measurements of photon statistics of classical and quantum...
Post on 18-Apr-2018
224 Views
Preview:
TRANSCRIPT
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
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
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
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= =
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
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 =
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
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= + −
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
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
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
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
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
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
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
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
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
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
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)
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.
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 :
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
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
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
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≡ = +
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 = +
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
Milano 19/04/2005 Maria Bondani 28
• 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
Milano 19/04/2005 Maria Bondani 29
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)
Milano 19/04/2005 Maria Bondani 30
• 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
Milano 19/04/2005 Maria Bondani 31
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∞
=
= − −∑
Milano 19/04/2005 Maria Bondani 32
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
Milano 19/04/2005 Maria Bondani 33
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
Milano 19/04/2005 Maria Bondani 34
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= =
Milano 19/04/2005 Maria Bondani 35
• 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
Milano 19/04/2005 Maria Bondani 36
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
Milano 19/04/2005 Maria Bondani 37
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
top related