarily goes into a higher energy state  When it returns to its original state, a different photon is emitted  This follows from the tri-vertex requirement of QED  QPI approximation only considers the change in photon path γ e- γ
  31. Two Photon Interference Two Photon Interference Quantum Path Integral Analysis

  32. (c) 2007 Performance Dynamics 32 HBT and Photon Bunching HBT

    and Photon Bunching  Quasi-monochromatic source  Beam splitter (creates 2 secondary sources)  Two detectors  Connected to correlation counter
  33. (c) 2007 Performance Dynamics 33 HBT Correlations HBT Correlations 

    Measure correlation in photon counts (classical intensity) between the 2 detectors  Time-dependent fluctuations in I(t)  < I(t) * I(t+τ) >  Normalized to g(2)(τ) – g(2)(0) = 1: uncorrelated – g(2)(0) > 1: correlated – g(2)(0) < 1: anticorrelated
  34. (c) 2007 Performance Dynamics 34 How to Calculate HBT with

    QPI How to Calculate HBT with QPI  BS replaced by 2-pinholes to provide secondary sources  Same as original Young interference experiment (c.1800)  But now introduce 2 detectors with coincidence counter (COR)  Direct paths are expected  Exchange paths must also be included. But how?  Need both types of QPI paths to understand HBT correlations COR D1 D2 S1 S2 BE source
  35. (c) 2007 Performance Dynamics 35 New Quantum Design Rule New

    Quantum Design Rule  Single photon rule – Single QPI path with interaction event (E) – Two path segments must be multiplied  Two photon rule – Path S1-D1 is AND- ed with path S2-D2 – Separate but simultaneous QPI paths are multiplied D1 D2 S1 S2 D1 S1 E
  36. (c) 2007 Performance Dynamics 36 QPI Calculation for HBT Detection

    QPI Calculation for HBT Detection ! "d1s1 AND "d 2s2 ( ) OR "d 2s1 AND "d1s2 ( ) "d1s1 # "d 2s2 ( )+ "d 2s1 # "d1s2 ( );superposition But "d1s1 = "d1s2 = "1 and "d 2s1 = "d 2s2 = "2 "1 "2 + "2 "1 2 = 2"1 "2 2 = 4 "1 2 "2 2 ;QPI "1 "2 2 + "2 "1 2 = 2"1 "2 2 = 2"1 2 "2 2 ;classical Direct Exchange D1 D2 S1 S2 D1 S1 D2 S2
  37. (c) 2007 Performance Dynamics 37 D1 D2 S1 S2 Laser

    Lens Can HBT Show YAW Bifurcation? Can HBT Show YAW Bifurcation?  Exchange paths are present due to lens refraction  QPI bifurcation paths correspond to direct HBT paths  Maximal interference with monochromatic Δω = 0 source (V=1)  But monochromatic laser has infinite coherence length τc ~ 1/Δω  No intensity fluctuations, so g(2)(0) = 1 (Poisson)  No HBT photon-pairs with monochromatic source
  38. (c) 2007 Performance Dynamics 38 Conventional Incoherent Imaging Conventional Incoherent

    Imaging  The other extreme of conventional imaging doesn’t help either  Extended incoherent primary source (BE photons)  But Young interference fringes are washed out (no twinning)  And HBT correlations are also washed out  Need something that is intermediate COR D1 D2 S1 S2 Extended source Lens Object Image
  39. (c) 2007 Performance Dynamics 39 Applying HBT to the YAW

    Applying HBT to the YAW  Quasi-monochromatic (Δω > 0) BE source (Hg discharge lamp)  Finite (τc ~ 1/Δω) coherence length ⇒ g(2)(0) > 1 (super-Poisson)  Partial Young interference supports phase twinning (V < 1)  If HBT photon-pairs then proof detectors see both sources (K << 1)  QM inequality V2 + K2 ≤ 1 is satisfied and Afshar is wrong S1 COR D1 D2 S2 Narrow BE source Lens
  40. (c) 2007 Performance Dynamics 40 Expected Outcome Expected Outcome 

    With coherent primary source (laser) in YAW, expect g(2)(0) = 1  If primary replaced by narrow BE emitter, expect g(2)(0) > 1 in YAW  Narrow BE supports both Young interference and photon bunching  Super-Poisson HBT means detectors see both sources  Due to phase twinning from both phase-coherent 2nd-ary sources.  QM: Photons get mixed up, but how? Direct HBT-like paths.  Invalidates Afshar claim: detectors see only one source (K=1) in presence of interference (V=1)
  41. Some Closing Speculations Some Closing Speculations From the Path-integral Standpoint

  42. (c) 2007 Performance Dynamics 42 Two Kinds of Photon Pairs

    Two Kinds of Photon Pairs  HBT – Bunched photon pairs  BBO – Entangled bi-photons  Much research on using thermal sources instead of BBO (e.g., Sun) – Simpler, less expensive – Military applications (e.g., satellites)
  43. (c) 2007 Performance Dynamics 43 QPI AND- QPI AND-ing ing

    of Paths of Paths  Each of the above involve AND-ing of QPI paths – Single photon scattering by event (E) – HBT bunched photon-pairs – Entangled pair where BBO source acts like event (E) D1 D2 S1 S2 D2 D1 BBO (E) idler signal D1 S1 E
  44. (c) 2007 Performance Dynamics 44 Future Work Future Work 

    Is there a symmetry between HBT bunched photons and BBO bi-photons?  If so, could we use entangled bi-photons to demonstrate phase twinning?  If not, what not exactly? – The conventional wisdom is that no first-order interference exists – Similarly, don’t see interference in HBT due to equal-arm length