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

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

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

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 F. De Matteis Quantum Optics

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 F. De Matteis Quantum Optics

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

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

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

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

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

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

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

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

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

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

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

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