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
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
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
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
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
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
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
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
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
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 ! ! ! !
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
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
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
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
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
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)
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.
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
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)
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
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?
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!
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
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
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
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
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
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
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
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)
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)
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
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
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