Photon Efficiency Measures

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_c
  ryptography.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 lt 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

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.

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 pgt0.
• 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

• 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, 203601 (2010).
11
Proving Conjectures

• 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, 203601 (2010).
12
Proving Conjectures

• 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, 203601 (2010).
13
Catalytic Processing
p
measurement
U(N)
Network of beam splitters and phase shifters
?
p
. . .
D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(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, arXiv1010.6302
(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, arXiv1010.6302
(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 K2 is 1.

D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(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 K2 is 2.

D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(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, arXiv1010.6302
(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 K1 is 1.

D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(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 K1 is 1/2.

D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(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, arXiv1010.6302
(2010).
22
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, arXiv1010.6302
(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, arXiv1010.6302
(2010).
24
Vacuum Components
discarded
interferometer
vacua
D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(2010).
25
Method of Proof
measurement
U(N)
. . .
D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(2010).
26
Method of Proof

• Each vacuum mode contributes to each output mode.

measurement
U(N)
. . .
D. W. Berry and A. I. Lvovsky, arXiv1010.6302
(2010).
27
Method of Proof

• 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, arXiv1010.6302
(2010).
28
Method of Proof

• 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, arXiv1010.6302
(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, arXiv1010.6302
  (2010).
• D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett.
  105, 203601 (2010).

References
30
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