Photon Efficiency Measures & Processing
Dominic W. BerryDominic W. BerryUniversity of WaterlooUniversity of Waterloo
Alexander I. LvovskyAlexander I. Lvovsky University of CalgaryUniversity of Calgary
State is incoherent superposition of 0 and 1 State is incoherent superposition of 0 and 1 photon:photon:
J. Kim J. Kim et alet al., Nature ., Nature 397397, 500 (1999)., 500 (1999). http://www.engineering.ucsb.edu/Announce/quantum_cryptography.htmlhttp://www.engineering.ucsb.edu/Announce/quantum_cryptography.html
Single Photon Sources
1100)1( ppp
Photon Processingout
. . .1 2 N
measurement
U(N)Network of beam splitters
and phase shifters
. . .
. . .out D 0 0
p
1/2
1/31/(N1)
2
A Method for Improvement
D. W. Berry, S. Scheel, B. C. Sanders, and P. L. Knight, Phys. Rev. A D. W. Berry, S. Scheel, B. C. Sanders, and P. L. Knight, Phys. Rev. A 6969, 031806(R) (2004)., 031806(R) (2004).
Works for Works for pp << 1/2.1/2.
A multiphoton A multiphoton component is component is introduced.introduced.
p p p
Conjectures
1.1. It is impossible to increase the probability It is impossible to increase the probability of a single photon without introducing of a single photon without introducing multiphoton components.multiphoton components.
2.2. It is impossible to increase the single It is impossible to increase the single photon probability for photon probability for pp ≥ 1/2.≥ 1/2.
Generalised Efficiency
Choose the initial Choose the initial state state 00 and loss and loss channel to get channel to get ..
Find minimum Find minimum transmissivity of transmissivity of channel.channel.
Ep
loss
0
D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Generalised Efficiency
Example: Example: incoherent single incoherent single photon.photon.
Minimum Minimum transmissivity is for transmissivity is for pure input photon.pure input photon.
Efficiency is p.Efficiency is p.
Ep
loss
1
D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
(1 ) 0 0 1 1p p
Generalised Efficiency
Example: coherent Example: coherent state.state.
Can be obtained Can be obtained from another from another coherent state for coherent state for any any pp>0.>0.
Efficiency is 0.Efficiency is 0.
Ep
loss
p
D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Proving Conjecturesout
. . .1 2 N
measurement
U(N)
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Proving Conjectures Inputs can be obtained via Inputs can be obtained via
loss channels from some loss channels from some initial states.initial states.
out
1 2 N
measurement
Ep
U(N)
Ep Ep
. . .01 0
2 0N
Ep Ep
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Proving Conjectures Inputs can be obtained via Inputs can be obtained via
loss channels from some loss channels from some initial states.initial states.
The equal loss channels The equal loss channels may be commuted through may be commuted through the interferometer.the interferometer.
outmeasurement
Ep
U(N)
Ep Ep
. . .01 0
2 0N
Ep Ep
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Proving Conjectures Inputs can be obtained via Inputs can be obtained via
loss channels from some loss channels from some initial states.initial states.
The equal loss channels The equal loss channels may be commuted through may be commuted through the interferometer.the interferometer.
The loss on the output may The loss on the output may be delayed until after the be delayed until after the measurement.measurement.
The output state can have The output state can have efficiency no greater than efficiency no greater than pp..
out
measurement
Ep
U(N)
Ep Ep
. . .01 0
2 0N
Ep Ep
0out
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. Phys. Rev. Lett. 105105, 203601 (2010)., 203601 (2010).
Catalytic Processingout
. . .
U(N)Network of beam splitters
and phase shifters
1 2 N
measurement
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
??pp
pp
Option 0Option 0 We have equal loss on We have equal loss on
the modes.the modes. The efficiency is the The efficiency is the
transmissivity transmissivity pp.. We take the infimum of We take the infimum of
pp..
0
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
pE pE pEpEpE
Option 1Option 1 We have independent We have independent
loss on the modes.loss on the modes. The efficiency is the The efficiency is the
maximum sum of maximum sum of KK of of the transmissivities the transmissivities ppjj..
We take the infimum of We take the infimum of this over schemes.this over schemes.
0
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
1pE 2pE NpE4pE3pE
Option 1Option 1 Example: a single photon in Example: a single photon in
one mode and vacuum in the one mode and vacuum in the other.other.
We can have complete loss We can have complete loss in one mode, starting from in one mode, starting from two single photons.two single photons.
The multimode efficiency for The multimode efficiency for KK=2 is 1.=2 is 1.
1 0
1 1
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
1E 0E
Option 1Option 1 Example: The same state, Example: The same state,
but a but a different basisdifferent basis.. We cannot have any loss in We cannot have any loss in
either mode.either mode. The multimode efficiency The multimode efficiency
for for KK=2 is 2.=2 is 2.
