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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
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  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

    View Slide

  2. (c) 2007 Performance Dynamics 2
    Collaboration
    Collaboration
     Professor Edoardo
    Charbon, EPFL
     Dr. Dmitri Boiko,
    EPFL
     Dr. Giordano Beretta,
    HP Labs

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  3. Splitting the Photon
    Splitting the Photon

    View Slide

  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

    View Slide

  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

    View Slide

  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

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  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

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  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

    View Slide

  9. Quantum Path Integral
    Quantum Path Integral
    QPI Applied to the Photon

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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 !
    !
    !
    !

    View Slide

  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

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  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

    View Slide

  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

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  17. What About the Double Slit?
    What About the Double Slit?

    View Slide

  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

    View Slide

  19. Young-
    Young-Afshar
    Afshar-Wheeler
    -Wheeler
    (YAW) Interferometer
    (YAW) Interferometer
    Quantum Path Integral Analysis

    View Slide

  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

    View Slide

  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)

    View Slide

  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.

    View Slide

  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

    View Slide

  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)

    View Slide

  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

    View Slide

  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?

    View Slide

  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!

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  31. Two Photon Interference
    Two Photon Interference
    Quantum Path Integral Analysis

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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)

    View Slide

  41. Some Closing Speculations
    Some Closing Speculations
    From the Path-integral
    Standpoint

    View Slide

  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)

    View Slide

  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

    View Slide

  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

    View Slide