Light-Matter Interactions
Of a Two level System (Up to SLIDE # 45)
Phy 4253/6293
Text Book: Ch - 9, Quantum Optics, Mark Fox
Study Guides to Mark Fox only:
We will start by ‘asking questions as usual’
Hopefully these QUESTIONS will guide to learn certain specific issues on the
importance of photon statistics in Quantum Optics.
You certainly need to follow it up by reading the relevant sections from Mark
Fox (Uploaded)
PLEASE EMAIL ME YOUR DOUBTS/QUESTIONS.
1
Recapitulation - Spontaneous Emission & Zero-Point
Fluctuations of Quantized Electromagnetic Field of Light
From Ch.9 of Mark Fox, Quantum Optics
2
Recapitulation • Although, spontaneous emission seems more trivial
compared to stimulated emission at the first glance!
• However, it is much more delicate and necessarily
requires the Zero-point Quantum Fluctuations of
Electromagnetic field.
• Therefore, SPONTANEOUS EMISSION is the best
evidence of the Photon concept or the Quantization of
Electromagnetic field of Light.
3
Next – Strength of light intensity determines
whether Light-Matter interactions are weak or
strong!
***Ch-9: Light Matter interactions where Atom is a Quantum Object and Light is
classical Electromagnetic Field – basically a Semi- Classical Approach only
***** What is that departure
from “conventional picture of absorption of photons” ????
4
Recapitulation: Optical Transitions in a Two Level System
THINK  Will there always (at all times?) be optical
absorption when the energy of photon matches with the
energy difference between the two levels ?????? 5
Optical Transitions in a Two Level System
6
Optical Transitions in a Two Level System
Much earlier than the formal development of Quantum Mechanics !
7
Optical Transitions in a Two Level System
8
Optical Transitions in a Two Level System
9
What is a Two Level System?
Please Try Exercise Problem 9.1
10
Off-Resonant Transition and Damping
Please Try Exercise Problem 9.1
11
Role of Coherent Superposition in Two-Level Transitions
& Quantum Control
12
Coherent Superposition in a Two-Level system
13
Not Coherent Superposition  Statistical Mixture
In this case
𝝋𝟏 > = 𝒄𝟏 𝟏 >
&
𝝋𝟐 > = 𝒄𝟐 𝟐 >
are independent
14
Analogy of Coherent Superposition as in Interference
15
Coherent Superposition as in Interference
16
Coherent Superposition as in Interference
17
Coherent Manipulation of Quantum Superposition in a
Two-Level System
18
Quantum Superposition & Density Matrix
Non-zero Off diagonal (Cross) Terms in the
Density Matrix indicates Coherent
Superposition leading to possibilities of
interference of two quantum states |1> and
|2>. Absence of the Off-diagonal terms
indicate Statistical Mixture.
𝝆 = |><|
19
Quantum Superposition & Density Matrix
Home Work :
Please try
Exercise Problem
9.2 & 9.3
***Please read Ch-3 of Modern Quantum Mechanics by Sakurai & other reading
materials on Density Operators & Pure/Mixed Ensembles
20
Quantum mechanics of a Two-Level System
21
Quantum mechanics of a Two-Level System
22
Quantum mechanics of a Two-Level System
23
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
24
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
This equation makes it clear that the time evolution
of c1 and c2 are interdependent as well as
෡
dependent of the Light Matter interaction 𝑽
This is qualitatively understandable as Light-Matter
෡ mediate both upward transition (Optical
interaction 𝑽
Absorption) and downward transition (Optical Emission)
and thereby it also change the probabilities of finding the
25
atom in either |1> or |2>
Dipole Matrix of a Two-Level Transition
26
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
These equations again make it clear that the time evolution of c1 and c2 are
෡
interdependent as well as dependent of the Light Matter interaction 𝑽
෡ mediate both
This is qualitatively understandable as Light-Matter interaction 𝑽
upward transition (Optical Absorption) and downward transition (Optical Emission)
and thereby it also change the probabilities of finding the atom in either |1> or27|2>
Dipole Matrix of a Two-Level Transition
28
Lowest order Electric Dipole Transition only & Dipole Matrix of a Two-Level Transition
29
Rabi Frequency of a Two-Level Transition
Probability amplitudes c1(t) and c2(t) are interdependent
30
Rabi Frequency of a Two-Level Transition
31
Rabi Frequency of a Two-Level Transition
32
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
These equations again make it clear that the time evolution of c1 and c2 are
෡
interdependent as well as dependent of the Light Matter interaction 𝑽
෡ mediate both
This is qualitatively understandable as Light-Matter interaction 𝑽
upward transition (Optical Absorption) and downward transition (Optical Emission)
and thereby it also change the probabilities of finding the atom in either |1> or33|2>
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
These equations again make it clear that the time evolution of c1 and c2 are
෡
interdependent as well as dependent of the Light Matter interaction 𝑽
෡ mediate both
This is qualitatively understandable as Light-Matter interaction 𝑽
upward transition (Optical Absorption) and downward transition (Optical Emission)
and thereby it also change the probabilities of finding the atom in either |1> or |2>
Time evolution of c1 and c2 are oscillating at two different frequencies
34
of (-0) & (+0)
Weak Field Limit of a Two-Level Transition
We will see Einstein’s Empirical Theory can be recovered only when
Light-Matter interactions are weak or Intensity of light is less!35
Weak Field Limit of a Two-Level Transition
36
Weak Field Limit of a Two-Level Transition
Transition Probability of
Excitation P2(t)
37
Transition Probability is ~ t2 or t?
