Lecture 38
Quantum Theory of Light
38.1 Quantum Theory of Light
38.1.1 Historical Background
Quantum theory is a major intellectual achievement of the twentieth century, even though
we are still discovering new knowledge in it. Several major experimental findings led to the
revelation of quantum theory or quantum mechanics of nature. In nature, we know that
many things are not infinitely divisible. Matter is not infinitely divisible as vindicated by the
atomic theory of John Dalton (1766-1844) [221]. So fluid is not infinitely divisible: as when
water is divided into smaller pieces, we will eventually arrive at water molecule, H2O, which
is the fundamental building block of water.
In turns out that electromagnetic energy is not infinitely divisible either. The electromag-
netic radiation out of a heated cavity would obey a very different spectrum if electromagnetic
energy is infinitely divisible. In order to fit experimental observation of radiation from a
heated electromagnetic cavity, Max Planck (1900s) [222] proposed that electromagnetic en-
ergy comes in packets or is quantized. Each packet of energy or a quantum of energy E is
associated with the frequency of electromagnetic wave, namely
E = ℏω = ℏ2πf = hf (38.1.1)
where ℏ is now known as the Planck constant and ℏ = h/2π = 6.626 × 10−34 J·s (Joule-
second). Since ℏ is very small, this packet of energy is very small unless ω is large. So it
is no surprise that the quantization of electromagnetic field is first associated with light, a
very high frequency electromagnetic radiation. A red-light photon at a wavelength of 700
nm corresponds to an energy of approximately 2 eV ≈ 3 × 10−19J ≈ 75 kB T , where kB T
denotes the thermal energy. A microwave photon has approximately 1 × 10−5 eV.
The second experimental evidence that light is quantized is the photo-electric effect [223].
It was found that matter emitted electrons when light shined on it. First, the light frequency
has to correspond to the “resonant” frequency of the atom. Second, the number of electrons
emitted is proportional to the number of packets of energy ℏω that the light carries. This
387
388 Electromagnetic Field Theory
was a clear indication that light energy traveled in packets or quanta as posited by Einstein
in 1905.
That light is a wave has been demonstrated by Newton’s ring effect [224] in the eighteenth
century (1717) (see Figure 38.1). In 1801, Thomas Young demonstrated the double slit
experiment [225] that further confirmed the wave nature of light (see Figure 38.2). But by
the beginning of the 20-th century, one has to accept that light is both a particle, called a
photon, carrying a quantum of energy with momentum, as well as a particle endowed with
wave-like behavior. This is called wave-particle duality.
Figure 38.1: A Newton’s rings experiment (courtesy of [224]).
Quantum Theory of Light 389
Figure 38.2: A Young’s double-slit experiment (courtesy of [226]).
This concept was not new to quantum theory as electrons were known to behave both like
a particle and a wave. The particle nature of an electron was confirmed by the measurement
of its charge by Millikan in 1913 in his oil-drop experiment. (The double slit experiment for
electron was done in 1927 by Davison and Germer, indicating that an electron has a wave
nature [225].) In 1924, De Broglie [227] suggested that there is a wave associated with an
electron with momentum p such that
p = ℏk (38.1.2)
where k = 2π/λ, the wavenumber. All this knowledge gave hint to the quantum theorists of
that era to come up with a new way to describe nature.
Classically, particles like an electron moves through space obeying Newton’s laws of mo-
tion first established in 1687 [228]. Old ways of describing particle motion were known as
classical mechanics, and the new way of describing particle motion is known as quantum
mechanics. Quantum mechanics is very much motivated by a branch of classical mechanics
called Hamiltonian mechanics. We will first use Hamiltonian mechanics to study a simple
pendulum and connect it with electromagnetic oscillations.
38.1.2 Connecting Electromagnetic Oscillation to Simple Pendulum
The quantization of electromagnetic field theory was started by Dirac in 1927 [3]. In the be-
ginning, it was called quantum electrodynamics important for understanding particle physics
390 Electromagnetic Field Theory
phenomena and light-matter interactions [229]. Later on, it became important in quantum
optics where quantum effects in electromagnetics technologies first emerged. Now, microwave
photons are measurable and are important in quantum computers. Hence, quantum effects
are important in the microwave regime as well.
Maxwell’s equations can be regarded as for describing an infinite set of coupled harmonic
oscillators. In one dimension, when a wave propagates on a string, or an electromagnetic
wave propagating on a transmission line, they can be regarded as propagating on a set of
coupled harmonic oscillators as shown in Figure 38.3. Maxwell’s equations describe the waves
travelling in 3D space due to the coupling between an infinite set of harmonic oscillators.
