Testing Born's Rule in Quantum Mechanics with 3 Slits

1. Testing Born’s rule in quantum mechanics with 3 slits Christophe Couteau Laboratoire de Nanotechnologie et d’Instrumentation Optique (LNIO) Urbasi Sinha, Thomas Jennewein, Raymond Laflamme, Gregor Weihs Saclay, SPhN, 05th of November 2010

2. University of Technology in Troyes Location : city of Troyes History… Birth place of the Templar’s order Timbered houses Province name : « Champagne »

3. General context: UV photonics Photon confinement to optimize the emission/absorption properties… scale involved /4n = 38nm !! • Applications: • UV laser (HR lithography and data storage) • Lighting and LEDs • Photovoltaic • Environment (decontamination) • Sensing • Transparent electronics • High rate future telecommunications… • Quantum plasmonics Note: Electronic and optical confinements are expected

4. The Institute for Quantum Computing -Quantum information and communication -Quantum algorithm and complexity -Quantum cryptography -Quantum error correction -Spin-based qautnum information processing -Nanoelectronics-based quantum information -Optical quantum information processing -Quantum sensors -Foundations of quantum mechanics -21 faculties -28 post-docs -90 students (50% PhDs) -Visitors, adjuncts, associates… You are here Must double still… -$35M for 2009 -Over 140 publications in 2009 Waterloo: land of Blackberry, Maple… … and the Perimeter Institute

5. Contents Quantum mechanics Probabilities from wavefunctions Kolmogorov / Sorkin sum rules Three slits Experiment Laser Single photon source Results Alternatives Implications

6. Born’s rule Nobel Prize in 1954

7. Born’s rule The quantum mechanical state is specified by a wavefunction Ψ(r,t). • The probability density dp to find the particle at time t located at rand t is given by • The probability to find an eigenvalue of an observable with eigenvector ϕi • The expectation value of an observable A: • Using the density operator ρ:

8. Interference From N. Bohr 1949/1983: Discussions with Einstein on Epistemological Problems in Atomic Physics, p. 27 Two possible paths Probability Interference

9. Double slit experiments 1805 Young: Is light is composed of particles or waves? 1909 Taylor et al: Double slit with “feeble light” 1961 Jönsson & Möllenstedt: Electrons and double slit 1974 Merli et al.: One electron at a time 1986 Grangier et al.: Single photon interference 1989 Hitachi group: More convincing single electron detection 1998 Dopfer, Weinfurter, Zeilinger: Englert inequality for indistinguishability and visibility verified by a double slit experiment

10. Photons Double slit experiments Electrons

11. Interference and sum rule Rafael D. Sorkin, Quantum Mechanics as Quantum Measure Theory, Modern Physics Letters A9, 3119-3127 (1994). • Interference describes the deviation from the classical additivity of the probabilities of mutually exclusive events. • If additivity holds, we call that a sum rule. • A sum rule says that an interference term I = 0 • Define a hierarchy of interference terms:

12. Interference and sum rule Zeroeth sum rule violated (I(A)  0) for non-trivial measure. If first sum rule holds (I(A,B) = 0) one gets regular probability theory, e.g. for classical stochastic processes. Violation of the first sum (I(A,B)  0) rule yields quantum mechanics. A sum rule always entails that the higher ones in the hierarchy hold. As far as we know, second sum rule holds in known physics: I(A,B,C) = 0 i.e. triadditivity of mutually exclusive possibilities is always true! Can we test this?

13. Three slits A B C Particles upon 3 slits, assumed to be mutually exclusive possibilities / paths (no loops)

14. Testing the 2nd sum rule 7 probabilities to detect the particles at a given point + dark counts! we then calculate U.Sinha, C.Couteau, Z.Medendorp, I.Sollner, R.Laflamme, R.Sorkin and G.Weihs in Foundations of Probability and Physics-5, A.I.P Conference Proceedings, Vol. 1101, pp 200-207, NY (2009). Arxiv: 0811.2068

15. Precision measurement How much is “O”? • Two examples • Michelson-Morley: Illingworth’s version [1927] • Expected fringe shift – 0.07 • Observed fringe shift – 0.0002± 0.0006 • Photon mass: Luo et al [2003] • Limit on photon mass: <10-51 g, or 6x10-19 eV, or m/m(e) = 1.1x10-24 • Null experiment defined to be one that results of “0” if all is well. Assuming Born’s rule, • ε = 0…….or ????

