january 2006chuck dimarzio, northeastern university10842-1c-1 eceg398 quantum optics course notes...

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-1

ECEG398 Quantum Optics Course Notes

Part 1: Introduction

Prof. Charles A. DiMarzio

and Prof. Anthony J. Devaney

Northeastern University

Spring 2006

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-2

Lecture Overview• Motivation

– Optical Spectrum and Sources

– Coherence, Bandwidth, and Fluctuations

– Motivation: Photon Counting Experiments

– Classical Optical Noise

– Back-Door Quantum Optics

• Background– Survival Quantum Mechanics

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-3

Classical Maxwellian EM Waves

E E

E

x

y

z H

HH

λ

v=c

λ=c/υ

c=3x108 m/s (free space)

υ = frequency (Hz)

Thanks to Prof. S. W.McKnight

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-4

Electromagnetic Spectrum (by λ)

1 μ 10 μ 100 μ = 0.1mm

0.1 μ10 nm =100Å

VIS=

0.40-0.75μ

1 mm 1 cm 0.1 m

IR=

Near: 0.75-2.5μ

Mid: 2.5-30μ

Far: 30-1000μ

UV=

Near-UV: 0.3-.4 μ

Vacuum-UV: 100-300 nm

Extreme-UV: 1-100 nm

MicrowavesX-Ray Mm-waves

10 Å1 Å0.1 Å

Soft X-Ray RFγ-Ray

(300 THz)

Thanks to Prof. S. W.McKnight

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-5

Coherence of Light

• Assume I know the amplitude and phase of the wave at some time t (or position r).

• Can I predict the amplitude and phase of the wave at some later time t+(or at r+)?

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-6

Coherence and Bandwidth

Pure Cosinef=1

Pure Cosinef=1.05

3 CosinesAveragedf=0.93, 1, 1.05

Same as at left, and a delayed copy. Note Loss of coherence.

0 5 10-1

-0.5

0

0.5

1

0 5 10-1

-0.5

0

0.5

1

0 5 10-1

-0.5

0

0.5

1

0 5 10-1

-0.5

0

0.5

1

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-7

Realistic Example

50 Random Sine Waves with Center Frequency 1 and Bandwidth 0.8.

f

0 1 2 3 4 5 6 7 8-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8-0.4

-0.2

0

0.2

0.4

Long Delay: Decorrelation

Short Delay

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-8

Correlation Function

I1+I2

21II

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-9

Controlling CoherenceMaking Light Coherent Making Light Incoherent

Spatial Filter forSpatial Coherence

Wavelength Filterfor Temporal Coherence

Ground Glass toDestroy Spatial Coherence

Move it toDestroy Temporal Coherence

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-10

A Thought Experiment

• Consider the most coherent source I can imagine.

• Suppose I believe that light comes in quanta called photons.

• What are the implications of that assumption for fluctuations?

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-11

Photon Counting Experiment

0 5

Clock

GateCounter

tClock Signal

t

Photon Arrival

t

Photon Count3 1 2

Probability Density

n

Experimental Setup to measure the probability distribution of photon number.

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-12

The Mean Number

• Photon Energy is h• Power on Detector is P

• Photon Arrival Rate is =P/h – Photon “Headway” is 1/

• Energy During Gate is PT

• Mean Photon Count is n=PT/h• But what is the Standard Deviation?

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-13

What do you expect?

• Photons arrive equally spaced in time.– One photon per time 1/– Count is T +/- 1 maybe?

• Photons are like the Number 39 Bus.– If the headway is 1/5 min...– Sometimes you wait 15 minutes and get three

of them.

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-14

Back-Door Quantum Optics (Power)

• Suppose I detect some photons in time, t

• Consider a short time, dt, after that– The probability of a photon is P(1,dt)=dt– dt is so small that P(2,dt) is almost zero– Assume this is independent of previous history– P(n,t+dt)=P(n,t)P(0,dt)+P(n-1,t)P(1,dt)

• Poisson Distribution: P(n,t)=exp(-at)(at)n/n!

• The proof is an exercise for the student

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-15

Quantum CoherenceHere are some results: Later we will prove them.

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-16

Question for Later: Can We Do Better?

• Poisson Distribution– – Fundamental Limit on Noise

• Amplitude and• Phase

– Limit is On the Product of Uncertainties

• Squeezed Light– Amplitude Squeezed (Subpoisson Statistics) but larger

phase noise– Phase Squeezed (Just the Opposite)

Stopped here 9 Jan 06

n2

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-17

Back-Door Quantum Optics (Field)

• Assume a classical (constant) field, Usig

• Add a random noise field Unoise

– Complex Zero-Mean Gaussian

• Compute as function of <| Unoise|2>

• Compare to Poisson distribution

• Fix <| Unoise|2> to Determine Noise Source Equivalent to Quantum Fluctuations

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-18

Classical Noise Model

Add Field Amplitudes

Re U

Im U

Us

Un

10842-1.tex:2

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-19

Photon Noise

10842-1.tex:3 10842-1.tex:5=10842-1-5.tif

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-20

Noise Power

• One Photon per Reciprocal Bandwidth

• Amplitude Fluctuation– Set by Matching Poisson Distribution

• Phase Fluctuation– Set by Assuming

• Equal Noise in Real and Imaginary Part

• Real and Imaginary Part Uncorrelated

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-21

The Real Thing! Survival Guide

• The Postulates of Quantum Mechanics

• States and Wave Functions

• Probability Densities

• Representations

• Dirac Notation: Vectors, Bras, and Kets

• Commutators and Uncertainty

• Harmonic Oscillator

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-22

Five Postulates• 1. The physical state of a system is described by a

wavefunction.

