Quantum Systems: Radiation, Absorption, and Emission

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome back to your study of quantum dynamics.

In our last lesson, we derived Fermi's Golden Rule, a powerful tool for calculating the rate of transitions into a continuum of states. We saw that for a harmonic perturbation, such as an oscillating field, two processes are possible: absorption of energy from the field and stimulated emission of energy into the field. The rates for these processes are given by:
$w_{i \to f} = \frac{2 π}{ℏ} ∣ ⟨ f ∣ H^{'} ∣ i ⟩ ∣^{2} ρ (E_{f})$
with energy conservation conditions $E_{f} = E_{i} \pm ℏ ω$ .

Today, we will apply this rule to one of the most fundamental phenomena in physics: the interaction of light with matter. Our goal is to analyze how quantum systems, like atoms or the electrons in a solid, absorb and emit radiation. We will do this by first modeling the light-matter interaction to find the perturbing Hamiltonian, $H^{'}$ , and then applying the Golden Rule to find the transition rates. This will lead us to the dipole approximation, a cornerstone of quantum optics and spectroscopy.

The three main processes we will consider are visualized below. Our semi-classical treatment will naturally explain absorption and stimulated emission, and we will use a classic argument by Einstein to account for spontaneous emission.

Photon-Matter Interaction Processes in Two-Level Systems — CAPTION: The three fundamental types of transitions between two energy levels, $E_1$ and $E_2$. In absorption, a photon is consumed to excite the system. In spontaneous emission, the system decays on its own, releasing a photon. In stimulated emission, an incoming photon causes the system to decay, producing a second, identical photon.

1. The Atom-Light Interaction Hamiltonian

Our first step is to model the interaction between a charged particle (like an electron in an atom) and a classical electromagnetic wave. We will find that under very common conditions, this complex interaction simplifies to a beautiful and intuitive form.

The following video from MIT OpenCourseWare provides a clear derivation of the interaction Hamiltonian. It justifies the key approximations we'll be making.

L12.5 Atom-light interactions: dipole operator

This video, 'Atom-light interactions: dipole operator', derives the interaction Hamiltonian between an atom and an electric field. It carefully explains the physical reasoning behind the dipole approximation, which is central to this topic.

Please watch from the beginning to 07:35. Focus on two key justifications: Why we can neglect the magnetic field's effect. Why we can treat the electric field as spatially uniform across the atom (the dipole approximation). Pay attention to how the final Hamiltonian is expressed in terms of the electric dipole operator.

Let's summarize the key steps and results from the video.

An electromagnetic wave is described by oscillating electric and magnetic fields. However, the force from the magnetic field on an electron is proportional to its velocity, $F_{B} \propto v \times B$ . For electrons in atoms, the ratio of magnetic to electric forces is of order $v / c$ , which is related to the fine-structure constant $α \approx 1/137$ . We can therefore neglect the magnetic interaction to a very good approximation.

We are left with the interaction of the electron's charge with the electric field, which we can write as $E (r, t) = E_{0} cos (k \cdot r - ω t)$ .

The Electric Dipole Approximation

The crucial insight is to compare the wavelength of the light, $λ$ , to the size of the atom, $a_{0}$ .

For optical or UV light, $λ \sim 100 - 1000$ nm.
For a hydrogen atom, $a_{0} \approx 0.05$ nm.

Since $λ ≫ a_{0}$ , the phase factor $k \cdot r$ (where $∣ k ∣ = 2 π / λ$ ) is extremely small over the volume of the atom. We can therefore approximate $k \cdot r \approx 0$ . This means the electric field is essentially uniform across the atom at any instant in time:
$E (r, t) \approx E (0, t) = E_{0} cos (ω t)$
This is the electric dipole approximation.

