measurements of photon statistics of classical and quantum...
TRANSCRIPT
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Milano 19/04/2005 Maria Bondani 1
MARIA BONDANIIstituto Nazionale per la Fisica della Materia - Unità di Como
Measurements of photon statistics of classical and quantum fields: fundamentals and
applicationsMatteo G.A. Paris
Alessandro FerraroStefano OlivaresAndrea R. Rossi
Giovanni De Cillis
Marco Genovese
Giorgio BridaMarco Gramegna
Alessandra Andreoni
Andrea AgliatiAlessia AlleviFabio Paleari
Emiliano PudduGuido Zambra
Eleonora GevintiPaola Rindi
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Milano 19/04/2005 Maria Bondani 2
Why measuring photon number statistics?
Incomplete description of the optical state!
• characterization of the optical state• discriminate between classical and non-classical statistics• evaluation of the Fano factor
⇒ capability
⇒ applications
• conditional measurement• squeezing in number of photons
• new schemes for Bell measurements• generation of optical states on demand
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Milano 19/04/2005 Maria Bondani 3
RADIATION-FIELD STATES
2,n ph ph
nσ =
, !ph
nn ph
n ph
nP e
n−
=• Coherent state Poissonian statistics
variance 2, 1n ph
ph
Fn
σ= = Fano factor
filtersphotodetectorlaser
Continuous-wave Nd:YAG ; l = 532 nm ; 100 mW He:Ne ; l = 632.8 nm ; 5 mW
Pulsed Nd:VAN ; l = 532 nm ; 113 MHz; 6.4 ps
Nd:YLF ; l = 527 nm ; 500 Hz; 5.5 ps
Nd:YAG ; l = 532 nm ; 10 Hz ; 5.5 ns
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Milano 19/04/2005 Maria Bondani 4
• Thermal state
( )2, 1n ph ph ph
n nσ = +
( ), 11
n
phn ph n
ph
nP
n+=
+
2, 1n ph
phph
F nn
σ= = +
22,n ph ph
nσ =
,
phn n
n phph
ePn
−
=
2,n ph
phph
F nn
σ= =
1ph
n >>
• Multi-thermal state : m equally populated independent thermal modes
2, 1ph
n ph ph
NNσ
µ
⎛ ⎞= +⎜ ⎟⎜ ⎟
⎝ ⎠2, 1n ph ph
ph
NF
nσ
µ= = +
1ph
N >>
( ) ( )
( ) ( ),
1 ! ! 1 !
1 1n ph n
ph ph
n nP
N Nµ
µ µ
µ µ
+ − −⎡ ⎤⎣ ⎦=+ +
2
2,
phn ph
Nσ
µ=
( ) ( )1
,1 !
phn N
n ph
ph
n ePN
µµ
µµ µ
−−
=−
2,n ph ph
ph
NF
nσ
µ= =
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Milano 19/04/2005 Maria Bondani 5
⇒ classical : pseudo-thermal speckle-field
photodetectorlaserfilters
rotatingground
glass
pin-hole
Nd:YLF III harmonics
BBO
photodetector⇒ quantum : twin-beam
converginglens
α = 34,7o
pin-hole
laser
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Milano 19/04/2005 Maria Bondani 6
the modes depend from the coherence properties and from the detection process
temporal modes
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,90
5
10
15
20
25
30 I1
I2
I3
I3 > I2 > I1
Num
ero
di m
odi
Frequenze (ωUV)
detectort
coher
TT
µ =
spatial modes
detectors
coher
SS
µ =
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Milano 19/04/2005 Maria Bondani 7
Different operation regimes :
• continuous-wave/pulsed
• low/high mean photon number
set the choice for the proper photodetector
photodetectors generate a current or voltage when illuminated by light
information on the photon number distribution ,n phP
• direct information : direct detection
• indirect information : homodyne detection + quantum tomography
• indirect information : ON/OFF detection + maximum likelihood
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Milano 19/04/2005 Maria Bondani 8
DIRECT DETECTION
ideal photodetector:- photoelectric effect: 1 photon 1 electron (quantum efficiency η = 1)
real photodetector- ideal photodetector + beam splitter (T =η <1)
( ) ( )1 n mmm
n m
nn n
mΠ η η η
∞−
=
⎛ ⎞= −⎜ ⎟
⎝ ⎠∑
BS: T =η
