Measurements of photon statistics of classical and quantum ...qinf.fisica.unimi.it/~paris/QSlides/MBondani_19Apr05.pdf · Measurements of photon statistics of classical and quantum

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

http://www.ien.it/index_i.shtml


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


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

http://www.coherentinc.com/index.cfm

http://www.newport.com/Lasers/1/productmain.aspx


• 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= =


⇒ 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

http://images.google.it/imgres?imgurl=http://www-okabon.ces.kyutech.ac.jp/~okamoto/okamoto/photo/speckle.gif&imgrefurl=http://www-okabon.ces.kyutech.ac.jp/~okamoto/&h=152&w=201&sz=15&tbnid=1uNkmFc4xpcJ:&tbnh=74&tbnw=98&start=4&prev=/images%3Fq%3Dspeckle%26hl%3Dit%26lr%3D%26sa%3DN


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 =


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


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= + −


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


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


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


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


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


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


( ) ( )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


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


K = 1 Average = 2.68


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



Det

ectio

n pr

obab

ility



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




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)


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.


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 :


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


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


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


⇒ 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≡ = +


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 = +


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


• 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


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)


• 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


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∞

=

= − −∑


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


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


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


ρ n

n

experimental data best fit

f ν

ην

• quantum multi thermal light

6.17 5n µ= =


• 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


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


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


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

Measurements of photon statistics of classical and quantum ...qinf.fisica.unimi.it/~paris/QSlides/MBondani_19Apr05.pdf · Measurements of photon statistics of classical and quantum

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