Under this approximation, the scalar potential associated with the uniform field is $ϕ (r, t) = - r \cdot E (t)$ . The interaction energy of an electron with charge $- e$ is the perturbing Hamiltonian:
$H_{p} (t) = (- e) ϕ (r, t) = e r \cdot E (t) = (e r) \cdot E_{0} cos (ω t)$
We identify the electric dipole operator as $d = e r$ . The Hamiltonian becomes:
$H_{p} (t) = - d \cdot E_{0} cos (ω t)$
(Note: Some conventions define $d = - e r$ for an electron. The physics remains the same.)

This is exactly the harmonic perturbation form we studied, $H_{p} (t) = 2 H^{'} cos (ω t)$ , with the time-independent part being:
$H^{'} = - \frac{1}{2} d \cdot E_{0}$

2. Absorption and Stimulated Emission Rates

Now that we have identified $H^{'}$ , we can use Fermi's Golden Rule to find the transition rates. The physical distinction between absorption and stimulated emission was discussed in the last module, but let's quickly review it before calculating the rates.

L11.2 Transition rates for stimulated emission and absorption processes

This video, 'Transition rates for stimulated emission and absorption processes', reinforces the physical picture of these two processes and connects them to Fermi's Golden Rule.

First, watch the brief conceptual explanations from 0:00 to 4:20. Then, watch the segment from 8:13 to 17:07, which shows how integrating over a continuum of final states for a harmonic perturbation leads to a constant transition rate. This provides a solid foundation for the formulas we are about to use.

The rate of transition from an initial state $∣ i ⟩$ to a final state $∣ f ⟩$ is given by plugging our $H^{'}$ into the Golden Rule formula. Let the polarization of the light be $ϵ^$ , so $E_{0} = E_{0} ϵ^$ . The rate of absorption ( $E_{f} = E_{i} + ℏ ω$ ) is:
$w_{i \to f}^{(abs)} = \frac{2 π}{ℏ} ⟨ f ∣ - \frac{1}{2} (E_{0} ϵ^\cdot d) ∣ i ⟩^{2} ρ (E_{f})$
$w_{i \to f}^{(abs)} = \frac{π E _{0}^{2}}{2ℏ} ∣ ϵ^\cdot ⟨ f ∣ d ∣ i ⟩ ∣^{2} ρ (E_{f})$
The vector matrix element $d_{f i} = ⟨ f ∣ d ∣ i ⟩ = e ⟨ f ∣ r ∣ i ⟩$ is called the transition dipole moment. Its magnitude squared determines the intrinsic strength of the transition.

The rate depends on the alignment between the light's polarization $ϵ^$ and the direction of the transition dipole $d_{f i}$ . If $ϵ^⊥ d_{f i}$ , the transition is forbidden for that polarization. This is the origin of polarization selection rules in spectroscopy.

The rate for stimulated emission ( $E_{f} = E_{i} - ℏ ω$ ) is identical, just with a different energy conservation condition:
$w_{f \to i}^{(stim)} = \frac{π E _{0}^{2}}{2ℏ} ∣ ϵ^\cdot ⟨ i ∣ d ∣ f ⟩ ∣^{2} ρ (E_{i})$
Since $d$ is a Hermitian operator, $∣ ⟨ i ∣ d ∣ f ⟩ ∣^{2} = ∣ ⟨ f ∣ d ∣ i ⟩ ∣^{2}$ . If the density of states is similar, the rates for stimulated emission and absorption between two levels are equal.

3. Incoherent Light and Spontaneous Emission

The rate we derived is for coherent, monochromatic, polarized light. In many situations, such as an atom in a cavity filled with blackbody radiation, the light is incoherent (randomly polarized and spanning a continuum of frequencies).

Chapter 2: Time-Dependent Approximation Methods

To get a more generally applicable formula, we need to average our result over all possible polarizations and integrate over the frequency spectrum of the light. This document provides a concise derivation.

Please read the subsection '2.1 Incoherent light'. Follow the symmetry argument used to average the polarization term ⟨(P·d)²⟩. Then, note how the final rate is expressed in terms of the field's energy density, U(ω).