Ideal Photodetector
, ph i phi
p i iρ∞
= ∑
( ) ( ), ,1 n mmm el ph m n ph
n m
np Tr p
mρ Π η η η
∞−
=
⎛ ⎞⎡ ⎤= = −⎜ ⎟⎣ ⎦
⎝ ⎠∑
photoelectron statistics ≠
photon statistics
el phm nη= ( )2 2 2
, , 1m el n ph phnσ η σ η η= + −
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Milano 19/04/2005 Maria Bondani 9
PHOTODETECTORS
photoemissive devices solid-state devicesexcited charge is transported in the solid by holes or electrons
photoconductive or photovoltaicphotoelectrons are emitted into a vacuum tube
photoelectic effect
advantages
drawbacks
relatively large sensitive arealow noise
photon counting capabilityhigh quantum efficiency
low quantum efficiency (< 40 %)limited dynamics (few hundreds photons)
small sensitive arearelatively high noise
• vacuum photodiodes• p-n-p phototransistors• p-n junction photovoltaic
generate a current or voltage when illuminated by light
• hybrid photodetectors• photomultipliers
• p-i-n photodetectors• avalanche photodiodes• hybrid photodetectors
• photomultipliers
• p-i-n photodetectors• avalanche photodiodes• hybrid photodetectors
• photomultipliers• avalanche photodiodes
photodetectors with internal gain : low mean photon number
p-i-n : high mean photon number
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Milano 19/04/2005 Maria Bondani 10
Photodetector
LightDETECTION CHAIN
out el phm nα αη= =v
( )2 2 2 2 2 2, , 1m el n ph ph
nσ α σ α η σ η η⎡ ⎤= = + −⎣ ⎦v
onda.jpg
2 ns/div
50 m
V/d
iv-5 0 5 10 15 20 25 30 35 40
-0.5
0.0
0.5
1.0
1.5
Vou
t (V)
Vin (mV)
module 1 module 2 module 3
calibration
Gated integrator
DELAY
GATE WIDTH
SENSITIVITY
OFFSET
EXT TRIGGER IN
ANALOG INPUT
DIGITAL OUTPUT
GATE 50Ω
BUSY OUTPUT
SIGNAL INPUT
typical single-shotoutput
gated integration+
amplification gate (60 ns)
20 ns/div
50 m
V/d
iv
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Milano 19/04/2005 Maria Bondani 11
DATA ANALYSIS
out el phm nα αη= =v
( )2 2 2 2 2 2, , 1m el n ph ph
nσ α σ α η σ η η⎡ ⎤= = + −⎣ ⎦v
( )( )
2 2 22, 1
1n ph ph
out ph
nF F
n
α η σ η ησ αη α ηαη
⎡ ⎤+ −⎣ ⎦= = = + −vv v
F α=vcoherent : 1F =
thermal : 1ph
F n= + ph outF nαη α α= + = +v v
1phN
Fµ
= +multi thermal : ph outN
F αη α αµ µ
= + = +v
V
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Milano 19/04/2005 Maria Bondani 12
FEATURES
• Low Dark Noise• High Gain• High-Stability Dynodes
APPLICATIONS
detection of extremely low-light levels
applications in the blue region of the spectrum
• single photon counting,• pollution monitoring, radiometry• Raman spectroscopy, scintillation counting• nuclear “time-of-flight” measurements, and astronomy
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Milano 19/04/2005 Maria Bondani 13
FEATURES
• Able to discriminate multi-photon events• Low excess noise• High Q.E. from 450 nm to 650 nm (H8236-40)• Simple operation• Built-in high voltage power supply and pre-amplifier• Low after pulseAPPLICATIONS
• Photon counting application• Low intensity pulse detection• Laser scanning microscope• Particle counter
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Milano 19/04/2005 Maria Bondani 14
High peak-intensity measurementLow peak-intensity measurement
• Source: Nd:VAN, λ=532 nm @ 110 MHz; t = 6.4 ps
• Source: Nd:YLF, λ=523 nm @ 500 Hz; t = 5.5 ps
• H.V. 2.8 kV (dark count rate: 2.8 kHz)
• Single photon per laser pulse
• Light counting rate: 667.5 kHz (τ ≈ 1.5 µs)
• H.V. 2.3 kV (dark count rate: 400 Hz)
• I2=2.15 I1; I3 = 4.3 I1; I4 = 10.4 I1
-20 0 20 40 60 80 100 1200.0
0.2
0.4
0.6
0.8
1.0
-20 0 20 40 60 80 100 1200.0
0.2
0.4
0.6
0.8
1.0
Cou
nts (
a. u
.)
