1. Photon Statistics Light beam  stream of photons
Difference with classical waves relies on different statistical properties
Average count rate corresponds to Intensity of light beam
Actual count rate fluctuates from measurement to measurement
The pulse nature of the photon count is the evidence of the discreteness of photon?
Or consequence of detector properties? t
Detection of a light beam (low intensity) by means of photomultiplier tube (PMT) or an avalanche photodiode (APD)
Average number of photons through a cross-section in unit time
Average number of counts registered by detector in counting time t
F. De Matteis
Quantum Optics

2. Photon Counting t
Max count rate determined by recovery time of detector («dead time» ~ 1ms) J eV Hz nm
cm-1 1
6,242x1018
1,509x1033
1,986x10-16
5,034x1022
1,602x10-19
2,418x1014
1,240x103
8,066x103
6,626x10-34
4,136x10-15
2,998x1017
3,336x10-11
1,986x10-25
1,240x10-6
0,01
1,986x10-23
1,240x10-4
2,998x1010
107
HeNe attenuated by 10-6
The response consists of short voltage pulses counted in a certain time interval
The count rate, deriving from a decay process which is discrete by nature, will intrinsically show fluctuations
The average number determines the intensity value around which the counting rate fluctuates.
The magnitude of such fluctuation is determined by the nature of the different type of light beams we can take.
F. De Matteis
Quantum Optics

3. Coherent light statistic
Statistic of coherent light is Poisson distribution
Probability for n photons
Independent random events
Root mean square deviation
Relative amplitude of fluctuations
F. De Matteis
Quantum Optics

4. Classification by statistics
A coherent beam of constant intensity is represented by Poisson statistics
Intensity fluctuation Super-Poissonian statistics
Sub-Poissonian distribution non-classic light beam
Not classic
Perfectly coherent light
Partially coherent, uncoherent, thermal emission
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Quantum Optics

5. Super-Poissonian light
Every time an intensity fluctuation is present Super-Poissonian distribution.
Most of experimental light sources
Blackbody emission, e.g. thermal radiation at equilibrium
Bose-Einstein distribution
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Quantum Optics

6. Super-Poissonian light
Every time an intensity fluctuation is present Super-Poissonian distribution.
Most of experimental light sources
Blackbody emission, e.g. thermal radiation at equilibrium
Wave Noise
Typical of the discrete (particle) nature of the light
F. De Matteis
Quantum Optics

7. Super-Poissonian light
Light from a single spectral line of a discharge lamp, chaotic light, has a partial coherence due to the phase jumps consequent to the collisions, the total intensity fluctuate in time around the mean value. Thermal radiation Off-Equilibrium Ī
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
Photon flux not constant due to fluctuations on a time scale of the order of coherence time tc
If t ≤ tc additional fluctuations are significant (Quantum + thermal + coherence)
Otherwise coherence fluctuations disappear in the average
F. De Matteis
Quantum Optics

8. Super-Poissonian light
Light from a single spectral line of a discharge lamp, chaotic light, has a partial coherence due to the phase jumps consequent to the collisions, the total intensity fluctuate in time around the mean value. Thermal radiation Off-Equilibrium
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
Wt count rate
t detection time interval
If no intensity fluctuations and F constant in time, it would revert to a Poissonian
First contribution
= Quantum Noise
Second contribution (thermal + coherence noise)
= Classical/Wave Noise
F. De Matteis
Quantum Optics

9. Sub-Poissonian light Typical quantistic state
Sub-Poissonian light is «more stable» than perfectly coherent beam.
No classical equivalence
Typical quantistic state
Represented by number state |n>
Maximum phase indetermination
Minimum intensity indetermination
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
Loss sources in the detection processes degrades the regularity of the photon flux making difficult the observation of sub-Poissonian light
F. De Matteis
Quantum Optics

10. Semi-classical theory of photodetection
Photomultiplier: Essential process  generation of primary electrons on the cathode
Dt’ such that I(t’)=I(t’+ Dt’)
Emission probability of a photo-electron in short time interval
Negligible probability for emission of 2 photo-electrons in Dt’
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
Events at different times are independent
Chain of recursive equations
F. De Matteis
Quantum Optics

11. Semi-classical theory of photodetection
Photomultiplier: Essential process  generation of primary electrons on the cathode
Dt’ such that I(t’)=I(t’+ Dt’)
Chain of recursive equations
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
F. De Matteis
Quantum Optics

12. Semi-classical theory of photodetection
Photomultiplier: Essential process  generation of primary electrons on the cathode
A set of measurements yields different values
It can fluctuate with starting time t
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
Average of the count probability over a large number starting times (ensemble average = time average)
F. De Matteis
Quantum Optics

13. Semi-classical theory of photodetection
Photomultiplier: Essential process  generation of primary electrons on the cathode
1) Constant intensity
Classical stable wave equivalent to quantum mechanical coherent state
2) Chaotic light with tr<<tc
Random walk case
Single mode of a thermal source
Also for chaotic light if tr>>tc
F. De Matteis
Quantum Optics

14. Quantum theory of photodetection
Variance of number of photon-counts
Variance of number of photons
detected photon lost photons
If h=1 then DN=Dn
If (Dn)2=n̅ then (DN)2=hn̅=N̅
If h<<1 then (DN)2=N̅ whatever light
Detector’s quantum efficiency = key element to faithfully measure light radiation statistic
Any optical loss in light detection process can be seen like a casual sampling in a beam-splitter.
Optical losses determine a degradation of the regularity of photon flux.
Anche in questo caso si parla di rumore quantistico e rumore d’onda (classico)
F. De Matteis
Quantum Optics

15. Shot-noise in photodiodes
Photon flux > 106 phot/s  photodiode in reverse bias V0
Photocurrent
Responsivity
Poissonian beam (Di)2∞ (Dn)2=<n>
(Di)2=2eDf <i> (Wiener-Kintchine)
Pnoise(f)=2eRLDf <i>
White noise
Shot-Noise
Always present a classical fluctuation component (electric noise of the circuit, mechanical noise of the laser cavity, wave fluctuation)
At high frequeny, detector response limit 1/tD
F. De Matteis
Quantum Optics

16. Shot-noise in photodiodes
Photocurrent
Responsivity
Pnoise(f)=2eRLDf <i>
Ti:sapphire
930nm
@50 ±3 Mhz
Classical noise can be reduced
Quantum=shot noise cannot
Both originate from source and detection
Optimize production and detection
Noise eater
Balanced detector
Signal from the two identical detectors are subtracted. All classical fluctuation are washed out
Negative feedback compensate for fluctuation (only classical contribution)
F. De Matteis
Quantum Optics

17. Observation of sub-Poissonian statistics
Light source with sub-Poissonian statistics
High quantum efficiency detector h~1
Excited state lifetime much shorter than time scale of the electrical current fluctuation used for generation of atomic
excitation
AlGaAs LED
875nm
LED or laser diodes have got much better efficiency.
Electrical noise determined by thermal noise of the resistence can be reduced well below the shot noise.
kbT~25meV
L’emissione termica degli elettroni dal catodo avviene, in generale, con una statistica Poissoniana.
A bassa tensione (regime di carica spaziale) il flusso di elettroni è più regolare
F. De Matteis
Quantum Optics