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Photon Efficiency Measures & Processing

Photon Efficiency Measures & Processing. Dominic W. Berry University of Waterloo Alexander I. Lvovsky University of Calgary. Single Photon Sources. State is incoherent superposition of 0 and 1 photon: J. Kim et al ., Nature 397 , 500 (1999).

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Photon Efficiency Measures & Processing

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  1. Photon Efficiency Measures & Processing Dominic W. Berry University of Waterloo Alexander I. Lvovsky University of Calgary

  2. Single Photon Sources • State is incoherent superposition of 0 and 1 photon: • J. Kim et al., Nature 397, 500 (1999). • http://www.engineering.ucsb.edu/Announce/quantum_cryptography.html

  3. Photon Processing measurement U(N) Network of beam splitters and phase shifters . . .

  4. A Method for Improvement . . . D 0 0 • Works for p<1/2. • A multiphoton component is introduced. 2 1/3 1/(N1) 1/2 . . . D. W. Berry, S. Scheel, B. C. Sanders, and P. L. Knight, Phys. Rev. A 69, 031806(R) (2004).

  5. Conjectures • It is impossible to increase the probability of a single photon without introducing multiphoton components. • It is impossible to increase the single photon probability for p≥ 1/2.

  6. Generalised Efficiency • Choose the initial state 0 and loss channel to get . • Find minimum transmissivity of channel. Ep loss D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  7. Generalised Efficiency • Example: incoherent single photon. • Minimum transmissivity is for pure input photon. • Efficiency is p. Ep loss D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  8. Generalised Efficiency • Example: coherent state. • Can be obtained from another coherent state for any p>0. • Efficiency is 0. Ep loss D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  9. Proving Conjectures measurement U(N) . . . D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  10. Proving Conjectures measurement • Inputs can be obtained via loss channels from some initial states. U(N) Ep Ep Ep Ep Ep . . . D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  11. Proving Conjectures measurement • Inputs can be obtained via loss channels from some initial states. • The equal loss channels may be commuted through the interferometer. Ep Ep Ep Ep Ep U(N) . . . D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  12. Proving Conjectures Ep • Inputs can be obtained via loss channels from some initial states. • The equal loss channels may be commuted through the interferometer. • The loss on the output may be delayed until after the measurement. • The output state can have efficiency no greater than p. measurement Ep Ep Ep Ep U(N) . . . D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  13. Catalytic Processing p measurement U(N) Network of beam splitters and phase shifters ? p . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  14. Multimode Efficiency Option 0 • We have equal loss on the modes. • The efficiency is the transmissivity p. • We take the infimum of p. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  15. Multimode Efficiency Option 1 • We have independent loss on the modes. • The efficiency is the maximum sum of K of the transmissivities pj. • We take the infimum of this over schemes. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  16. Multimode Efficiency Option 1 • Example: a single photon in one mode and vacuum in the other. • We can have complete loss in one mode, starting from two single photons. • The multimode efficiency for K=2 is 1. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  17. Multimode Efficiency Option 1 • Example: The same state, but a different basis. • We cannot have any loss in either mode. • The multimode efficiency for K=2 is 2. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  18. Multimode Efficiency Option 2 • We only try to obtain the reduced density operators. • The efficiency is the maximum sum of K of the transmissivities pj. • We take the infimum of this over schemes. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  19. Multimode Efficiency Option 2 • Example: a single photon in one mode and vacuum in the other. • We can have complete loss in one mode, starting from two single photons. • The multimode efficiency for K=1 is 1. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  20. Multimode Efficiency Option 2 • Example: the same state in a different basis. • We can have loss of 1/2 in each mode, starting from two single photons. • The multimode efficiency for K=1 is 1/2. D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  21. Multimode Efficiency Option 3 • We have independent loss on the modes. • This is followed by an interferometer, which mixes the vacuum between the modes. • The efficiency is the maximum sum of K of the transmissivities pj. • We take the infimum of this over schemes. interferometer D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  22. Loss via Beam Splitters • In terms of annihilation operators: • Model the loss via beam splitters. • Use a vacuum input, and NO detection on one output. NO detection NO detection vacuum D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  23. Vacuum Components • We can write the annihilation operators at the output as • Form a matrix of commutators • The efficiency is the sum of the K maximum eigenvalues. interferometer . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  24. Vacuum Components discarded interferometer vacua D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  25. Method of Proof measurement U(N) . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  26. Method of Proof measurement • Each vacuum mode contributes to each output mode. U(N) . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  27. Method of Proof measurement • Each vacuum mode contributes to each output mode. • We can relabel the vacuum modes so they contribute to the output modes in a triangular way. U(N) . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  28. Method of Proof measurement • Each vacuum mode contributes to each output mode. • We can relabel the vacuum modes so they contribute to the output modes in a triangular way. • A further interferometer, X, diagonalises the vacuum modes. X U(N) . . . D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  29. Conclusions • We have defined new measures of efficiency of states, for both the single-mode and multimode cases. • These quantify the amount of vacuum in a state, which cannot be removed using linear optical processing. • This proves conjectures from earlier work, as well as ruling out catalytic improvement of photon sources. • D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010). • D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010). References

  30. Positions Open • Macquarie University (Australia) • 1 Year postdoctoral position • 2 x PhD scholarships • Calculations on Tesla supercomputer!

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