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


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


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


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


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+)?


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


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


Correlation Function

I1+I2

21II


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


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?


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.


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?


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.


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


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


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


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


Classical Noise Model

Add Field Amplitudes

Re U

Im U

Us

Un

10842-1.tex:2


Photon Noise

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


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


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


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


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


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


More on Probability

• Where is the particle?

• Matrix Notation

dxxxxdxxPx )()()( *

Xx †

2

1

21 ** x

x

Xxxx


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


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


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


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


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

*

http://www-groups.dcs.st-and.ac.uk/%7Ehistory/PictDisplay/Schrodinger.html


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:


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


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?


Harmonic Oscillator

• Hamiltonian

• Frequency

22

2

1

2

1kxmvH

222

2

1

2

1mx

m

pH

m

k2

PotentialEnergy

x


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


Louisell’s Approach

• Harmonic Oscillator– Unit Mass

• New Operators

222

2

1qpH

ipqa 2

1 ipqa 2

1†

†

1, † aa


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


Energy Eigenvalues

• Number Operator

• Eigenvalues of the Hamiltonian

aaN †2

11 HN

EEEH || '|''| nnnN

2

1nEn


Creation and Anihilation (1)

• Note the Following Commutators

• Then

1, † aa aaaa †, †††, aaaa

1†† NaNa 1 NaNa

'|)1'('| nannaN '|)1'('| †† nannaN


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


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

'|)1('| †† nannaN

'|''| nnnN

1'|'|† nna

1'|'| nna

1'|)1'(1'| nnnN

1'|)1'(1'| nnnN


Reminder!

• All Observables are Represented by Hermitian Operators.

• Their Eigenvalues must be Real

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

Documents

mcknightchuck dimarzio

northeastern universitya

northeastern universityphoton

northeastern universitythe

northeastern universitywhat

studentchuck dimarzio

later time t t

short time