b)darklight
a)darklight
• 1.5 µs gate • 5 µs gate
Anodic-pulse charge (10-12C ) Anodic-pulse charge (10-12C )0 5 10 15 20 25 30 35
0
1000
2000
30002000040000
Cou
nts
Anodic-pulse charge (10-12C )
K1234
• 70 ns gate
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Milano 19/04/2005 Maria Bondani 15
( ) ( )max
0
M
m m mm
f q A y q q=
= −∑
( ) ( )1 1...m
m
y q y y q= ∗ ∗
n photoelectron leaving the cathode at the same timeindependent amplification (no saturation or spatial charge effects)
( )2
,1
2,2
2
1 2
0
0
l
l
q
q
e qy q
e q
σ
σ
−
−
⎧ ≤⎪= ⎨>⎪⎩
( )2
,120
lqy q e σ−=
Analysis method
• fit the charge distribution
• fit the 0-photon peak
• fit the 1-photon peak
• obtain the fit of the n-photon peak as the convolution of n-times the 1-photon peak fit
( )
( ),
m m m
m el
A y q q dqp
f q dq
−=
∫
∫D
D
( ), , 1 n mmm el n ph
n m
np p
mη η
∞
• evaluate
−
=
⎛ ⎞= −⎜ ⎟
⎝ ⎠∑• find that reproduce the results,n php
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Milano 19/04/2005 Maria Bondani 16
0 10 20 300.0
0.2
0.4
3.2
3.4
0 2 4 6 8 100.00
0.25
0.50
0.75
Cou
nts (
103 )
Anodic-pulse charge (10-12C)
Det
ectio
n pr
obab
ility
Number of photoelectrons
( )( )Kf q
( )jY q
Number of modes = 18
Average = 2.45
G. Zambra, M. Bondani, A.S. Spinelli, F. Paleari and A. Andreoni, Rev. Sci. Instrum., 75 (2004) 2762-2765
Multi-thermal distribution
Poissonian distribution
0 10 20 300.0
0.5
1.0
1.5
5.0
6.0
0 2 4 6 8 100.00
0.25
0.50
0.75
Cou
nts (
103 )
Anodic-pulse charge (10-12C)
K = 3
K = 2
K = 4
K = 3
Det
ectio
n pr
obab
ility
Number of photoelectrons
K = 1 Average = 2.68
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Milano 19/04/2005 Maria Bondani 17
LINEARITY OF PHOTON COUNTERS
Number of photoelectrons0 10 20 30 40
1000
2000
3000
4000
5000D
etec
tion
prob
abili
ty 0 2 4 6 8 10
50010001500200025003000
0 2 4 6 8 10
50010001500200025003000
0 2 4 6 8 10
50010001500200025003000
0 2 4 6 8 10
50010001500200025003000
0 2 4 6 8 10
50010001500200025003000
0 10 20 30 40 50
50010001500200025003000
Poissonian distribution
Number of photoelectrons
Det
ectio
n pr
obab
ility
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Milano 19/04/2005 Maria Bondani 18
LINEARITY OF PHOTON COUNTERS
0.00010.00020.00030.00040.00050.00060.0007
0.000020.000040.000060.000080.0001
Fano
Fac
tor
a
Mean output voltage
Mea
n ph
otoe
lect
ron
num
ber
Transmittance0.2 0.4 0.6 0.8 1
5
10
15
20
25
F α=v
Poissonian distribution
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Milano 19/04/2005 Maria Bondani 19
LINEARITY OF PHOTON COUNTERS
0.5 1 1.5 2 2.5 3 3.5 4
0.250.5
0.751
1.251.5
1.752
Mean output voltage
Fano
Fac
tor
a
Multi thermal distribution
b
outF αµ
= +v
V
m = 1/tan b
from the measurement of a known radiation, we get the response parameters of the system that can be used to measure an unknown light
M. Bondani, A. Agliati, A. Allevi, and A. AndreoniSelf-consistent characterization of light statistics, Submitted (2005)
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Milano 19/04/2005 Maria Bondani 20
FEATURES
• no internal gain, but can operate at much higher light levels than other detectors• low capacitance and dark current• low noise • high speed • High Q.E. APPLICATIONS
• precision photometry• medical instrumentation • analytical instruments • semiconductor tools • industrial measurement systems.