• 2. Every physical observable corresponds to a Hermitian operator.

• 3. The result of a measurement is an eigenvalue of the corresponding operator.

• 4. If we obtain the result ai in measuring A, then the system is in the corresponding eigenstate, i after making the measurement.

• 5. The time dependence of a state is given by

Ht

it

hi

2

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-23

State of a System

• State Defined by a Wave Function, – Depends on, eg. position or momentum– Equivalent information in different

representations. (x) and (p), a Fourier Pair

• Interpretation of Wavefunction– Probability Density: P(x)=|(x)|2

– Probability: P(x)dx=|(x)|2dx

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-24

Wave Function as a Vector

• List (x) for all x (Infinite Dimensionality)

• Write as superposition of vectors in a basis set. (x)

(x)

x

x

(x)=a11(x)+a22(x)+...

...2

1

a

a

...2

1

x

x

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-25

More on Probability

• Where is the particle?

• Matrix Notation

dxxxxdxxPx )()()( *

Xx †

2

1

21 ** x

x

Xxxx

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-26

Pop Quiz! (Just kidding)

• Suppose that the particle is in a superposition of these two states.

• Suppose that the temporal behaviors of the states are exp(i1t) and exp(i2t)

• Describe the particle motion.(x) (x)

x xStopped Wed 11 Jan 06

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-27

Dirac Notation

• Simple Way to Write Vectors– Kets– and Bras

• Scalar Products– Brackets

• Operators

2

1

|

*2

*1|

2

1*2

*1|

2

1

2

1*2

*1

0

0

00

||

x

x

xx

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-28

Commutators and Uncertainty

• Some operators commute and some don’t.

• We define the commutator as

[a b] = a b - b a

• Examples

[x p] = x p - p x = ih

xp h [x H] = x H - H x = 0

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-29

Recall the Five Postulates• 1. The physical state of a system is described by a

wavefunction.

• 2. Every physical observable corresponds to a Hermitian operator.

• 3. The result of a measurement is an eigenvalue of the corresponding operator.

• 4. If we obtain the result ai in measuring A, then the system is in the corresponding eigenstate, i after making the measurement.

• 5. The time dependence of a state is given by

H

ti

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-30

Shrödinger Equation

• Temporal Behavior of the Wave Function

– H is the Hamiltonian, or Energy Operator.

• The First Steps to Solve Any Problem:– Find the Hamiltonian– Solve the Schrödinger Equation– Find Eigenvalues of H

*http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html

Born: 12 Aug 1887 in Erdberg, Vienna, AustriaDied: 4 Jan 1961 in Vienna, Austria*

H

ti

*

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-31

Particle in a Box

• Before we begin the harmonic oscillator, let’s take a look at a simpler problem. We won’t do this rigorously, but let’s see if we can understand the results.

2

22

2 xmH

xip

m

pmv

22

1 22 Momentum

Operator:

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-32

Some Wavefunctions

Eigenvalue Problem

H=ESolution

2

22

8mL

hnEn

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1Shrödinger Equation

H

ti

2

22

2 xmti

Temporal Behavior 2

22

8mL

hn

ti

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-33

Pop Quiz 2 (Still Kidding)

• What are the energies associated with different values of n and L?

• Think about these in terms of energies of photons.

• What are the corresponding frequencies?

• What are the frequency differences between adjacent values of n?

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-34

Harmonic Oscillator

• Hamiltonian

• Frequency

22

2

1

2

1kxmvH

222

2

1

2

1mx

m

pH

m

k2

PotentialEnergy

x

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-35

Harmonic Oscillator Energy

• Solve the Shrödinger Equation

• Solve the Eigenvalue Problem

• Energy

– Recall that...

nnn EH ||

nnn EEEH ||

hnnEn

2

1

2

1

2

h 2

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-36

Louisell’s Approach

• Harmonic Oscillator– Unit Mass

• New Operators

222

2

1qpH

ipqa 2

1 ipqa 2

1†

†

1, † aa

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-37

The Hamiltonian

• In terms of a, a †

• Equations of Motion

2

1

2††† aaaaaaH

pp

H

dt

dq

qq

H

dt

dp 2

aiHaidt

da ,1

††

†

,1

aiHaidt

da

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-38

Energy Eigenvalues

• Number Operator

• Eigenvalues of the Hamiltonian

aaN †2

11 HN

EEEH || '|''| nnnN

2

1nEn

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-39

Creation and Anihilation (1)

• Note the Following Commutators

• Then

1, † aa aaaa †, †††, aaaa

1†† NaNa 1 NaNa

'|)1'('| nannaN '|)1'('| †† nannaN

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-40

Creation and Anihilation (2)

'|)1'('| nannaN

'|)1'('| †† nannaN

'|''| nnnN

Eigenvalue Equations States Energy Eigenvalues

'|† na

'| n

'| na

2

1'nh

2

1'nh

2

3'nh

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-41

Creation and Anihilation (3) '|)1('| nannaN

'|)1('| †† nannaN

'|''| nnnN

1'|'|† nna

1'|'| nna

1'|)1'(1'| nnnN

1'|)1'(1'| nnnN

January 2006 Chuck DiMarzio, Northeastern University 10842-1c-42

Reminder!

• All Observables are Represented by Hermitian Operators.

• Their Eigenvalues must be Real