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).

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

4. A Method for Improvement D
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, (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, (2010).

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

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

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

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

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

12. Multimode Efficiency interferometer
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: (2010).

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

14. Vacuum Components interferometer . . .
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: (2010).

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

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

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

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

19. Conclusions References
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: (2010).
D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, (2010).
References

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