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-1
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
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Classical Maxwellian EM Waves
v=c
λ
E
H
x
E
z
H
E
H
λ=c/υ
y
c=3x108 m/s (free space)
υ = frequency (Hz)
January 2006
Chuck DiMarzio, Northeastern University
Thanks to Prof. S. W.McKnight
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Electromagnetic Spectrum (by λ)
UV=
Near-UV: 0.3-.4 μ
VIS=
IR=
0.40-0.75μ
Near: 0.75-2.5μ
Vacuum-UV: 100-300 nm
Mid: 2.5-30μ
Extreme-UV: 1-100 nm
Far: 30-1000μ
10 nm
=100Å
0.1 Å
γ-Ray
January 2006
0.1 μ
1μ
(300 THz)
1Å
10 Å
X-Ray Soft X-Ray
1 mm
Mm-waves
Chuck DiMarzio, Northeastern University
Thanks to Prof. S. W.McKnight
10 μ
100 μ =
0.1mm
1 cm
0.1 m
Microwaves
RF
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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)?
January 2006
Chuck DiMarzio, Northeastern University
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Coherence and Bandwidth
Pure Cosine
f=1
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
0
3 Cosines
Averaged
f=
0.93, 1, 1.05
5
10
0
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
5
10
-1
0
January 2006
Pure Cosine
f=1.05
5
10
0
Chuck DiMarzio, Northeastern University
5
10
Same as at
left, and a
delayed copy.
Note Loss of
coherence.
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Realistic Example
Long Delay: Decorrelation
50 Random Sine Waves
with Center Frequency 1
and Bandwidth 0.8.
0.4
0.2
0
-0.2
-0.4
0
1
2
3
4
5
6
4
5
6
7
8
Short Delay
0.4
0.2
f
0
-0.2
-0.4
January 2006
0
1
2
Chuck DiMarzio, Northeastern University
3
7
8
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Correlation Function
 I1I 2
I1+I2
January 2006
Chuck DiMarzio, Northeastern University
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Controlling Coherence
Making Light Coherent
Making Light Incoherent
Ground Glass to
Destroy Spatial Coherence
Spatial Filter for
Spatial Coherence
Wavelength Filter
for Temporal Coherence
January 2006
Move it to
Destroy Temporal Coherence
Chuck DiMarzio, Northeastern University
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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
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Photon Counting Experiment
Clock Signal
t
Experimental Setup to
measure the probability
distribution of photon
number.
Photon Arrival
t
Photon Count
3
Gate
Counter
1
2 t
Probability Density
0 5
n
Clock
January 2006
Chuck DiMarzio, Northeastern University
10842-1c-11
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?
January 2006
Chuck DiMarzio, Northeastern University
10842-1c-12
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.
January 2006
Chuck DiMarzio, Northeastern University
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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
January 2006
Chuck DiMarzio, Northeastern University
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Quantum Coherence
Here are some results: Later we will prove them.
January 2006
Chuck DiMarzio, Northeastern University
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Question for Later:
Can We Do Better?
• Poisson Distribution
2
–  =n
– 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)
January 2006
Chuck DiMarzio, Northeastern University
Stopped here 9 Jan 06
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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
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Classical Noise Model
Add Field Amplitudes
Im U
Un
Us
January 2006
Re U
Chuck DiMarzio, Northeastern University
10842-1.tex:2
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Photon Noise
10842-1.tex:3
January 2006
10842-1.tex:5
=
10842-1-5.tif
Chuck DiMarzio, Northeastern University
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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
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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
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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
January 2006
h y
y
i
= i
= Hy
2 t
t
Chuck DiMarzio, Northeastern University
10842-1c-22
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
January 2006
Chuck DiMarzio, Northeastern University
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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)+...
y (x)
1
y  x1  


y = y  x2 
 ... 


January 2006
y2(x)
x
x
Chuck DiMarzio, Northeastern University
 a1 
 
y =  a2 
 ... 
 