16. Testing the 2nd sum rule In practice, we can’t easily measure a probability, so we use a normalization say δmeasures the “amount of interference” κmeasures any violation of the second sum rule vs. the expected violation of the first sum rule Since the rule supposedly holds point-wise it should also hold for finite detection area and anywhere on the interferogram

17. What if… Would we need to modify the dynamics? NLQM usually keeps the same definition for probabilities Spontaneous collapse models (GRW) use renormalization What about classical physics? Photon Electrical field energy density Superquantum computing? D. Meyer S. Aaronson [Proc. R. Soc. A 461, 3473 (2005)] Other consequences C. Ududec, H. Barnum, J. Emerson [Found. Phys. (2010)]

18. 0.025 0.020 0.015 0.010 0.005 0.000 400 200 0 200 400 k Photons Original suggestion for electrons Photons should work just fine Single Photons? Just a laser Photons defined by photon counting Single photon source Diffraction pattern at a multiple slit(width a, separation b), for transverse wavevector k

19. Slits Four iterations Homemade photoemulsion mask - ~5% spurious transmission - blocking pads Professional Chromium mask - ~3% spurious transmission - “unblocking openings” Professional Aluminium mask - less than 0.1% transmission - both sides AR coated Laser cut air slits in steel membrane - rough edges

20. Air slits

21. Air slits

22. Three slit set-up - 50 micron fiber as a point detector - Reference arm for power monitoring - Slits stay stationary - Openings can be switched to different combinations - Horizontal microscope for opening to slit alignment

23. <I(t+)I(t)> g(2)() = <I(t)>2 Γ=1/T1 T1 T1 g(2)(0)=0 Correlation function measurement «Split» intensity in 2 IDEA Stop Photon counting regime n(t+)≈I(t+) n(t) ≈I(t) R=50% Start T=50% But reflected OR transmitted Reflected and transmitted Construction of a time histogram function of τ

24. Single photon source Heralded single photon source: • PPKTP produces HV photon pairs • Detection of a photon in i channel heralds a photon in the s channel • - Statistics in s channel tested by Hanbury Brown – Twiss interferometry

25. Single photon source: antibunching Multimode theory reproduces conditional g(2) extremely well • Fitting Parameters: • - Pair production rate • - Detector time resolution E. Bocquillon et al., Phys. Rev.A 79, 035801 (2009). M. Razavi et al., J. Phys. B 42, 114013 (2009).

26. Experimental diffraction patterns

27. 3 slit pattern Using single photons (red) and attenuated laser (blue)

28. Main result

29. Position dependence No obvious spatial correlation

30. Main error contribution Culprit: opening mask error  |κ| ≤ 0.010

31. Alternatives Three-path interferometer

32. Implications How well do we know Born’s rule? Interesting consequences Tighter bounds needed  More accurate interferometer version More work for theoreticians Other experiments to test quantum mechanics

33. Thank you for the invitation…

34. Growing facilities: chemical growth of ZnO nanostructures (bottom up) E-beam facilities and RIE: structured patterns and etching (top-down) Optical lithography: ZnO photonic crystal by optical interferences Other aspects Simulations of light-matter interaction with ZnO

35. Application for sensing: ULTAFLU Detection of ultra-trace of TNT by optical sensors A. Rose et al. Nature 434, 876 (2005) ZnO as a passive or active substrate NWs of ZnO for increasing surface/volume ratio (random lasing)

36. Application for lighting: MATISSE Efficient UV extraction ZnO ZnO Photonic crystals Sapphire Efficient UV to visible conversion Non-radiative Forster Process CdSe NCs/ZnO

37. And more… Integrated (quantum) plasmonics