Transition
Rate
38
Transition Probability is ~ t2 or t?
OPTIONAL: WEAK LIGHT MATTER INTERACTION
IS NOT IN YOUR SYLLABUS…
I still kept my old slides (From Slide# 47 onwards),
may help you to understand how Quantum
mechanics can reproduce Einstein A & B coefficient
based understandings.
39
Quantum mechanics of a Two-Level System
𝝏𝝍
෡
𝑯𝝍 = 𝒊ℏ
𝝏𝒕
These equations again make it clear that the time evolution of c1 and c2 are
෡
interdependent as well as dependent of the Light Matter interaction 𝑽
෡ mediate both
This is qualitatively understandable as Light-Matter interaction 𝑽
upward transition (Optical Absorption) and downward transition (Optical Emission)
and thereby it also change the probabilities of finding the atom in either |1> or |2>
Time evolution of c1 and c2 are oscillating at two different frequencies
40
of (-0) & (+0)
The Strong-Field Limit !
41
1) Rotating wave Approximation
2) Resonant Transition (ℏ𝝎𝟎 = 𝑬𝟐 − 𝑬𝟏 )
3) C1(t) need not be 1 always as C2(t) can be significant
42
1) Rotating wave Approximation
2) Resonant Transition
3) C1(t) need not be 1 always, as C2(t) can be significant
Probability amplitudes of finding the Two Level Quantum systems at |1>, |2> are c1
43
and c2 respectively. Both evolve like Simple Harmonic Oscillators !
1) Rotating wave Approximation
2) Resonant Transition
3) C1(t) need not always be 1 as C2(t) can be significant
Quantum Probabilities of finding the Coherent
Superposition in Quantum States of |1> or |2> vary
Like Simple Harmonic Oscillators !
44
Strong Field Limit : Resonant Transition
Ideal Rabi Oscillation
Simple Harmonic Oscillator !
45
46
Weak Field Limit of a Two-Level Transition
We will see Einstein’s Empirical Theory can be recovered only when
Light-Matter interactions are weak or Intensity of light is less!47
Weak Field Limit of a Two-Level Transition
48
Weak Field Limit of a Two-Level Transition
By ignoring
terms of ()2
Transition Probability of
Excitation P2(t)
In the limit of →0
49
Transition Probability is ~ t2 or t?
Transition
Rate
50
Einstein’s A-B Theory before the advent
of Quantum Mechanics
Spontaneous Emission
Stimulated Emission
Absorption
51
Einstein’s A-B Theory before the advent
of Quantum Mechanics
52
Einstein’s A-B Theory before the advent
of Quantum Mechanics
53
Einstein’s A-B Theory before the advent
of Quantum Mechanics
54
Einstein’s A-B Theory before the advent
of Quantum Mechanics
In weak
interaction model,
N1 remains
constant
As absorption effectively increases N2  dN1/dt = (-1) dN2/dt
55
Recapitulation: Any practical Transitions are
with finite width- Non-monochromatic and in
weak field limit !
56
Recapitulation: Any practical Transitions are
with finite width- Non-monochromatic and in
weak field limit !
Compare equation 9.40 and 9.41  57
So any practical transition has a finite energy
width or Non-monochromatic !
- A direct consequence of Energy-Time Uncertainty principle
from Time-Dependent Perturbation Theory
58
Einstein’s A-B Theory using
Quantum Mechanics
Independent of time 
 |c2|2/t
First Estimate
12, then find
B12, then B21 and
finally A21
59
Recapitulation: Any practical Transitions are
with finite width - Non-monochromatic !
We recover Einstein’s AB theory’s Linear Time
variations for Broadband Light-matter(atom)
interactions only in the Weak Field Limit
60
Comparing with Einstein’s A-B Theory
61
Comparison with Weak-Field Limit
and Einstein’s A-B Theory using Broadband Light
**Compare with the similar treatments in Ch-9 of A Text Book of Quantum mechanics by
62
Mathews & Venkatesan
Exercise Problems on Two-Level Transition
A Classical Correspondence
In Quantum Picture
ℏ𝝎𝟎 = 𝑬𝟐 − 𝑬𝟏
Correspondingly here in
the Classical Picture
0 is the fundamental
oscillator frequency
Using a Classical Picture of electron in Atom as Dipole !
63
Classical Correspondence Also explain
broadening of transition !!!
Using a Classical Picture of electron in Atom as Dipole !
64
Exercise Problems on Two-Level Transition
Using a Semi-Classical Picture of electron in Atom as Dipole !
65
Exercise Problems on Two-Level Transition
Using a Classical Picture of electron in Atom as Dipole !
66
Exercise Problems on Density Matrices for
Two-Level Transitions
Interested Students please
read # 3.4 from Sakurai’s
Modern Quantum Mechanics
and other Study Materials in
Google Drive to fully
understand the concept of
Density Matrix or Density
Operator
- As some books/papers treat
light matter interactions using
Density matrix formalisms
67
Exercise Problems on Two-Level Transition
involving Statistical Mixtures
Statistical
Mixture
 No definite
Phase
relationship
between
Quantum States
of |1> and |2>
as these states
are not part of
any
superposition
68
Example:
Exercise Problems on Two-Level Transition to
Calculate Einstein’s A & B coefficients
69
Exercise Problems on Two-Level Transition
First Estimate 12,
then find B12, then
B21 and finally A21
70
Exercise Problems on Two-Level Transition
First Estimate 12,
then find B12, then
B21 and finally A21
71
Exercise Problems on Two-Level Transition
First Estimate
12, then find
B12, then B21
and finally
A21
72