Displaced-photon counting for
coherent optical communication
Shuro Izumi
1. Discrimination of phase-shift keyed coherent states
2. Super resolution with displaced-photon counting
3. Phase estimation for coherent state
1. Discrimination of phase-shift keyed coherent states
2. Super resolution with displaced-photon counting
3. Phase estimation for coherent state
Optical communication
Encode the information on the optical states
Sender
Laser
Receiver
Optical state
Detector
Decision
Non-classical states
Squeezed states
Entangled states
Photon number states
Super position states
Excellent properties
However..
Changed to mixed
states by losses
Non-classical states are not optimal for signal carriers
Optical communication with coherent states
Coherent state is the best signal carrier under the losses because
✓ Remain pure state under loss condition
✓ Easily generated compared with non-classical states
However
✓ It is impossible to discriminate coherent states without
error because of their non-orthogonality
Motivation
Achievable minimum Error Probability
Binary phase-shift keyed (BPSK)
0
0
1
Error
1
Standard Quantum Limit…. Achievable Error probability by measurement
of the observable which characterizes the states
Phase-shift keyed→ Homodyne measurement
Helstrom bound…. Achievable Error probability for given states
C. W. Helstrom, Quantum Detection and Estimation
Theory (Academic Press, New York, 1976).
Overcome the SQL and approach the Helstrom bound!!
Near-optimal receiver for BPSK signals
Displaced-photon counting
R. S. Kennedy, Research Laboratory of Electronics,
MIT, Technical Report No. 110, 1972
Displacement operation
Beam splitter
Photon
counter
off
on
Local Oscillator
Displaced-photon counting
Error
Experimental demonstration of near-optimal receiver for BPSK signals
k. Tsujino et al., Phys. Rev. Lett. 106, 250503(2011)
✓Detector with high detection efficiency
Transition edge sensor (TES)
→Detection efficiency : 95 % for 853nm
✓Displacement optimization→Optimize the amount of
Optimal receiver for BPSK signals
Displaced-photon counting with feedback operation
(Dolinar receiver) S. J. Dolinar, Research Laboratory of Electronics,
MIT, Quarterly Progress Report No. 111, 1973
or
Photon counter
Classical (electrical)
feedback
R. L. Cook, et al., Nature 446, 774, (2007)
QPSK signals
p
p
x
✓Displaced-photon counting
→ Near-optimal
✓Displaced-photon counting with feedback
→ Optimal
x
Near-optimal receiver for QPSK signals
R. S. Bondurant,5 Opt. Lett. 18, 1896 (1993)
Displaced-photon counting with feedback receiver
p
Photon counter
0.1
Error Probability
x
Classical (electrical)
feedback
0.001
10
5
10
7
0
5
10
15
Signal mean photon number
20
2
Evaluation for finite feedforward steps
S. Izumi et al., PRA. 86, 042328 (2012)
p
x
off
on
N→∞
⁼Bondurant receiver
M. Takeoka et al., PRA. 71, 022318 (2005)
Change the
displacement
operation depending
on previous results
Numerical evaluation
S. Izumi et al., PRA. 86, 042328 (2012)
Displaced-photon counting
without feedforward
Bondurant
N=∞
N=20
N=10
Displaced-photon counting with Feedforward operation(Dolinar receiver )
S. Izumi et al., PRA. 87, 042328 (2013)
p
x
Photon-number
resolving detector
*Symbol selection
Bayesian estimation
→The signal which maximizes
the posteriori probability
Change the
displacement
operation depending
on previous results
Numerical evaluation
S. Izumi et al., PRA. 87, 042328 (2013)
Photon-number-resolving detector
On-off detector
Numerical evaluation with detector’s imperfection
S. Izumi et al., PRA. 87, 042328 (2013)
Dark count ν :10−4 ~10−1 counts/pulse
On-off detector
PNRD
Experimental realization of feedforward receiver for QPSK
NIST demonstrated the feedforward (feedback) receiver
F. E. Becerra et al., Nature Photon. 7, 147 (2013)
F. E. Becerra et al., Nature Photon. 9, 48 (2015)
With on-off detector
With PNRD
Hybrid scheme from Max-Plank institute
C. R. Muller et al., New J. Phys. 14, 083009 (2012)
Homodyne + Displaced-photon counting
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 states
2. Super resolution with displaced-photon counting
3. Phase estimation for coherent state
Phase sensing with displaced-photon counting
Displaced-photon counting is near-optimal receiver for signal discrimination
✓ Better performance than homodyne measurement
✓ Approach the Helstrom bound
Can displaced-photon counting make improvements in phase sensing?
