Question

Consider a gas of atoms that might serve as a laser medium but that is in equilibrium, with no population inversions. A photon gas coexists with the atoms. Would a photon whose energy is precisely the difference between two atomic energy states be more likely to be absorbed or to induce a stimulated emission or neither? We expect that in equilibrium the numbers of atoms at different levels and the number of photons of a given energy should be stable. Is your answer compatible?

Short answer

A photon whose energy is precisely the difference between two atomic energy states is equally likely to be absorbed or to induce stimulated emission in equilibrium without population inversions. Such a scenario is compatible with stable numbers of atoms at different levels and number of photons of a given energy.

Step-by-step solution

Understanding Basic Atomic and Photon Interactions

Atoms can interact with photons in three possible ways. They can absorb the photon and go to an excited state, emit a photon spontaneously and come down from an excited state, or a passing by photon can stimulate an atom in excited state to come down and emit a photon of same energy, phase, direction and polarization. This is called stimulated emission. In equilibrium and without population inversions, the number of atoms in the ground and excited state would not change over time.

Analyzing Photon Behavior

Now considering the photon of energy equal to the atomic energy difference. An atom in lower energy state can absorb the photon and move to higher state or a photon can stimulate an atom in higher state to emit a photon and come down. Now, the probabilities of absorption and stimulated emission are equal. This is a consequence of Einstein coefficients for emission and absorption being equal when there is no population inversion.

Compatibility Check

In equilibrium, the number of photons of a certain energy and the number of atoms in different energy states should be stable. Given that absorption and stimulated emission are equally likely for the considered photon, these processes counteract each other, keeping the number of atoms in different states and the number of photons stable. This verifies that the explanation is indeed compatible with the principles of equilibrium.

Key concepts

Photon Gas

Imagine a collection of photons, called a 'photon gas', which behaves similarly to a gas composed of molecules. These photons move at the speed of light and can interact with atoms in various ways. The idea of a photon gas is particularly useful when discussing how light interacts with materials like the atoms in a laser medium.

In the context of our exercise, these photons have a specific energy that matches the difference between two atomic energy states. The behavior of this photon gas is dictated by the complex interplay between the photons and the atomic states they encounter. If the system is in equilibrium, meaning no external energy is being added or removed, the interactions between the photon gas and the atoms reach a steady state where the number of emissions and absorptions are balanced.

Atomic Energy States

Atoms exist in discrete energy levels or 'states'. These atomic energy states play a crucial role in the absorption and emission of photons. An electron in an atom can move to a higher energy state by absorbing a photon with the correct energy or, conversely, drop to a lower energy state by emitting a photon.

During this process, the energy difference between the two states defines the energy of the photon involved. This is the fundamental mechanism behind both photon absorption and stimulated emission, which is the cornerstone of laser operation. As illustrated in the exercise, a photon whose energy precisely matches this energy difference could either be absorbed by an atom or stimulate an emission, depending on the state of the atom it encounters.

Einstein Coefficients

The interaction between light and matter is quantitatively described by the Einstein coefficients. These are probabilities that define the rates of spontaneous emission, stimulated emission, and absorption. Specifically, the coefficient 'A' pertains to spontaneous emission, while 'B' refers to both stimulated emission and absorption.

In a state of equilibrium without population inversion, as the exercise proposes, the Einstein B coefficients for absorption and stimulated emission are equal. This equality implies that if a photon encounters an excited atom, it is just as likely to induce an emission as it is to be absorbed by an atom in the ground state, leading to a static situation where no net change in the population of the atomic energy states occurs over time.

Population Inversion

Population inversion is the condition required to achieve laser action. It occurs when more atoms are in an excited state, compared to the lower energy, or ground, state. Only by creating a population inversion, can stimulated emission dominate over absorption, allowing for the amplification of light.

However, in our exercise scenario, there is no population inversion as the system is in equilibrium. The lack of population inversion is precisely why our photon is equally likely to be absorbed or to induce stimulated emission. This equality of probability is crucial for maintaining the population of the energy states and enabling the dynamics of the photon gas to remain stable.

Photon Absorption

Photon absorption is the process by which an atom in a lower energy state captures a photon and transitions to a higher energy state. The energy of the absorbed photon must match the energy difference between the initial and final states of the atom for absorption to occur.

In the exercise, a photon gas coexists with atoms in equilibrium, meaning that the propensity for photon absorption is precisely balanced with the likelihood of stimulated emission. Since these processes are equal in probability and opposite in effect, they permit the photon gas and the atomic population to coexist in a steady state, with no net energy exchange altering the equilibrium.