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

State is incoherent superposition of 0 and 1 photon: State is incoherent superposition of 0 and 1 photon: J. Kim et al., Nature 397, 500 (1999). J. Kim et al., Nature 397, 500 (1999). Single Photon Sources

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

... D00 1/2 1/3 1/(N  1)  2 2 A Method for Improvement D. W. Berry, S. Scheel, B. C. Sanders, and P. L. Knight, Phys. Rev. A 69, (R) (2004). Works for p < 1/2. Works for p < 1/2. A multiphoton component is introduced. A multiphoton component is introduced.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Method of Proof Each vacuum mode contributes to each output mode. 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. 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. A further interferometer, X, diagonalises the vacuum modes. measurement U(N)... X D. W. Berry and A. I. Lvovsky, arXiv: (2010).

Conclusions We have defined new measures of efficiency of states, for both the single-mode and multimode cases. 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. 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. 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, arXiv: (2010). D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, (2010). D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105, (2010). References

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