When we average $∣ ϵ^\cdot d_{f i} ∣^{2}$ over all possible random polarization directions, a simple symmetry argument shows:
$⟨ ∣ ϵ^\cdot d_{f i} ∣^{2} ⟩_{pol} = \frac{1}{3} ∣ d_{f i} ∣^{2}$
The rate for unpolarized light is then proportional to the squared magnitude of the transition dipole moment, $∣ d_{f i} ∣^{2}$ . The final rate for absorption is often expressed in terms of the energy density of the radiation field at the transition frequency, $U (ω_{f i})$ :
$w_{i \to f}^{(abs)} = B_{i f} U (ω_{f i}) where B_{i f} = \frac{4 π ^{2}}{3 ℏ ^{2}} ∣ d_{f i} ∣^{2}$
Here, $B_{i f}$ is the Einstein B coefficient for absorption. The rate for stimulated emission is $w_{f \to i}^{(stim)} = B_{f i} U (ω_{f i})$ , where $B_{f i} = B_{i f}$ .

Spontaneous Emission

Our semi-classical model, where the field is classical, cannot explain why an excited atom in a perfect vacuum will still decay. This is spontaneous emission. A full derivation requires quantizing the electromagnetic field (QED), but in 1917, Einstein devised a brilliant argument based on thermodynamics to relate all three processes.

Chapter 2: Time-Dependent Approximation Methods

Einstein's argument is a masterclass in physical reasoning. By demanding that a collection of atoms in thermal equilibrium with a blackbody radiation field remains in equilibrium, he was able to deduce the rate of spontaneous emission and its relation to the other two processes.

Now read the subsection '2.2 Spontaneous emission'. Follow the logic of balancing the population rates Ṅ_b using the three processes and the known Boltzmann distribution and Planck's blackbody formula.

Einstein introduced the A coefficient for the rate of spontaneous emission, $w_{f \to i}^{(spon)} = A_{f i}$ . By insisting that the atomic populations at thermal equilibrium remain static, he derived the following relations:

$B_{i f} = B_{f i}$ (The coefficient for stimulated emission is the same as for absorption).
$A_{f i} = \frac{ℏ ω _{f i}^{3}}{π ^{2} c ^{3}} B_{f i}$

This is a profound result. The rate of spontaneous emission is not an independent parameter but is determined by the transition dipole moment (via $B_{f i}$ ) and scales as $ω^{3}$ . This is why high-energy (high $ω$ ) transitions have much shorter lifetimes than low-energy ones.

4. Application: Optical Absorption in Semiconductors

Your interest lies in condensed matter, and this framework is the foundation for understanding the optical properties of materials. Let's apply what we've learned to calculate the absorption of light in a direct bandgap semiconductor. Here, "absorption" means a photon excites an electron from the filled valence band to the empty conduction band.

Direct Bandgap Absorption and Emission in Semiconductors — CAPTION: Photon absorption and emission in a direct bandgap semiconductor. An electron in the valence band absorbs a photon and moves to the conduction band (left). An electron in the conduction band recombines with a hole, emitting a photon (right). Crucially, these transitions are 'vertical', meaning the crystal momentum $\vec{k}$ is conserved. This is because the photon's momentum is negligible compared to the scale of the Brillouin zone.

The "vertical" nature of the transition is a direct consequence of the dipole approximation. The photon's momentum $ℏ k_{o pt}$ is so small compared to the electron crystal momenta that momentum must be conserved: $k_{f ina l} \approx k_{ini t ia l}$ .

The following resource provides a complete walkthrough of this calculation. It is an excellent example of how the abstract tools of perturbation theory are applied to a real, measurable material property.

32 Optical absorption in semiconductors

Let's see how this all comes together in a core condensed matter problem: calculating the optical absorption coefficient of a semiconductor. This document connects the quantum mechanical transition rate to a macroscopic property.

