Displaced-photon counting for coherent optical communication

Shuro Izumi

1. Discrimination of phase-shift keyed coherent states2. Super resolution with displaced-photon counting3. Phase estimation for coherent state

1. Discrimination of phase-shift keyed coherent states2. Super resolution with displaced-photon counting3. Phase estimation for coherent state

Optical communicationEncode the information on the optical states

Squeezed statesEntangled statesPhoton number statesSuper position states

ReceiverSender

Laser DetectorOptical state

Excellent properties

Decision

However..

Changed to mixed states by losses

Non-classical states are not optimal for signal carriers

Non-classical states

MotivationDiscriminate phase-shift keyed coherent states with minimum error probability

Optical communication with coherent states

✓ Remain pure state under loss condition ✓ Easily generated compared with non-classical states

Coherent state is the best signal carrier under the losses because

However ✓ It is impossible to discriminate coherent states without

error because of their non-orthogonality

Achievable minimum Error Probability

Standard Quantum Limit…. Achievable Error probability by measurement of the observable which characterizes the states

Helstrom bound…. Achievable Error probability for given states

→How to realize optimal measurement?Overcome the SQL and approach the Helstrom bound!!

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).

0

1

0

1Helstrom bound

SQL

Error

Phase-shift keyed→ Homodyne measurement

Binary phase-shift keyed (BPSK)

on

off

Photon counter

Displacement operation

Near-optimal receiver for BPSK signals Displaced-photon counting

R. S. Kennedy, Research Laboratory of Electronics, MIT, Technical Report No. 110, 1972

Local Oscillator

Beam splitter

ErrorHelstrom bound

SQL (Homodyne measurement)

Displaced-photon counting

k. Tsujino et al., Phys. Rev. Lett. 106, 250503(2011)

Experimental demonstration of near-optimal receiver for BPSK signals

✓Detector with high detection efficiencyTransition edge sensor (TES)→Detection efficiency : 95 % for 853nm

Photon counter

Classical (electrical)feedback

or

Optimal receiver for BPSK signalsDisplaced-photon counting with feedback operation(Dolinar receiver) S. J. Dolinar, Research Laboratory of

Electronics, MIT, Quarterly Progress Report No. 111, 1973

R. L. Cook, et al., Nature 446, 774, (2007)

✓Displacement optimization→Optimize the amount of

p

x

QPSK signals

p

x

✓Displaced-photon counting → Near-optimal✓Displaced-photon counting with feedback → Optimal

How to realize near-optimal measurement for QPSK signals?

R. S. Bondurant,5 Opt. Lett. 18, 1896 (1993)

Near-optimal receiver for QPSK signals

Photon counter

Classical (electrical)feedback

p

x

Displaced-photon counting with feedback receiver

0 5 10 15 2010 7

10 5

0.001

0.1

S ignal m ean photon number 2

Error

Pro

bability

Helstrom

Heterodyne measurement (SQL)

Bondurant receiver

Infinitely fast feedback

More practical condition→finite feedback

x

p

on

off

Evaluation for finite feedforward steps

Change the displacement operation depending on previous results

S. Izumi et al., PRA. 86, 042328 (2012)

M. Takeoka et al., PRA. 71, 022318 (2005)

N→∞⁼Bondurant receiver

Displaced-photon countingwithout feedforward

Helstrom

Heterodyne measurement (SQL)BondurantN=∞N=20

N=10

N=5

N=4N=3

Numerical evaluation

Improve the error probability with increasing the feedforward steps

S. Izumi et al., PRA. 86, 042328 (2012)

x

p

Displaced-photon counting with Feedforward operation(Dolinar receiver )

Change the displacement operation depending on previous results

Photon-number resolving detector

*Symbol selectionBayesian estimation→The signal which maximizes the posteriori probability

S. Izumi et al., PRA. 87, 042328 (2013)

Heterodyne measurement

(SQL)

Helstrom

bound

N=

10

N=4

N=3

On-off detector

Photon-number-resolving detector

N=5

Numerical evaluation

Improve the error probability in small feedforward steps!!

S. Izumi et al., PRA. 87, 042328 (2013)

Numerical evaluation with detector’s imperfection

Dark count ν : counts/pulse

On-off detector PNRD

Robust against dark count noise

S. Izumi et al., PRA. 87, 042328 (2013)

Experimental realization of feedforward receiver for QPSK NIST demonstrated the feedforward (feedback) receiver

F. E. Becerra et al., Nature Photon. 7, 147 (2013)

C. R. Muller et al., New J. Phys. 14, 083009 (2012)

F. E. Becerra et al., Nature Photon. 9, 48 (2015)

With on-off detector With PNRD

Homodyne + Displaced-photon counting

Hybrid scheme from Max-Plank instituteReal time feedback with FPGA

Feedforward operation dependent on the result of homodyne measurement

Summary

✓ We propose and numerically evaluated the receiver for QPSK signals

✓ Displaced-photon counting with PNRD based feedforward operation improve the performance for QPSK discrimination

1. Discrimination of phase-shift keyed coherent states2. Super resolution with displaced-photon counting3. Phase estimation for coherent state

Phase sensing with displaced-photon counting

✓ Better performance than homodyne measurement

Displaced-photon counting is near-optimal receiver for signal discrimination

Can displaced-photon counting make improvements in phase sensing?

