photon efficiency measures & processing

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).


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


(1 ) 0 0 1 1p p


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


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




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




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


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


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



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



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


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..


1

0



1pE 2pE NpE4pE3pE

2 3 4 5



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



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..


interferometer

0



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


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


0

discarded

vacua

ˆ ˆˆi i ia B V

Vacuum Components

interferometer


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.


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.


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


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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