Shuro Izumi Discrimination of phaseshif t keyed coherent states Super resolution with displacedphoton counting Phase estimation for coherent state Discrimination of phaseshif t keyed coherent states ID: 574573
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Slide1
Displaced-photon counting for coherent optical communication
Shuro IzumiSlide2
Discrimination of phase-shif
t keyed coherent states
Super resolution with
displaced-photon counting
Phase estimation for coherent stateSlide3
Discrimination of phase-shif
t keyed coherent states
Super resolution with
displaced-photon counting
Phase estimation for coherent stateSlide4
Optical communication
Encode the information on the optical states
Squeezed states
Entangled states
Photon number states
Super position states
Receiver
Sender
Laser
Detector
Optical state
Excellent properties
Decision
However..
Changed to mixed states by losses
Non-classical states are not optimal for signal carriers
Non-classical statesSlide5
Motivation
Discriminate
p
hase-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 statesCoherent state is the best signal carrier under the losses because However
✓ It is impossible to discriminate coherent states without error because of their non-orthogonalitySlide6
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
1
Helstrom bound
SQL
Error
Phase-shift keyed
→
Homodyne measurement
Binary phase-shift keyed (BPSK)Slide7
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
Error
Helstrom bound
SQL (Homodyne measurement)
Displaced-photon countingSlide8
k
.
Tsujino
et al., Phys. Rev.
Lett
. 106, 250503(2011)Experimental demonstration of near-optimal receiver for BPSK signals
✓
Detector
with high detection efficiency
Transition
edge
sensor (TES)→Detection efficiency : 95 % for 853nm
Photon counterClassical (electrical)
feedback
or
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
R.
L. Cook,
et al.,
Nature 446, 774, (2007)
✓
Displacement
optimization
→
Optimize the amount of Slide9
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
?Slide10
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
Helstrom
Heterodyne measurement (SQL)
Bondurant receiver
Infinitely fast feedback
More practical condition
→
finite feedback Slide11
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 receiverSlide12
Displaced-photon counting
without feedforward
Helstrom
Heterodyne measurement (SQL)
Bondurant
N
=
∞
N
=20
N
=10
N=5N=4
N=3Numerical evaluation
Improve the error probability with increasing the feedforward
steps
S. Izumi et al., PRA.
86
, 042328
(2012)Slide13
x
p
Displaced-photon counting
with Feedforward operation(Dolinar receiver )
Change the displacement operation depending on previous results
Photon-number resolving detector
*Symbol selection
Bayesian estimation
→
The signal which maximizes the posteriori probability
S. Izumi et al., PRA.
87
, 042328
(2013)Slide14
Heterodyne measurement (SQL)
Helstrom
bound
N
=10
N
=4
N
=3
On-off detector
Photon-number-resolving detector
N=5
Numerical evaluationImprove the error probability in small feedforward steps!!
S. Izumi et al., PRA. 87
, 042328
(2013)Slide15
Numerical evaluation with detector’s imperfection
Dark count
ν
:
counts/pulse
On-off detector
PNRD
Robust against dark count noiseS. Izumi et al., PRA. 87
, 042328 (2013)Slide16
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 institute
Real time feedback with FPGA
Feedforward
operation
dependent
on
the result of homodyne measurementSlide17
Summary
✓
We propose and numerically evaluated the receiver
for QPSK signals
✓
Displaced-photon counting with PNRD based feedforward operation improve the performance for QPSK discriminationSlide18
Discrimination of phase-shif
t keyed coherent states
Super resolution with displaced-photon counting
Phase estimation for coherent stateSlide19
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 estimationSlide20
Super resolution and Sensitivity
Input 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 state
with particular quantum measurements
Super resolution
Narrower width
Non-classical states are not necessary
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)
K. J.
Resch et al
.,
Phys. Rev. Lett.
98
, 223601
(2007)Slide21
Standard two-port intensity difference monitoring
Input state
Intensity difference monitoring
-Slide22
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 resolutionSlide23
Super resolution with homodyne measurement
E.
Distant et al
., Phys. Rev. Lett.
111
, 033603(2013)
-
Homodyne measurement
Super resolutionSlide24
Super resolution with homodyne measurement
Threshold
h
omodyne 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 resolutionSlide25
Evaluatio
n of sensitivity
E.
Distant et al
., Phys. Rev. Lett.
111
, 033603(2013)Slide26
Super resolution with displaced-photon counting
Photon counter
Displacement
operation
Does displaced-photon counting show the super resolutio
n
?
General phase detection scheme
Mach-Zehnder
phase detection scheme
→
Count probability with the phase shift Slide27
Super resolution with displaced-photon counting
Homodyne measurement (with normalization)
Parity detection (same input power to the phase shifter)
Displaced-photon counting
Super resolution
Width
Width :
Width :
Width :Slide28
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 resolutionSlide29
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 limitSlide30
Discrimination of phase-shif
t keyed coherent states
Super resolution with displaced-photon counting
Phase estimation for coherent stateSlide31
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 Slide32
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)Slide33
Fisher information (FI)
Quantum FI
Classical FI
depends 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 measurementSlide34
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)
Homodyne
HeterodyneSlide35
Fisher information for coherent state with displaced-photon counting (PNRD)
PNRD
Displaced-PNRD
Fisher information for discrete variableSlide36
Displaced-photon
counting decrease
slowly
Fisher information
Homodyne decrease r
apidlySlide37
Experimental
setup ~Preliminary experiment
99:1 BS
LO
probe
PNRD
PZT
Transition edge sensor (TES)
✓
Photon-number resolving up to 8-photon
✓
Detection efficiency 92%
Fukuda et al., (AIST)
Metrologia
,
46
,
S288 (2009)
Laser
1550 nmSlide38
Experimental
condition
Probe amplitude
Displacement amplitude
Detection efficiency
Visibility Slide39
Experimental
results ~Preliminary experiment
Heterodyne
Displaced-PNRD Experiment
Displaced-PNRD Theory
Homodyne
Displaced-PNRD Theory with imperfections
# of measurement
Expectation value
# of measurement
Variance
Expectation value
VarianceSlide40
Summary
✓
D
isplaced-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