In fact, methods have been developed to solve Maxwell’s equations using transmission-line-
matrix (TLM) method [230], or the partial element equivalent circuit (PEEC) method [231].
Figure 38.3: Maxwell’s equations describe the coupling of harmonic oscillators in a 3D space.
This is similar to waves propagating on a string or a 1D transmission line, or a 2D array of
coupled oscillators. The saw-tooth symbol in the figure represents a spring.
The cavity modes in electromagnetics are similar to the oscillation of a pendulum. To
understand the quantization of electromagnetic field, we start by connecting these cavity
modes oscillations to the oscillations of a simple pendulum. It is to be noted that funda-
mentally, electromagnetic oscillation exists because of displacement current. Displacement
current exists even in vacuum because vacuum is polarizable, namely that D = εE. Further-
more, displacement current exists because of the ∂D/∂t term in the generalized Ampere’s
law added by Maxwell, namely,
∇ × H = ∂D
∂t + J (38.1.3)
Quantum Theory of Light 391
Together with Faraday’s law that
∇ × E = − ∂B
∂t (38.1.4)
(38.1.3) and (38.1.4) together allow for the existence of wave. The coupling between the two
equations gives rise to the “springiness” of electromagnetic fields.
Wave exists due to the existence of coupled harmonic oscillators, and at a fundamental
level, these harmonic oscillators are electron-positron (e-p) pairs. The fact that they are
coupled allows waves to propagate through space, and even in vacuum.
Figure 38.4: A one-dimensional cavity solution to Maxwell’s equations is one of the simplest
way to solve Maxwell’s equations.
To make life simpler, we can start by looking at a one dimensional cavity formed by
two PEC (perfect electric conductor) plates as shown in Figure 38.4. Assume source-free
Maxwell’s equations in between the plates and letting E = ˆxEx, H = ˆyHy , then (38.1.3) and
(38.1.4) become
∂
∂z Hy = −ε ∂
∂t Ex (38.1.5)
∂
∂z Ex = −μ ∂
∂t Hy (38.1.6)
The above are similar to the telegrapher’s equations. We can combine them to arrive at
∂2
∂z2 Ex = με ∂2
∂t2 Ex (38.1.7)
There are infinitely many ways to solve the above partial differential equation. But here, we
use separation of variables to solve the above by letting Ex(z, t) = E0(t)f (z). Then we arrive
392 Electromagnetic Field Theory
at two separate equations that
d2E0(t)
dt2 = −ω2
l E0(t) (38.1.8)
and
d2f (z)
dz2 = −ω2
l μεf (z) (38.1.9)
where ω2
l is the separation constant. There are infinitely many ways to solve the above
equations which are also eigenvalue equations where ω2
l and ω2
l με are eigenvalues for the first
and the second equations, respectively. The general solution for (38.1.9) is that
E0(t) = E0 cos(ωlt + ψ) (38.1.10)
In the above, ωl, which is related to the separation constant, is yet indeterminate. To make ω2
l
determinate, we need to impose boundary conditions. A simple way is to impose homogeneous
Dirchlet boundary conditions thatf (z) = 0 at z = 0 and z = L. This implies that f (z) =
sin(klz). In order to satisfy the boundary conditions at z = 0 and z = L, one deduces that
kl = lπ
L , l = 1, 2, 3, . . . (38.1.11)
Then,
∂2f (z)
∂z2 = −k2
l f (z) (38.1.12)
where k2
l = ω2
l μ. Hence, kl = ωl/c, and the above solution can only exist for discrete
frequencies or that
ωl = lπ
L c, l = 1, 2, 3, . . . (38.1.13)
These are the discrete resonant frequencies ωl of the modes of the 1D cavity.
The above solutions for Ex(z, t) can be thought of as the collective oscillations of coupled
harmonic oscillators forming the modes of the cavity. At the fundamental level, these oscil-
lations are oscillators made by electron-positron pairs. But macroscopically, their collective
resonances manifest themselves as giving rise to infinitely many electromagnetic cavity modes.
The amplitudes of these modes, E0(t) are simple harmonic oscillations.
The resonance between two parallel PEC plates is similar to the resonance of a trans-
mission line of length L shorted at both ends. One can see that the resonance of a shorted
transmission line is similar to the coupling of infnitely many LC tank circuits. To see this, as
shown in Figure 38.3, we start with a single LC tank circuit as a simple harmonic oscillator
with only one resonant frequency. When two LC tank circuits are coupled to each other, they
will have two resonant frequencies. For N of them, they will have N resonant frequencies. For
a continuum of them, they will be infinitely many resonant frequencies or modes as indicated
by (38.1.11).
