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DAVID SHWA AND HAGAI EISENBERG

SUPER. RESOLUTION. DAVID SHWA AND HAGAI EISENBERG. Outline. Interferometer sensitivity. Super sensitivity and Super Resolution. Path entanglement and the creation of N00N states. N=3,4,5 N00N states experiments. Interferometer description.

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DAVID SHWA AND HAGAI EISENBERG

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  1. SUPER RESOLUTION DAVID SHWA AND HAGAI EISENBERG

  2. Outline Interferometer sensitivity. Super sensitivity and Super Resolution. Path entanglement and the creation of N00N states. N=3,4,5 N00N states experiments.

  3. Interferometer description • Michelson interferometer Mach-Zehnder interferometer‏ • LIGO -Laser interferometer gravitational wave observatory.‏ - 100 kW, 4 Km interferometer.

  4. Classic Interferometer • Measuring the difference in the intensity.‏ The classical phase uncertainty :

  5. What happen in the quantum world – Heisenberg uncertainty • Number- phase uncertainty:‏ • laser light can be treated as a coherent state I>‏ having Poissonian statistics. This is the shot noise limit !‏

  6. Can we do better ? - Super Sensitivity • Let us take a non classical light – a squeezed state.‏ • The most elongated state will be when . • Applying it to the Heisenberg uncertainty principle : ‏ This is called the Heisenberg limit !‏

  7. Comparison should be able to detect gravity waves.‏

  8. Super Resolution • Classically the best resolution between close objects is given by the Rayleigh Criterion :‏ • There is a major interest in bringing the resolution to sub wavelength and beating this criterion. • This will be helpful for : • Lithography • Microscopy The objects are resolved when the first minima of one object lies on the maxima of the other.

  9. Paths to super resolution There are many ways that super resolution can be achieved : Non linear absorbing material. Non linear harmonic generation. Near field optics. Super resolving N00N states. Classical light Super resolution using nonlinear multiphoton absorption 420 nm Interference pattern Rb level scheme Normal resolution Super resolution A. Pe'er et al., "Quantum lithography by coherent control of classical light pulses," Opt. Express 12, 6600-6605 (2004)

  10. super resolution due to nonlinear detection Experimental set up : Detector sensitive to number of photons φ Palarization mach zehnder K=7 with different phase shifting Super resolution But bad sensitivity G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, Phys. Rev. Lett.96, 203601 (2006)

  11. Path entanglement and N00N states • Kinds of entanglement - Polarization (spin 1/2) -‏ - Path (physical direction) -‏ • We already saw examples for entangled states with spin ½ – e.g. Bell states.‏ • N00N states –

  12. Creating path entanglement BeamSplitter - The same number of photons that enter the BS also exit, hence – no losses – a unitary matrix.‏ Generally – rotation +phase

  13. One photon, two photon, bunching (HOM) revisited.‏ • For a 50-50 beamsplitter the results are very simple.‏ • One photon enter the BS : : • the result is an entangled state ! • What happen when two photons hit the BS from one side :‏ Up to global phasefgf BS

  14. Bunching • Lets see what happen when two photons – one from each side hit the BS – Bunching !‏ • We get the N=2 N00N state ! • But what will happen if we take higher N ?‏ The task – a setup that creates only N00N for arbitrary N.‏

  15. Super Resolution and super sensitivity Can N00N states help ? • Applying phase shifter to a N00N state : Phase shift operator : For N photons :‏ And for N00N state :‏ For simplicity, let us look on detector for only N photons : The detection will be :

  16. Super Resolution – The effective wave length is reduced by N This leads to Super sensitivity N00N states are maximal uncertainty states as required by Heisenberg limit. N00N states – leads to super sensitivity

  17. Quantum lithography A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams and J. P. Dowling., Phys. Rev. Lett. 85, 2733-2736 (2000)

  18. The sensitivity depends on meaning, that visibility is important Resch et al. have found that the threshold for super sensitivity : Super Resolution super sensitivity M 0  K. J. Resch et al., Phys. Rev. Lett.98, 223601 (2007)

  19. Set up for creation of N00N states - N=3 • Polarization Mach-Zehnder – • path=polarization and Beamsplitter =Wave plate. • Experimental Setup : M. W. Mitchell, J. S. Lundeen, and J. S. Steinberg, Nature (London)429, 161 (2004)