This reading is comprehensive. I recommend scanning through it to see how the pieces fit together. Focus on these key ideas: Section 1 ('Matrix element'): Notice how the matrix element is calculated with Bloch wavefunctions. The authors use the A·p gauge, which is more common in solid-state theory but is equivalent to our d·E approach in the dipole approximation. Section 2 ('Total transition rate'): See how summing over all possible initial (valence band) and final (conduction band) k-states leads to the concept of a joint density of states. Section 3 ('Absorption coefficient'): The final step connects the total transition rate W_TOT to the experimentally measurable absorption coefficient α.

This calculation demonstrates several key concepts in practice:

The initial and final states are Bloch waves, $ψ_{k} (r) = u_{k} (r) e^{i k \cdot r}$ .
The matrix element $⟨ ψ_{c, k^{'}} ∣ H^{'} ∣ ψ_{v, k} ⟩$ is evaluated, and the dipole approximation leads to the momentum conservation rule $k^{'} \approx k$ .
To get the total absorption rate, one must integrate over all possible initial states in the valence band that can be excited by a photon of energy $ℏ ω$ . This involves the joint density of states, which counts the number of pairs of valence and conduction band states separated by a given energy.
For a typical parabolic band structure, the absorption coefficient $α (ω)$ near the band edge is found to have a characteristic frequency dependence:
$α (ω) \propto \frac{ℏ ω - E _{g}}{ℏ ω}$
where $E_{g}$ is the bandgap energy. Measuring this dependence is a standard technique for determining the bandgap of a semiconductor.

Test your understanding!

The dipole approximation assumes the electric field is uniform over the size of the "atom". In the semiconductor example, the "atom" is effectively the crystal's unit cell. We justified this by arguing $λ_{l i g h t} ≫ a_{a t o m}$ . Now consider X-rays, with $λ \sim 0.1$ nm, which is comparable to the size of a unit cell. Would you expect the dipole approximation to hold for X-ray absorption spectroscopy? What physical effect that we neglected might become important?

Show answer

No, the dipole approximation would break down. The condition $λ ≫ a$ is no longer met, so we cannot approximate the phase factor $e^{i k \cdot r}$ as 1 over the unit cell. The spatial variation of the field becomes crucial.

The physical effect that becomes important is the momentum of the photon. In the dipole approximation, we assumed the photon transfers energy but negligible momentum ( $k_{f ina l} \approx k_{ini t ia l}$ ). When the photon's momentum $ℏ k_{o pt}$ is significant, it must be accounted for in the conservation law: $k_{f ina l} = k_{ini t ia l} + k_{o pt}$ . This allows for "non-vertical" transitions in the E-k diagram. Expanding the exponential $e^{i k \cdot r} = 1 + i k \cdot r - \frac{1}{2} (k \cdot r)^{2} + \dots$ gives rise to higher-order multipole transitions (electric quadrupole, magnetic dipole, etc.) which have different selection rules.

Conclusion

In this lesson, we have connected the formal machinery of time-dependent perturbation theory to the physical processes of light-matter interaction.

Key Takeaways:

The interaction between an atom and light is dominated by the electric field.
The electric dipole approximation ( $λ ≫ a_{0}$ ) allows us to model this interaction with the simple Hamiltonian $H_{p} = - d \cdot E (t)$ .
Applying Fermi's Golden Rule gives rates for absorption and stimulated emission that are proportional to the squared transition dipole moment, $∣ d_{f i} ∣^{2}$ . Transitions are "allowed" if this matrix element is non-zero, giving rise to selection rules.
Spontaneous emission can be understood through Einstein's A and B coefficients, which relate it to stimulated emission. Its rate scales with $ω^{3}$ .
This framework is directly applicable to problems in condensed matter physics, such as calculating the optical absorption coefficient of semiconductors, where the key concepts are Bloch wavefunctions and the joint density of states.

Preview of the next lesson:
In our analysis, we assumed a slowly varying Hamiltonian to derive the transition rate. But what happens if the Hamiltonian changes very slowly over the course of the system's evolution, allowing the system to continuously adapt? This is the subject of the adiabatic theorem. We will explore its content, its proof, and its connection to the geometric Berry phase, a deep and beautiful concept in quantum mechanics.