Quantum Theory of Light 393
What is more important is that the resonance of each of these modes is similar or ho-
momorphic to the resonance of a simple pendulum or a simple harmonic oscillator. For a
fixed point in space, the field due to this oscillation is similar to the oscillation of a simple
pendulum.
As we have seen in the Drude-Lorentz-Sommerfeld mode, for a particle of mass m attached
to a spring connected to a wall, where the restoring force is like Hooke’s law, the equation of
motion of a pendulum by Newton’s law is
m d2x
dt2 + κx = 0 (38.1.14)
where κ is the spring constant, and we assume that the oscillator is not driven by an external
force, but is in natural or free oscillation. By letting1
x = x0e−iωt (38.1.15)
the above becomes
−mω2x0 + κx0 = 0 (38.1.16)
Again, a non-trivial solution is possible only at the resonant frequency of the oscillator or
that when ω = ω0 where
ω0 =
√ κ
m (38.1.17)
38.2 Hamiltonian Mechanics
Equation (38.1.14) can be derived by Newton’s law but it can also be derived via Hamiltonian
mechanics. Since Hamiltonian mechanics motivates quantum mechanics, we will look at the
Hamiltonian mechanics view of the equation of motion (EOM) of a simple pendulum given
by (38.1.14).
Hamiltonian mechanics, developed by Hamilton (1805-1865) [232], is motivated by energy
conservation. The Hamiltonian H of a system is given by its total energy, namely that
H = T + V (38.2.1)
where T is the kinetic energy and V is the potential energy of the system.
For a simple pendulum, the kinetic energy is given by
T = 1
2 mv2 = 1
2m m2v2 = p2
2m (38.2.2)
where p = mv is the momentum of the particle. The potential energy, assuming that the
particle is attached to a spring with spring constant κ, is given by
V = 1
2 κx2 = 1
2 mω2
0 x2 (38.2.3)
1For this part of the lecture, we will switch to using exp(−iωt) time convention as is commonly used in
optics and physics literatures.
394 Electromagnetic Field Theory
Hence, the Hamiltonian is given by
H = T + V = p2
2m + 1
2 mω2
0 x2 (38.2.4)
At any instant of time t, we assume that p(t) = mv(t) = m d
dt x(t) is independent of x(t).2
In other words, they can vary independently of each other. But p(t) and x(t) have to time
evolve to conserve energy to keep H, the total energy, constant or independent of time. In
other words,
d
dt H [p(t), x(t)] = 0 = dp
dt
∂H
∂p + dx
dt
∂H
∂x (38.2.5)
Therefore, the Hamilton equations of motion are derived to be3
dp
dt = − ∂H
∂x , dx
dt = ∂H
∂p (38.2.6)
From (38.2.4), we gather that
∂H
∂x = mω2
0 x, ∂H
∂p = p
m (38.2.7)
Applying (38.2.6), we have4
dx
dt = p
m , dp
dt = −mω2
0 x (38.2.8)
Combining the two equations in (38.2.8) above, we have
m d2x
dt2 = −mω2
0 x = −κx (38.2.9)
which is also derivable by Newton’s law.
A typical harmonic oscillator solution to (38.2.9) is
x(t) = x0 cos(ω0t + ψ) (38.2.10)
The corresponding p(t) = m dx
dt is
p(t) = −mx0ω0 sin(ω0t + ψ) (38.2.11)
Hence
H = 1
2 mω2
0 x2
0 sin2(ω0t + ψ) + 1
2 mω2
0 x2
0 cos2(ω0t + ψ)
= 1
2 mω2
0 x2
0 = E (38.2.12)
And the total energy E very much depends on the amplitude x0 of the oscillation.
2p(t) and x(t) are termed conjugate variables in many textbooks.
3Note that the Hamilton equations are determined to within a multiplicative constant, because one has
not stipulated the connection between space and time, or we have not calibrated our clock [233].
4We can also calibrate our clock here so that it agrees with our definition of momentum in the ensuing
equation.
Quantum Theory of Light 395
38.3 Schrodinger Equation (1925)
Having seen the Hamiltonian mechanics for describing a simple pendulum which is homomor-
phic to a cavity resonator, we shall next see the quantum mechanics description of the same
simple pendulum: In other words, we will look at a quantum pendulum. To this end, we will
invoke Schrodinger equation.
Schrodinger equation cannot be derived just as in the case Maxwell’s equations. It is
a wonderful result of a postulate and a guessing game based on experimental observations
[63, 64],. Hamiltonian mechanics says that
H = p2
2m + 1
2 mω2
0 x2 = E (38.3.1)
where E is the total energy of the oscillator, or pendulum. In classical mechanics, the position
x of the particle associated with the pendulum is known with great certainty. But in the
quantum world, this position x of the quantum particle is uncertain and is fuzzy.