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More on Probability
• Where is the particle?
x =  P( x )dx = y ( x ) xy ( x )dx
*
• Matrix Notation
x = y † Xy
x = y *  x1  y *  x2  
January 2006
X
Chuck DiMarzio, Northeastern University
y  x1  


y  x2 
  


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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
January 2006
Chuck DiMarzio, Northeastern University
x
Stopped Wed 11 Jan 06
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Dirac Notation
• Simple Way to Write Vectors
– Kets
– and Bras
 y |= y 1* y 2* 
• Scalar Products
– Brackets
• Operators
January 2006
y 1 
 
|y = y 2 
  
 
 y |y = y 1* y 2*
x = y | x |y = y 1* y 2*
y 1 
 
y 2 
  
 
 x1

 0
0

Chuck DiMarzio, Northeastern University
0
x2

0 y 1 
 
y 2 
  
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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
xp  h/2
[x H] = x H - H x = 0
January 2006
Chuck DiMarzio, Northeastern University
10842-1c-28
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
January 2006
y
i
= Hy
t
Chuck DiMarzio, Northeastern University
10842-1c-29
Born: 12 Aug 1887 in Erdberg, Vienna, Austria
Died: 4 Jan 1961 in Vienna, Austria*
Shrödinger Equation
• Temporal Behavior of the Wave Function
y
i
= Hy
t
– 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
January 2006
Chuck DiMarzio, Northeastern University
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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.
1 2 p2
mv =
2
2m
Momentum
Operator:
 
p=
i x
January 2006
 2  2
H=
2m x 2
Chuck DiMarzio, Northeastern University
10842-1c-31
Some Wavefunctions
1
Shrödinger Equation
y
i
= Hy
t
y   2  2y
i
=
t
2m x 2
Eigenvalue Problem
Hy=Ey
Solution
2 2
nh
En =
2
8mL
January 2006
0.5
0
-0.5
-1
0
0.2
0.4
0.6
0.8
y n h
Temporal Behavior i
=
y
2
t 8mL
Chuck DiMarzio, Northeastern University
2 2
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1
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-33
Harmonic Oscillator
• Hamiltonian
1 2 1 2
H = mv  kx
2
2
1 p2 1 2 2
H=
 w mx
2 m 2
Potential
Energy
• Frequency
k
w =
m
2
January 2006
x
Chuck DiMarzio, Northeastern University
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Harmonic Oscillator Energy
• Solve the Shrödinger Equation
• Solve the Eigenvalue Problem
• Energy
H |y n = En |y n 
H | En = En | En 
1
1


En =  n  w =  n  hn
2
2


– Recall that...
h
=
2
January 2006
w = 2n
Chuck DiMarzio, Northeastern University
10842-1c-35
†
Louisell’s Approach
• Harmonic Oscillator
– Unit Mass
1 2
H =  p  w 2q2 
2
• New Operators
1
wq  ip 
a=
2w
1
wq  ip 
a =
2w
†
a, a  = 1
†
January 2006
Chuck DiMarzio, Northeastern University
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The Hamiltonian
• In terms of a, a †
w
1
 †
†
†

H=
aa  a a  = w  a a  
2
2

• Equations of Motion
dq H
=
=p
dt p
da 1
= a , H  = iwa
dt i
January 2006
dp
H
=
= w 2 q
dt
q
da † 1 †
= a , H = iwa †
dt
i
Chuck DiMarzio, Northeastern University


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Energy Eigenvalues
• Number Operator
1
1
N=
H
w
2
N =a a
†
• Eigenvalues of the Hamiltonian
H | E = E | E 
N | n' = n ' | n ' 
1

En = w  n  
2

January 2006
Chuck DiMarzio, Northeastern University
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Creation and Anihilation (1)
• Note the Following Commutators
a, a  = 1
†
a, aa  = a
• Then
Na = aN  1
†
†
†
Na = a N  1
†
N a | n'  = (n'1)a | n' 
January 2006
a , aa  = a
†
†
N a † | n' = (n'1)a † | n' 
Chuck DiMarzio, Northeastern University
10842-1c-39
Creation and Anihilation (2)
Eigenvalue Equations
N a | n'  = (n'1)a | n' 
N a | n' = (n'1)a | n' 
January 2006
Energy
Eigenvalues
a | n' 
1

hn  n '  
2

| n' 
N | n' = n ' | n ' 
†
States
†
a † | n' 
Chuck DiMarzio, Northeastern University
1

hn  n '  
2

3

hn  n '  
2

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Creation and Anihilation (3)
N a | n'  = (n  1)a | n' 
a | n ' =| n '1 
N | n'1 = (n'1) | n'1 
N | n' = n ' | n ' 
N a † | n' = (n  1)a † | n' 
†
a | n' =| n'1 
N | n'1 = (n'1) | n'1 
January 2006
Chuck DiMarzio, Northeastern University
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Reminder!
• All Observables
are Represented
by Hermitian
Operators.
• Their Eigenvalues
must be Real
January 2006
Chuck DiMarzio, Northeastern University
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