displaced-photon counting for coherent optical communication shuro izumi
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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