To build this uncertainty into a quantum harmonic oscillator, we have to look at it from
the quantum world. The position of the particle is described by a wave function,5 which
makes the location of the particle uncertain. To this end, Schrodinger proposed his equation
which is a partial differential equation. He was very much motivated by the experimental
revelation then that p = ℏk from De Broglie and that E = ℏω from Planck’s law. Equation
(38.3.1) can be written more suggestively as
ℏ2k2
2m + 1
2 mω2
0 x2 = ℏω (38.3.2)
To add more depth to the above equation, one lets the above become an operator equation
that operates on a wave function ψ(x, t) so that
− ℏ2
2m
∂2
∂x2 ψ(x, t) + 1
2 mω2
0 x2ψ(x, t) = iℏ ∂
∂t ψ(x, t) (38.3.3)
If the wave function is of the form
ψ(x, t) ∼ eikx−iωt (38.3.4)
then upon substituting (38.3.4) into (38.3.3), we retrieve (38.3.2).
Equation (38.3.3) is Schrodinger equation in one dimension for the quantum version of
the simple harmonic oscillator. In Schrodinger equation, we can further posit that the wave
function has the general form
ψ(x, t) = eikx−iωtA(x, t) (38.3.5)
where A(x, t) is a slowly varying function of x and t, compared to eikx−iωt.6 In other words,
this is the expression for a wave packet. With this wave packet, the ∂2/∂x2 can be again
5Since a function is equivalent to a vector, and this wave function describes the state of the quantum
system, this is also called a state vector.
6This is similar in spirit when we study high frequency solutions of Maxwell’s equations and paraxial wave
approximation.
396 Electromagnetic Field Theory
approximated by −k2 as has been done in the paraxial wave approximation. Furthermore, if
the signal is assumed to be quasi-monochromatic, then iℏ∂/∂tψ(x, t) ≈ ℏω, we again retrieve
the classical equation in (38.3.2) from (38.3.3). Hence, the classical equation (38.3.2) is a
short wavelength, monochromatic approximation of Schrodinger equation. However, as we
shall see, the solutions to Schrodinger equation are not limited to just wave packets described
by (38.3.5).
For this course, we need only to study the one-dimensional Schrodinger equation. The
above can be converted into eigenvalue problem, just as in waveguide and cavity problems,
using separation of variables, by letting7
ψ(x, t) = ψn(x)e−iωnt (38.3.6)
By so doing, (38.3.3) becomes
[
− ℏ2
2m
d2
dx2 + 1
2 mω2
0 x2
]
ψn(x) = Enψn(x) (38.3.7)
where En = ℏωn is the eigenvalue of the problem while ψn(x) is the eigenfunction.
The parabolic x2 potential profile is also known as a potential well as it can provide the
restoring force to keep the particle bound to the well classically. The above equation is also
similar to the electromagnetic equation for a dielectric slab waveguide, where the second term
is a dielectric profile (mind you, varying in the x direction) that can trap a waveguide mode.
Therefore, the potential well is a trap for the particle both in classical mechanics or wave
physics.
The above equation (38.3.7) can be solved in closed form in terms of Hermite-Gaussian
functions (1864) [234], or that
ψn(x) =
√
1
2nn!
√ mω0
πℏ e− mω0
2ℏ x2
Hn
(√ mω0
ℏ x
)
(38.3.8)
where Hn(y) is a Hermite polynomial, and the eigenvalues are
En =
(
n + 1
2
)
ℏω0 (38.3.9)
Here, the eigenfunction or eigenstate ψn(x) is known as the photon number state (or just
a number state) of the solution. It corresponds to having n photons in the oscillation. If
this is conceived as the collective oscillation of the e-p pairs in a cavity, there are n photons
corresponding to energy of nℏω0 embedded in the collective oscillation. The larger En is,
the larger the number of photons there is. However, there is a curious mode at n = 0. This
corresponds to no photon, and yet, there is a wave function ψ0(x). This is the zero-point
energy state. This state is there even if the system is at its lowest energy state.
It is to be noted that in the quantum world, the position x of the pendulum is random.
Moreover, this position x(t) is mapped to the amplitude E0(t) of the field. Hence, it is the
7Mind you, the following is ωn, not ω0.
Quantum Theory of Light 397
amplitude of an electromagnetic oscillation that becomes uncertain and fuzzy as shown in
Figure 38.5.