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Milano 19/04/2005 Maria Bondani 21
Measurement of a single component of a twin-beam
high mean photon number : phn 8@ 10
SIMPLE MODEL : the radiation field is made of mindependent equally-populated thermal modes
rebinning
of the outputout el ph phM N x N
qηα αη= = ⎯⎯⎯⎯⎯→ =∆
V
2 22
12 2 2 2, 2
phN ph phrebinningel xof the output
N Nq
ησ α η σµ µ
>>⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯→ =
∆v
2
2x
xµ
σ= pulse
coherence
TT
µ =number of independent modes :
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Milano 19/04/2005 Maria Bondani 22
0.0 0.2 0.4 0.6 0.8 1.0
0
250
500
750
Mea
n ch
anne
l, <x>
ND-filter transmittance
0 500 10000.00
0.01
0.02
0.03
0.04
noneOD = 0.3
OD = 0.6
blank
OD = 0.9
Phot
on d
etec
tion
prob
abili
ty
Channel, x
Channel, xPh
oton
det
ectio
n pr
obab
ility
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Milano 19/04/2005 Maria Bondani 23
Analysis method• fit the experimental data by using :
⇒ the impulse response of the system
( ) ( )1
1 !
x x
x
ph
n eP
x
µµ
µµ µ
−−
=−
F. Paleari , A. Andreoni, G. Zambra and M. Bondani, Opt. Express, 12 (2004) 2816-2824
convoluted with
⇒ the theoretical multithermal function
⇒ the fitting parameter is the
number of modes m0 100 200 300 400 500 600
0.00
0.01
0.02
0.03
0.04
Phot
on d
etec
tion
prob
abili
tyChannel, x
impulse response
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Milano 19/04/2005 Maria Bondani 24
BALANCED HOMODYNE DETECTION
• is a tool for measuring FIELD QUADRATURES
Signal a
LO aLO
( )2 LO12
a a α= +
( )1 LO12
a a α= − 2 1D n n∝ −
f iL O L O L Oa e φα α→ =
1L Oα > >
† †2 1 2 2 1 1
† * † †
2 2
ˆ ˆ
2 2
LO LO
i iLO LO
LO
n n a a a aD
a a a e aeφ φ
α α
α αα
−
− −∝ =
+ += =
( ) ( )†12
i iLOTr D a e ae Xφ φ
φ−≅ + = field quadrature
By varying the phase of the local oscillator, φ, we can measure every quadrature of the field
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Milano 19/04/2005 Maria Bondani 25
⇒ The marginal distributions of the Wigner function, give the distribution of the quadrature
( ) ( ), cos sin , sin cosp X dY W X Y X Y X Xφ φφ φ φ φ φ ρ= − + =∫
( ) ( ),W X Y W X iYα≡ = +
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Milano 19/04/2005 Maria Bondani 26
DISTRIBUTION FOR QUADRATURES
Vacuum state
( )22
22
10 exp2
Xp X Xφ φ σπσ⎛ ⎞
= = −⎜ ⎟⎝ ⎠
0 0 0x Xφ= = 2 2 10 04
Xφσ = ∆ =
Thermal state
( ) ( )2
22
1 exp2
T TXp X Tr X Xφ φ φρσπσ
⎛ ⎞= = −⎜ ⎟
⎝ ⎠
0x = ( )2 1 2 14 ph
nσ = +
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Milano 19/04/2005 Maria Bondani 27
BALANCED HOMODYNE DETECTION
• ultra-low noise amplification profile
⇒ measurement of sub-nanowatt pulses in the entire frequency range
• p-i-n photodiodes
• high quantum efficiency
• pulsed operation
BHD linear gain
Vin (mV)
Vou
t(m
V)
problems :
• matching of the spatio-temporal modes
• stability of the LO phase• long-term stability of the experimental setup
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• measurements with random-phase LO
OPTICAL DELAY LINE
-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80
2000400060008000
1000012000140001600018000200002200024000260002800030000
coun
ts
homodyne current (V)
vacuum state thermal state
0 50000 100000 150000 200000 250000
-1.0
-0.5
0.0
0.5
1.0 thermal state vacuum state
hom
odyn
e ou
tput
(V)
number of measurements
LASER
BBO
BSBS
BHD
LO BOX CAR
PC
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QUANTUM TOMOGRAPHYtomography of a 2D-object is the ensemble of the 1D-projections taken at different angles : starting from this partial information the entire knowledge of the object can be recovered
⇒ Inverse Radon transform
The collection of is the Radon transform of the two-dimensional image
( )p Xφ
( ),W X Y
( ) ( ) ( ), , exp cos sin4
k dkW X Y d dqp q ik q X Yφ φ φ φ= − −⎡ ⎤⎣ ⎦∫ ∫ ∫⇒ the integral diverges if implemented on the experimental data.
