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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

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ECEG398 Quantum Optics Course NotesPart 1: Introduction

Prof. Charles A. DiMarzio

and Prof. Anthony J. Devaney

Northeastern University

Spring 2006

Chuck DiMarzio, Northeastern University

lecture overview
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

Chuck DiMarzio, Northeastern University

classical maxwellian em waves
Classical Maxwellian EM Waves

v=c

λ

E

H

H

x

E

E

z

H

λ=c/υ

y

c=3x108 m/s (free space)

υ = frequency (Hz)

Chuck DiMarzio, Northeastern University

Thanks to Prof. S. W.McKnight

electromagnetic spectrum by

VIS=

0.40-0.75μ

γ-Ray

RF

Electromagnetic Spectrum (by λ)

UV=

Near-UV: 0.3-.4 μ

Vacuum-UV: 100-300 nm

Extreme-UV: 1-100 nm

IR=

Near: 0.75-2.5μ

Mid: 2.5-30μ

Far: 30-1000μ

10 nm =100Å

0.1 μ

1 μ

10 μ

100 μ = 0.1mm

(300 THz)

0.1 Å

1 Å

10 Å

1 mm

1 cm

0.1 m

X-Ray

Soft X-Ray

Mm-waves

Microwaves

Chuck DiMarzio, Northeastern University

Thanks to Prof. S. W.McKnight

coherence of light
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+t (or at r+r)?

Chuck DiMarzio, Northeastern University

coherence and bandwidth

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

0

5

10

0

5

10

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

0

5

10

0

5

10

Coherence and Bandwidth

Pure Cosine

f=1

Pure Cosine

f=1.05

3 Cosines

Averaged

f=

0.93, 1, 1.05

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

Chuck DiMarzio, Northeastern University

realistic example

0.4

0.2

0

-0.2

-0.4

0

1

2

3

4

5

6

7

8

f

0.4

0.2

0

-0.2

-0.4

0

1

2

3

4

5

6

7

8

Realistic Example

Long Delay: Decorrelation

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

Short Delay

Chuck DiMarzio, Northeastern University

correlation function

I1+I2

Correlation Function

Chuck DiMarzio, Northeastern University

controlling coherence
Controlling Coherence

Making Light Coherent

Making Light Incoherent

Ground Glass to

Destroy Spatial Coherence

Spatial Filter for

Spatial Coherence

Wavelength Filter

for Temporal Coherence

Move it to

Destroy Temporal Coherence

Chuck DiMarzio, Northeastern University

a thought experiment
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?

Chuck DiMarzio, Northeastern University

photon counting experiment

Clock Signal

t

0

5

Photon Arrival

t

Photon Count

3

1

2

t

Counter

Gate

Clock

n

Photon Counting Experiment

Experimental Setup to measure the probability distribution of photon number.

Probability Density

Chuck DiMarzio, Northeastern University

the mean number
The Mean Number
  • Photon Energy is hn
  • Power on Detector is P
  • Photon Arrival Rate is a=P/hn
    • Photon “Headway” is 1/a
  • Energy During Gate is PT
  • Mean Photon Count is n=PT/hn
  • But what is the Standard Deviation?

Chuck DiMarzio, Northeastern University

what do you expect
What do you expect?
  • Photons arrive equally spaced in time.
    • One photon per time 1/a
    • Count is aT +/- 1 maybe?
  • Photons are like the Number 39 Bus.
    • If the headway is 1/a=5 min...
    • Sometimes you wait 15 minutes and get three of them.

Chuck DiMarzio, Northeastern University

back door quantum optics power
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)=adt
    • 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

Chuck DiMarzio, Northeastern University

quantum coherence
Quantum Coherence

Here are some results: Later we will prove them.