✓ Super resolution

✓ Approach the Helstrom bound

✓ Phase estimation

Super resolution and SensitivityInput state Quantum measurement

Phase shift

Super sensitivity

N00N state

Coherent state

Nagata et al., Science 316, 726 (2007)Xiang et al., Nature Photonics 5, 268 (2010)

Sensitivity

Resolution→Interference pattern

Coherent statewith particular quantum measurements

Super resolution

Narrower width

Non-classical states are not necessaryY. Gao et al., J. Opt. Soc. Am. B. 27, No.6 (2010)E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)K. Jiang et al., J. Appl. Phys. 114, 193102(2013)

K. J. Resch et al., Phys. Rev. Lett. 98, 223601 (2007)

Standard two-port intensity difference monitoring

Input state

Intensity difference monitoring

－

Super resolution with parity detection

Input state

Parity detection

Even

Odd

PNRD

Y. Gao et al., J. Opt. Soc. Am. B. 27, No.6 (2010)

Super resolution

Super resolution with homodyne measurementE. Distant et al., Phys. Rev. Lett. 111, 033603(2013)

－

Homodyne measurement

Super resolution

Super resolution with homodyne measurement

Threshold homodyne measurement POVM

Normalized

E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)

→Count probability with the phase shift

a=0.1

a=1.0

a=2.0

a=0.1

a=1.0

a=2.0

Trade off between sensitivity (variance) and resolution

Evaluation of sensitivityE. Distant et al., Phys. Rev. Lett. 111, 033603(2013)

Super resolution with displaced-photon counting

Photon counter

Displacement operation

Does displaced-photon counting show the super resolution?

General phase detection scheme Mach-Zehnder phase detection scheme

→Count probability with the phase shift

Super resolution with displaced-photon counting

Homodyne measurement (with normalization) Parity detection (same input power to the phase shifter)

Displaced-photon counting

Super resolution

WidthWidth :

Width : Width :

Evaluation of resolution and sensitivity

a=0.1

Displaced-photon counting

a=1.0

Parity detection

Resolution Sensitivity

a=0.1

Displaced-photon counting

a=1.0

Parity detection

Shot noise limit

Displaced-photon counting shows better performance

Displaced-photon counting also shows super resolution

Summary

✓Displaced-photon counting shows both super resolution and good sensitivity

Super resolution can be observed with coherent state and quantum measurement→parity detection, homodyne measurement

a=0.1

Displaced-photon counting

a=1.0

Parity detection

Shot noise limit

1. Discrimination of phase-shift keyed coherent states2. Super resolution with displaced-photon counting3. Phase estimation for coherent state

Phase estimation

Quantum measurement

Estimator

Input state

Phase shift

Optimal input state Optimal measurement

Optimize for good estimation

Figure of merit →Variance of the estimator

Cramer-Rao boundCramer-Rao bound

The variance of estimator must be larger than inverse of Fisher information.

For M states , B.R.Frieden, “Science from Fisher Information” ,CAMBRIDGE UNI.PRESS(2004)

Fisher information (FI)

Quantum FI Classical FIdepends only on input state.

depends on input state　and measurement.

Possible to derive the minimum variance for given state

S.L.Braunstein and C.M.Caves, PRL, 72, 3439 (1994)

Possible to derive the minimum variance for given state and measurement

How much information state has How much information we can extract from the state by measurement

Fisher information for coherent state

Phase shift

Quantum FI

Quantum measurement

Homodyne measurement

Heterodyne measurement

Classical FI

S.Olivares et, al., J.Phys.B, Mol. Opt. Phys, 42(2009) 3 2 1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

Relative p hase radFIQFI

Homodyne

Heterodyne

Fisher information for coherent state with displaced-photon counting (PNRD)

PNRD

Displaced-PNRD

Fisher information for discrete variable

Displaced-photon counting decrease slowly

Fisher information

Homodyne decrease rapidly

Experimental setup ~Preliminary experiment

99:1 BS

LO

probePNRD

PZT

Transition edge sensor (TES)

✓ Photon-number resolving up to 8-photon

✓Detection efficiency 92%Fukuda et al., (AIST)Metrologia, 46, S288 (2009)

Laser1550 nm

Experimental condition

Probe amplitude

Displacement amplitude

Detection efficiency

Visibility

Experimental results ~Preliminary experiment

Heterodyne

Displaced-PNRD Experiment

Displaced-PNRD Theory

Homodyne

Displaced-PNRD Theory with imperfections

# of measurement

Expe

ctati

on v

alue

# of measurementVa

rianc

e

Expectation value Variance

Summary

✓ Displaced-photon counting gives higher fisher information than homodyne measurement around Θ=0→Is it possible to use this result for phase sensing?

✓ We demonstrated preliminary experiment →We experimentally show that displaced-photon counting gives better performance in particular condition→Adjustment of the experimental setup more carefully is required