⇒ quantum inversion algorithm that applies to experimental data without any regularization M.G.A. Paris and J. Reachek Ed.s
Lectures notes in physics, 649 (2004)
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• tomographic reconstruction of the Wigner function
• tomographic reconstruction of the photon number distribution
-2-1
01
2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2-1
01
2
Wig
ne fu
nctio
n, W
(q,p
)
qp
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
0.8
1.0
phot
on n
umbe
r di
stri
butio
n, P
(n)
number of photons, n
vacuum state thermal state theoretical thermal
distribution
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ON/OFF DETECTION
( ) ( ),0
1 nOFF el n
n
p η η ρ∞
=
= −∑
( ) ( )0
1 nOFF
n
n nΠ η η∞
=
= −∑
BS: T =η
ON/OFF photodetector
ph ii
i iρ ρ∞
= ∑
( ) ( )ON OFFΠ η Π η= −
⇒ reconstruction of the photon number statistics starting from the minimum possible information : the statistics of the "no-click" and "click" events from an ON/OFF detector
⇒ ON/OFF detectors : photon counters, avalanche photodiodes
( ) ( ),0
1 1 nON el n
n
p η η ρ∞
=
= − −∑
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RECONSTRUCTION ALGORITHM
⇒ measure for a collection of different quantum efficiencies( ),OFF elp νη νη
( ) 00 , nf f
nν
νν
η ν= =⇒ evaluate the collection of frequencies
nρ
( ) ( ),0
1 nOFF el n
n
p η η ρ∞
=
= −∑⇒ apply the maximum-likelihood estimation toto find
1
1
Ki i nn n i
m nm
A fA p
ν ν
ν ν ν
ρ ρρ
+
=
=⎡ ⎤⎣ ⎦
∑∑ ( )0p pν νη=
( )1 nnAν νη= −
⇒ iterative solution
0
Ki i
nf pν νν
ε ρ=
⎡ ⎤= − ⎣ ⎦∑⇒ measure of the convergence
A.R. Rossi, S. Olivares, M.G.A. ParisPhys. Rev. A , 70 (2004) 055801
⇒ numerical simulations give good results also in the presence of noise
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rotatingground
glass
ON/OFF detectorlaserfilters
pin-hole
• photomultiplier (BURLE)
• insert neutral filters to change the quantum efficiency
• use the mean value to evaluate the effective quantum efficiency
• typically 104 - 105 acquisitions for each value of h
• relatively high mean photon number
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0.00 0.05 0.10 0.15 0.20
0.6
0.8
1.0
0 5 10 15 20 25 300.00
0.04
0.08
0.12
0.16
reconstruction best fit
ρ n
n
experimental data best fit
f ν
ην
5.33n =
• classical thermal light
0.00 0.05 0.10 0.15 0.20 0.250.25
0.50
0.75
1.00
0 5 10 15 20 25 300.00
0.04
0.08
0.12
reconstruction best fit
ρ n
n
experimental data best fit
f ν
ην
• quantum multi thermal light
6.17 5n µ= =
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• gaussian light(laser with thermal noise)
0.00 0.05 0.10 0.15 0.200.4
0.6
0.8
1.0
0 5 10 15 200.00
0.04
0.08
0.12
0.16
reconstruction best fit
ρ n
n
experimental data bst fit
f ν
ην ( )
( )( )
2
, 22
1 exp22
n teo
n nnn
ρσπ σ
⎡ ⎤−⎢ ⎥= −
+⎢ ⎥+ ⎣ ⎦24.88 0.63n σ= =
0 200 400 600 800 100010-4
10-3
10-2
10-1
100
ε(i)
Iteration number, i
Fock Coherent Thermal Multithermal• the convergence criterium is satisfied
• long term drift of the reconstructed distribution
⇒increase number of acquisitionsto decrease noise
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0.00 0.05 0.10 0.15 0.200.80
0.85
0.90
0.95
1.00
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0ρ n
n
f ν
ην
0.02n =
G. Zambra, A. Andreoni, M. Bondani, M. Gramegna,M. Genovese, G. Brida, A.R. Rossi, M.G.A. ParisExperimental reconstruction of photon statistics without photon counting Submitted (2005)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.985
0.990
0.995
1.000
• Fock state n = 1
0 1 2 30.0
0.2
0.4
0.6
0.8
1.0
ρ nn
f ν
ην
• poissonian light
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CONCLUSIONS
⇒ importance of determining photon number statistics
• diagnostics of the nature of light• preparation of conditional states of light
⇒ direct detection
• photon counting low mean numbers low quantum efficiency
noise• intensity measurements
⇒ indirect detection
low mean numbers mode matching• homodyne detection
• ON/OFF instability of the algorithmlong acquisition time