✓ Super resolution
✓ Phase estimation
Super resolution and Sensitivity
Input state
Quantum measurement
Phase shift
Resolution
→Interference pattern
Sensitivity
Super resolution
Coherent state
Narrower width
Coherent state
with particular quantum measurements
N00N state
Nagata et al., Science 316, 726 (2007)
Xiang et al., Nature Photonics 5, 268 (2010)
K. J. Resch et al., Phys. Rev. Lett. 98, 223601 (2007)
Y. 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)
Standard two-port intensity difference monitoring
Input state
－
Intensity difference
monitoring
Super resolution with parity detection
Y. Gao et al., J. Opt. Soc. Am. B. 27, No.6 (2010)
Input state
PNRD
Even
Parity detection Odd
Super resolution
Super resolution with homodyne measurement
E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)
Homodyne
measurement
－
Super resolution
Super resolution with homodyne measurement
E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)
Threshold homodyne measurement POVM
→Count probability with
the phase shift
Normalized
a=2.0
a=2.0
a=1.0
a=1.0
a=0.1
a=0.1
Evaluation of sensitivity
E. Distant et al., Phys. Rev. Lett. 111, 033603(2013)
Super resolution with displaced-photon counting
General phase detection scheme
Mach-Zehnder phase detection scheme
Displacement
operation
Photon
counter
→Count probability with the phase shift
Super resolution with displaced-photon counting
Displaced-photon counting
Width :
Width
Super resolution
Homodyne measurement (with normalization)
Width :
Parity detection (same input power to the phase shifter)
Width :
Evaluation of resolution and sensitivity
Sensitivity
Resolution
Parity detection
Parity detection
a=0.1
Displacedphoton counting
a=1.0
a=1.0
Displacedphoton counting
a=0.1
Shot noise limit
Summary
Super resolution can be observed with coherent
state and quantum measurement
→parity detection, homodyne measurement
✓Displaced-photon counting shows both
super resolution and good sensitivity
Parity detection
a=0.1
a=1.0
Displaced-photon counting
Shot noise limit
1. Discrimination of phase-shift keyed coherent states
2. Super resolution with displaced-photon counting
3. Phase estimation for coherent state
Phase estimation
Estimator
Phase shift
Input state
Optimal input state
Quantum measurement
Optimal measurement
Optimize for good estimation
Figure of merit →Variance of the estimator
Cramer-Rao bound
Cramer-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)
S.L.Braunstein and C.M.Caves, PRL, 72, 3439 (1994)
Quantum FI
depends only on input state.
How much information state has
Possible to derive the minimum
variance for given state
Classical FI
depends on input state
and measurement.
How much information we can extract
from the state by measurement
Possible to derive the minimum variance for
given state and measurement
Fisher information for coherent state
Phase shift
Quantum measurement
Quantum FI
1.0
Classical FI
Homodyne measurement
Homodyne
FI QFI
0.8
0.6
Heterodyne
0.4
Heterodyne measurement
0.2
0.0
S.Olivares et, al., J.Phys.B, Mol. Opt. Phys, 42(2009)
3
2
1
0
Relative phase
1
rad
2
3
Fisher information for coherent state with
displaced-photon counting (PNRD)
PNRD
Fisher information for discrete variable
Displaced-PNRD
Fisher information
Experimental setup ~Preliminary experiment
probe
PNRD
99:1 BS
PZT
Transition edge sensor (TES)
LO
✓ Photon-number resolving
up to 8-photon
✓Detection efficiency 92%
Fukuda et al., (AIST)
Metrologia, 46, S288 (2009)
Laser
1550 nm
Experimental condition
Probe amplitude
Displacement amplitude
Detection efficiency
Visibility
Experimental results ~Preliminary experiment
Variance
Expectation value
Expectation value
Variance
Displaced-PNRD Experiment
Displaced-PNRD Theory with imperfections
Heterodyne
Homodyne
Displaced-PNRD Theory
# of measurement
# of measurement
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