1 0 0 1 2Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
1E 1E
1 0 0 1 2
Option 2Option 2 We only try to obtain We only try to obtain
the reduced density the reduced density operators.operators.
The efficiency is the The efficiency is the maximum sum of maximum sum of KK of of the transmissivities the transmissivities ppjj..
We take the infimum of We take the infimum of this over schemes.this over schemes.
1
0
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
1pE 2pE NpE4pE3pE
2 3 4 5
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
Option 2Option 2 Example: a single photon in Example: a single photon in
one mode and vacuum in the one mode and vacuum in the other.other.
We can have complete loss We can have complete loss in one mode, starting from in one mode, starting from two single photons.two single photons.
The multimode efficiency for The multimode efficiency for KK=1 is 1.=1 is 1.
1
1E 0E
0
11
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
Option 2Option 2 Example: the same state in a Example: the same state in a
different basis.different basis. We can have loss of 1/2 in We can have loss of 1/2 in
each mode, starting from two each mode, starting from two single photons.single photons.
The multimode efficiency for The multimode efficiency for KK=1 is 1/2.=1 is 1/2.
12 0 0 1 1
1/2E 1/2E
1 1
12 0 0 1 1
Option 3Option 3 We have independent loss We have independent loss
on the modes.on the modes. This is followed by an This is followed by an
interferometer, which interferometer, which mixes the vacuum mixes the vacuum between the modes.between the modes.
The efficiency is the The efficiency is the maximum sum of maximum sum of KK of the of the transmissivities transmissivities ppjj..
We take the infimum of We take the infimum of this over schemes.this over schemes.
interferometer
0
Multimode Efficiency
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
1pE 2pE NpE4pE3pE
Loss via Beam Splitters Model the loss via beam Model the loss via beam
splitters.splitters. Use a vacuum input, and Use a vacuum input, and
NO detection on one NO detection on one output.output.
0
vacuumvacuumNO NO detectiondetection
In terms of annihilation In terms of annihilation operators:operators:
ˆˆ ˆ1a pb pv
NO NO detectiondetection
a
b
v
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
We can write the We can write the annihilation operators at annihilation operators at the output asthe output as
Form a matrix of Form a matrix of commutatorscommutators
The efficiency is the The efficiency is the sum of the sum of the KK maximum maximum eigenvalues.eigenvalues.
interferometer
ˆ ˆˆi i ia B V
†ˆ ˆ,ij i jM B B
Vacuum Components
1b 2b ˆNb
1v2v
. . .
ˆNv
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
0
discarded
vacua
ˆ ˆˆi i ia B V
Vacuum Components
interferometer
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).
Method of Proofout
1a
measurement
U(N)
. . .1b 2b ˆ
Nb
2a ˆNa
1v2v
ˆNv
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).arXiv:1010.6302 (2010).
Method of Proofout
measurement
U(N)
. . .1b 2b ˆ
Nb
1v2v
ˆNv
Each vacuum mode Each vacuum mode contributes to each contributes to each output mode.output mode.
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).arXiv:1010.6302 (2010).
Method of Proofout
measurement
U(N)
. . .1b 2b ˆ
Nb
1u2u
ˆNu
Each vacuum mode Each vacuum mode contributes to each contributes to each output mode.output mode.
We can relabel the We can relabel the vacuum modes so they vacuum modes so they contribute to the output contribute to the output modes in a triangular modes in a triangular way.way.
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).arXiv:1010.6302 (2010).
Method of Proof Each vacuum mode Each vacuum mode
contributes to each contributes to each output mode.output mode.
We can relabel the We can relabel the vacuum modes so they vacuum modes so they contribute to the output contribute to the output modes in a triangular modes in a triangular way.way.
A further A further interferometer, X, interferometer, X, diagonalises the vacuum diagonalises the vacuum modes.modes.
out
measurement
U(N)
. . .1b 2b ˆ
Nb
1w2w
ˆNu
X
D. W. Berry and A. I. Lvovsky, D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).arXiv:1010.6302 (2010).
Conclusions We have defined new measures of efficiency We have defined new measures of efficiency
of states, for both the single-mode and of states, for both the single-mode and multimode cases.multimode cases.
These quantify the amount of vacuum in a These quantify the amount of vacuum in a state, which cannot be removed using linear state, which cannot be removed using linear optical processing.optical processing.
This proves conjectures from earlier work, as This proves conjectures from earlier work, as well as ruling out catalytic improvement of well as ruling out catalytic improvement of photon sources.photon sources.
D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010).D. W. Berry and A. I. Lvovsky, arXiv:1010.6302 (2010). D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. D. W. Berry and A. I. Lvovsky, Phys. Rev. Lett. 105105, ,
203601 (2010).203601 (2010).
References
Positions Open Macquarie University Macquarie University
(Australia)(Australia)
1 Year postdoctoral position1 Year postdoctoral position 2 x PhD scholarships2 x PhD scholarships
Calculations on Tesla Calculations on Tesla supercomputer!supercomputer!
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