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Conservation of Vacuum in an Interferometer

Conservation of Vacuum in an Interferometer. 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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Conservation of Vacuum in an Interferometer

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  1. Conservation of Vacuum in an Interferometer 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. Proving Conjectures measurement U(N) . . . D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, 203601 (2010).

  8. 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).

  9. 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).

  10. 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).

  11. 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).

  12. Multimode Efficiency • 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 the transmissivities pj. • We take the infimum of this over schemes. interferometer D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).

  13. 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).

  14. 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).

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

  16. 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).

  17. 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).

  18. 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).

  19. Conclusions • We have defined new measures of efficiency of sources, 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

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

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