Information in a Photon OPTI 495B/595BLecture 1

Saikat GuhaJanuary 16, 2020

Associate ProfessorCollege of Optical SciencesMeinel 523

Logistical information

• Course instructor: Saikat Guha, saikat@optics.arizona.edu, OSC 523, (520) 621-7595

• Course assistant: Brianna Moreno, bmoreno@optics.arizona.edu, OSC 501B, (520) 626-7080

• Teaching Assistant: Michael Grace, michaelgrace@email.arizona.edu, OSC 501

• Lectures 12:30 pm – 1:45 pm Tuesdays and Thursdays, OSC 305• Office hours 10 am – 11 am, Wednesdays, OSC 447• Problem sets – problems assigned on both days of the week, solutions

due following Thursday in class• Finals week – 15-minute student presentations on one or more

“advanced” homework problem(s) of your own choice– Undergraduate students can work in groups– Graduate students expected to submit a written report

• Grading: problem sets (70%), presentation (30%)

Preamble

• The fundamental limits of efficiency with which one can extract information encoded in light– Light is used in communication, sensing, computing– Optical detection must add noise (fundamental)– This noise degrades quality of information extraction– Information-bearing light may be manipulated in the

optical domain without adding appreciable noise. This can result in the inevitable detection noise to affect the information extraction efficiency less or more favorably

– Light is a fundamentally quantum mechanical object– Finding optimal ways to detect (a.k.a., optimal pre-

detection optical-domain manipulations) requires quantum information and quantum estimation theory

Mathematical pre-requisites

• Basic undergraduate mathematics– Complex numbers– Basic linear algebra (matrices, eigenvalues, etc.)– Probability – Calculus– MATLAB (or, equivalent)

Course outline

• Module 1: Shot-noise in photodetection and novel optical receiver design [10 lectures]

• Module 2: Information theory and quantum limits of optical communications [10 lectures]

• Module 3: Estimation theory, and quantum limits of optical sensing [10 lectures]

Random variables

• Discrete random variable, – Example probability mass function (p.m.f.)

• Continuous random variable, – Example probability distribution function (p.d.f.)

Poisson

Mean, Variance,

Gaussian

Variance,Mean,

Two more Discrete Random Variables

• Bernoulli

– Prove that:

• Binomial

– Evaluate E[N] and Var[N]

Problem 1

Problem 2

Convergence of Binomial to Poisson

• Take constant, while

– Prove that in the above limit,

Problem 3

Random process describing Discreteevents in continuous time and/or space

• Where such random processes might occur– Spiking patterns in neurons, arrival processes, photon detection, etc.

• Arrival process, ; counting process,– I(t): random arrivals (each arrival denoted by a delta function)– N(t): number of occurrences (arrivals) before time t,

Arrival rate:

Constant rate:

Mean # arrivals:

Poisson Point Process (PPP)

• Let us watch this short lecture video…

Poisson point process (PPP)

• Counting process that satisfies the following is a PPP–– For all , S.I. of – , s.t. for any ,– If then

• Let us generate a PPP with constant rate, T = 1e-9;N = 10;lambda = N/T;dT = 1e-14;t = 0:dT:T-dT;p = lambda*dT;n = 0;clicks = -(sign(rand(1,length(t))-p)-1)/2;plot(t,clicks,'LineWidth',2);

Probability of one arrival in a interval,

Independent increments (memoryless)

Stationary increments

No more than one arrival in a small increment

Inter-arrival times and number of arrivals

Probability of one arrival in a interval,

Let us denote by t, the time of first arrival;

(large)

c.d.f.,

p.d.f.,

Prove that the Probability distribution of the total number of arrivals K is given by, Problem 5

Problem 4

Prove this last step:

• Quasi-mono-chromatic laser light pulse: in √(photons/m2-sec) units

• Mean photon number,

A pulse of laser light

Spatial mode Temporal mode

Spatial and temporal dependence may not be factorable in general

Re

Im

Phase space picture: once we identify a spatio-temporal-polarization mode, a complex number describes the state of the laser pulse

No detector can accurately measure the field

“coherent state”

Ideal photon detection on a laser pulse

• Consider the √(photons/sec) unit pulse,• Detector will produce clicks as PPP with rate,• Mean photon number in pulse,

This is the model for an “ideal” photon (direct) detection, i.e.,

Infinite bandwidth / zero dead timeUnity quantum efficiencyZero dark clicksZero timing jitterZero after-pulsing rate

Direct detection has no information about the phase,

Square (flat top) pulse

• We will be using this abstraction quite a lot for a flat-top pulse going forward– E: √(photons/sec) unit field– Mean photon number,

Rate of photon arrivals, when detected with an ideal direct detection receiver

Coming next…

• Coherent states• On-off keying (OOK) and binary phase-shift keying

(BPSK) modulation formats