  20. Results N=3 N00N state with super resolution is visible. Problems: Visibility=42%. According to Resch – no super sensitivity. Post selection only at detectors – destructive measurement. statistical bottleneck

  21. Multiphoton path entanglement by nonlocal bunching Creation of pdc and using one path in order to create N00N states in the other path.‏ The state created with two pairs production : After coincidence measurement in mod a, Mode b transfers into an N=2 N00N state. Detected in path a H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, Phys. Rev. Lett.94, 090502 (2005)

  22. Output State PBS Pump Beam Mirror BBO Crystal BS APD l/2@45o Experimental setup mode a mode b PBS+λ/2 Coincidence in both detectors can be created only with or with 50% chance. BS Bunching ! No. H pol. photons No. V pol. photons

  23. Output State l/2@22.5o (or l/4@45o) a b Results Twofold coincidence Fourfold coincidence Visibility= 83% Variable λ/2

  24. Optional way for higher N Poincare sphere Proposed schemes for higher N Problem – much lower efficiency for high N

  25. N=4 N00N with post selection Experimental setup Input : PDC state BS φ Coincidence post selection 0 T. Nagata, R. Okamoto, J. O’Brien, K. Sasaki, and S. Takeuchi, Science316, 726 (2007)

  26. Results Visibility of 4 photon coincidence = 87% Calculated Visibility threshold = 81% Super resolution and super sensitivity ! Problem : Destructive post selection. Possible solution : Dowling, JP, CONTEMPORARY PHYSICS, 49 (2): 125-143 2008

  27. Different approach : mixing quantum and classical light The Schwinger representation of a mach-Zehnder : Half photon number difference between paths when φ=0. This is exactly to same relationship as a spin N/2 algebra. Phase shift is just a rotation around : Coherent state Squeezed state Hofmann, H. F. & Ono, T., phys. Rev. A 76, 031806 (2007)

  28. mixing Squeezed PDC with coherent light PDC Laser The N photon component of the product : The squeezed annihilation operator is Hence, , , is the N photon component.

  29. Dawn and morning states For η<1 : almost all photons from laser light - and The N photon state will be : For 1<η: Now and the average input difference is : is split with values Morning state Dawn state

  30. “almost” N00N states In the special case η=2 : and This is almost a N00N state ! η=2 For high N:

  31. Experimental realization = phase retarder at 45 deg Polarization Mach Zehnder I. Afek, O. Ambar, and Y. Silberberg, arXiv:0912.4009v2 [quant-ph]

  32. Results V=95% (N1,N2)=1,1 (N1,N2)=2,1 V=86% Input states : V=88% (N1,N2)=0,2 (N1,N2)=0,3 V=80% Theoretical simulation V=74% (N1,N2)=3,1 (N1,N2)=2,3 V=42% V=73% (N1,N2)=2,2 Squeezing amplitude to coherent amplitude ratio :

  33. Summary • N00N states, as the maximal path entangled states have the ability to achieve super resolution and super sensitvity. • Schemes for receiving N=3,4 N00N states using only non classical light were presented. • A new and exciting scheme for achieving higher N00N states using mixing of classical and non classical light was introduced.

  34. Bibliography • Dowling, JP, CONTEMPORARY PHYSICS, 49 (2): 125-143 2008 • A. Pe'er et al., Opt. Express 12, 6600-6605 (2004). • G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, • Phys. Rev. Lett.96, 203601 (2006) • A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams and J. P. Dowling., • Phys. Rev. Lett. 85, 2733-2736 (2000) • K. J. Resch et al., Phys. Rev. Lett.98, 223601 (2007) • M. W. Mitchell, J. S. Lundeen, and J. S. Steinberg, Nature (London)429, 161 (2004) • H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, • Phys. Rev. Lett.94, 090502 (2005) • T. Nagata, R. Okamoto, J. O’Brien, K. Sasaki, and S. Takeuchi, Science316, 726 (2007) • Hofmann, H. F. & Ono, T., phys. Rev. A 76, 031806 (2007) • Hofmann, H. F. & Ono, T., arXiv:0711.0047v1 [quant-ph] 1 Nov 2007 • I. Afek, O. Ambar, and Y. Silberberg, arXiv:0912.4009v2 [quant-ph] 27 Dec 2009

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