Figure 38.5: Schematic representation of the randomness of measured electric field. The elec-
tric field amplitude maps to the displacement (position) of the quantum harmonic oscillator,
which is a random variable (courtesy of Kira and Koch [235]).
398 Electromagnetic Field Theory
Figure 38.6: Plots of the eigensolutions of the quantum harmonic oscillator (courtesy of
Wiki [236]).
38.4 Some Quantum Interpretations–A Preview
Schrodinger used this equation with resounding success. He derived a three-dimensional ver-
sion of this to study the wave function and eigenvalues of a hydrogen atom. These eigenvalues
En for a hydrogen atom agreed well with experimental observations that had eluded scientists
for decades. Schrodinger did not actually understand what this wave function meant. It was
Max Born (1926) who gave a physical interpretation of this wave function.
Given a wave function ψ(x, t), then |ψ(x, t)|2∆x is the probability of finding the particle
in the interval [x, x + ∆x]. Therefore, |ψ(x, t)|2 is a probability density function (PDF), and
it is necessary that
∞
−∞
dx|ψ(x, t)|2 = 1 (38.4.1)
The position x of the particle is uncertain and is now a random variable. The average value
or expectation value of x is given by
∞
−∞
dxx|ψ(x, t)|2 = 〈x(t)〉 = ¯x(t) (38.4.2)
This is not the most ideal notation, since although x is not a function of time, its expectation
value with respect to a time-varying function, ψ(x, t), can be time-varying.
Quantum Theory of Light 399
Notice that in going from (38.3.1) to (38.3.3), or from a classical picture to a quantum
picture, we have let the momentum become p, originally a scalar number in the classical
world, become a differential operator, namely that
p → ˆp = −iℏ ∂
∂x (38.4.3)
The momentum of a particle also becomes uncertain, and its expectation value is given by
∞
∞
dxψ∗(x, t)ˆpψ(x, t) = −iℏ
∞
−∞
dxψ∗(x, t) ∂
∂x ψ(x, t) = 〈ˆp(t)〉 = ¯p(t) (38.4.4)
The expectation values of position x and the momentum operator ˆp are measurable in the
laboratory. Hence, they are also called observables.
One more very important aspect of quantum theory is that since p → ˆp = −iℏ ∂
∂x , ˆp and
x do not commute. In other words, it can be shown that
[ˆp, x] =
[
−iℏ ∂
∂x , x
]
= −iℏ (38.4.5)
In the classical world, [p, x] = 0, but not in the quantum world. In the equation above, we
can elevate x to become an operator by letting ˆx = x ˆI, where ˆI is the identity operator. Then
both ˆp and ˆx are now operators, and are on the same footing. In this manner, we can rewrite
equation (38.4.5) above as
[ˆp, ˆx] = −iℏ ˆI (38.4.6)
It can be shown easily that when two operators share the same set of eigenfunctions,
they commute. When two operators ˆp and ˆx do not commute, it means that the expectation
values of quantities associated with the operators, 〈ˆp〉 and 〈ˆx〉, cannot be determined to
arbitrary precision simultaneously. For instance, ˆp and ˆx correspond to random variables,
then the standard deviation of their measurable values, or their expectation values, obey the
uncertainty principle relationship that8
∆p∆x ≥ ℏ/2 (38.4.7)
where ∆p and ∆x are the standard deviation of the random variables p and x.
38.5 Bizarre Nature of the Photon Number States
The photon number states are successful in predicting that the collective e-p oscillations are
associated with n photons embedded in the energy of the oscillating modes. However, these
number states are bizarre: The expectation values of the position of the quantum pendulum
associated these states are always zero. To illustrate further, we form the wave function with
a photon-number state
ψ(x, t) = ψn(x)e−iωnt
8The proof of this is quite straightforward but is outside the scope of this course.
400 Electromagnetic Field Theory
Previously, since the ψn(x) are eigenfunctions, they are mutually orthogonal and they can be
orthonormalized meaning that
∞
−∞
dxψ∗
n(x)ψn′ (x) = δnn′ (38.5.1)
Then one can easily show that the expectation value of the position of the quantum pendulum
in a photon number state is
〈x(t)〉 = ¯x(t) =
∞
−∞
dxx|ψ(x, t)|2 =
∞
−∞
dxx|ψn(x)|2 = 0 (38.5.2)
because the integrand is always odd symmetric. In other words, the expectation value of the
position x of the pendulum is always zero. It can also be shown that the expectation value
of the momentum operator ˆp is also zero for these photon number states. Hence, there are
no classical oscillations that resemble them. Therefore, one has to form new wave functions
by linear superposing these photon number states into a coherent state. This will be the
discussion next.
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