Chuck DiMarzio, Northeastern University

question for later can we do better
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

Chuck DiMarzio, Northeastern University

back door quantum optics field
Back-Door Quantum Optics (Field)
  • Assume a classical (constant) field, Usig
  • Add a random noise field Unoise
    • Complex Zero-Mean Gaussian
  • Compute s as function of <| Unoise|2>
  • Compare to Poisson distribution
  • Fix <| Unoise|2> to Determine Noise Source Equivalent to Quantum Fluctuations

Chuck DiMarzio, Northeastern University

classical noise model
Classical Noise Model

Add Field Amplitudes

Im U

Un

10842-1.tex:2

Us

Re U

Chuck DiMarzio, Northeastern University

photon noise
Photon Noise

10842-1.tex:3

10842-1.tex:5

=

10842-1-5.tif

Chuck DiMarzio, Northeastern University

noise power
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

Chuck DiMarzio, Northeastern University

the real thing survival guide
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

Chuck DiMarzio, Northeastern University

five postulates
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, yi after making the measurement.
  • 5. The time dependence of a state is given by

Chuck DiMarzio, Northeastern University

state of a system
State of a System
  • State Defined by a Wave Function, y
    • Depends on, eg. position or momentum
    • Equivalent information in different representations. y(x) and f(p), a Fourier Pair
  • Interpretation of Wavefunction
    • Probability Density: P(x)=|y(x)|2
    • Probability: P(x)dx=|y(x)|2dx

Chuck DiMarzio, Northeastern University

wave function as a vector

y1(x)

y2(x)

x

x

Wave Function as a Vector
  • List y(x) for all x (Infinite Dimensionality)
  • Write as superposition of vectors in a basis set.

y(x)=a1y1(x)+a2y2(x)+...

Chuck DiMarzio, Northeastern University

more on probability
More on Probability
  • Where is the particle?
  • Matrix Notation

Chuck DiMarzio, Northeastern University

pop quiz just kidding
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(iw1t) and exp(iw2t)
  • Describe the particle motion.

y2(x)

y1(x)

x

x

Stopped Wed 11 Jan 06

Chuck DiMarzio, Northeastern University

dirac notation
Dirac Notation
  • Simple Way to Write Vectors
    • Kets
    • and Bras
  • Scalar Products
    • Brackets
  • Operators

Chuck DiMarzio, Northeastern University

commutators and uncertainty
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

sxsp > h/2

[x H] = x H - H x = 0

Chuck DiMarzio, Northeastern University

recall the five postulates
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, yi after making the measurement.
  • 5. The time dependence of a state is given by

Chuck DiMarzio, Northeastern University

shr dinger equation

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

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

Chuck DiMarzio, Northeastern University

particle in a box
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.

Momentum

Operator:

Chuck DiMarzio, Northeastern University

some wavefunctions

1

0.5

0

-0.5

-1

0

0.2

0.4

0.6

0.8

1

Some Wavefunctions

Shrödinger Equation

Eigenvalue Problem

Hy=Ey

Solution

Temporal Behavior

Chuck DiMarzio, Northeastern University

pop quiz 2 still kidding
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?

Chuck DiMarzio, Northeastern University

harmonic oscillator
Harmonic Oscillator
  • Hamiltonian
  • Frequency

Potential

Energy

x

Chuck DiMarzio, Northeastern University

harmonic oscillator energy
Harmonic Oscillator Energy
  • Solve the Shrödinger Equation
  • Solve the Eigenvalue Problem
  • Energy
    • Recall that...

Chuck DiMarzio, Northeastern University

louisell s approach

Louisell’s Approach
  • Harmonic Oscillator
    • Unit Mass
  • New Operators

Chuck DiMarzio, Northeastern University

the hamiltonian
The Hamiltonian
  • In terms of a, a †
  • Equations of Motion

Chuck DiMarzio, Northeastern University

energy eigenvalues
Energy Eigenvalues
  • Number Operator
  • Eigenvalues of the Hamiltonian

Chuck DiMarzio, Northeastern University

creation and anihilation 1
Creation and Anihilation (1)
  • Note the Following Commutators
  • Then

Chuck DiMarzio, Northeastern University

creation and anihilation 2
Creation and Anihilation (2)

Energy

Eigenvalues

Eigenvalue Equations

States

Chuck DiMarzio, Northeastern University

creation and anihilation 3
Creation and Anihilation (3)

Chuck DiMarzio, Northeastern University

reminder
Reminder!
  • All Observables are Represented by Hermitian Operators.
  • Their Eigenvalues must be Real

Chuck DiMarzio, Northeastern University