In physics, specifically statistical mechanics, a population inversion occurs when a system (such as a group of atoms or molecules) exists in a state in which more members of the system are in higher, excited states than in lower, unexcited energy states. It is called an "inversion" because in many familiar and commonly encountered physical systems in thermal equilibrium, this is not possible. This concept is of fundamental importance in laser science because the production of a population inversion is a necessary step in the workings of a standard laser.
Condition To understand the concept of a population inversion, it is necessary to understand some thermodynamics and the way that light interacts with matter. To do so, it is useful to consider a very simple assembly of atoms forming a laser medium.
Assume there is a group of atoms, each of which is capable of being in one of two energy states: either The ground state, with energy ; or The excited state, with energy , with . The number of these atoms which are in the ground state is given by , and the number in the excited state . Since there are atoms in total,
The energy difference between the two states, given by
determines the characteristic frequency of light which will interact with the atoms; This is given by the relation
being the Planck constant.
If the group of atoms is in thermal equilibrium, it can be shown from Maxwell–Boltzmann statistics that the ratio of the number of atoms in each state is given by the ratio of two Boltzmann distributions, the Boltzmann factor:
where is the thermodynamic temperature of the group of atoms, is the Boltzmann constant and and are the degeneracies of each state.
Calculable is the ratio of the populations of the two states at room temperature ( ≈ 300 K) for an energy difference that corresponds to light of a frequency corresponding to visible light ( ≈ ). In this case ≈ 2.07 eV, and ≈ 0.026 eV. Since , it follows that the argument of the exponential in the equation above is a large negative number, and as such is vanishingly small; i.e., there are almost no atoms in the excited state. When in thermal equilibrium, then, it is seen that the lower energy state is more populated than the higher energy state, and this is the normal state of the system. As increases, the number of electrons in the high-energy state () increases, but never exceeds for a system at thermal equilibrium; rather, at infinite temperature, the populations and become equal. In other words, a population inversion () can never exist for a system at thermal equilibrium. To achieve population inversion therefore requires pushing the system into a non-equilibrated state.
Interaction of light with matter
There are three types of possible interactions between a system of atoms and light that are of interest:
Absorption
If light (photons) of frequency passes through the group of atoms, there is a possibility of the light being absorbed by electrons which are in the ground state, which will cause them to be excited to the higher energy state. The rate of absorption is proportional to the radiation density of the light, and also to the number of atoms currently in the ground state, .
Spontaneous emission
If atoms are in the excited state, spontaneous decay events to the ground state will occur at a rate proportional to , the number of atoms in the excited state. The energy difference between the two states is emitted from the atom as a photon of frequency as given by the frequency-energy relation above.
The photons are emitted stochastically, and there is no fixed phase relationship between photons emitted from a group of excited atoms; in other words, spontaneous emission is incoherent. In the absence of other processes, the number of atoms in the excited state at time , is given by
where is the number of excited atoms at time , and is the mean lifetime of the transition between the two states.
Stimulated emission
If an atom is already in the excited state, it may be agitated by the passage of a photon that has a frequency corresponding to the energy gap of the excited state to ground state transition. In this case, the excited atom relaxes to the ground state, and it produces a second photon of frequency . The original photon is not absorbed by the atom, and so the result is two photons of the same frequency. This process is known as stimulated emission.
Specifically, an excited atom will act like a small electric dipole which will oscillate with the external field provided. One of the consequences of this oscillation is that it encourages electrons to decay to the lowest energy state. When this happens due to the presence of the electromagnetic field from a photon, a photon is released in the same phase and direction as the "stimulating" photon, and is called stimulated emission.
The rate at which stimulated emission occurs is proportional to the number of atoms in the excited state, and the radiation density of the light. The base probability of a photon causing stimulated emission in a single excited atom was shown by Albert Einstein to be exactly equal to the probability of a photon being absorbed by an atom in the ground state. Therefore, when the numbers of atoms in the ground and excited states are equal, the rate of stimulated emission is equal to the rate of absorption for a given radiation density.
The critical detail of stimulated emission is that the induced photon has the same frequency and phase as the incident photon. In other words, the two photons are coherent. It is this property that allows optical amplification, and the production of a laser system. During the operation of a laser, all three light–matter interactions described above are taking place. Initially, atoms are energized from the ground state to the excited state by a process called pumping, described below. Some of these atoms decay via spontaneous emission, releasing incoherent light as photons of frequency, . These photons are fed back into the laser medium, usually by an optical resonator. Some of these photons are absorbed by the atoms in the ground state, and the photons are lost to the laser process. However, some photons cause stimulated emission in excited-state atoms, releasing another coherent photon. In effect, this results in optical amplification.
If the number of photons being amplified per unit time is greater than the number of photons being absorbed, then the net result is a continuously increasing number of photons being produced; the laser medium is said to have a gain of greater than unity.
Recall from the descriptions of absorption and stimulated emission above that the rates of these two processes are proportional to the number of atoms in the ground and excited states, and , respectively. If the ground state has a higher population than the excited state (), then the absorption process dominates, and there is a net attenuation of photons. If the populations of the two states are the same (), the rate of absorption of light exactly balances the rate of emission; the medium is then said to be optically transparent.
If the higher energy state has a greater population than the lower energy state (), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that ; In other words, a population inversion is required for laser operation.
Selection rules
Many transitions involving electromagnetic radiation are strictly forbidden under quantum mechanics. The allowed transitions are described by so-called selection rules, which describe the conditions under which a radiative transition is allowed. For instance, transitions are only allowed if , being the total spin angular momentum of the system. In real materials, other effects, such as interactions with the crystal lattice, intervene to circumvent the formal rules by providing alternate mechanisms. In these systems, the forbidden transitions can occur, but usually at slower rates than allowed transitions. A classic example is phosphorescence where a material has a ground state with , an excited state with , and an intermediate state with . The transition from the intermediate state to the ground state by emission of light is slow because of the selection rules. Thus emission may continue after the external illumination is removed. In contrast fluorescence in materials is characterized by emission which ceases when the external illumination is removed.
Transitions that do not involve the absorption or emission of radiation are not affected by selection rules. The radiationless transition between levels, such as between the excited and states, may proceed quickly enough to siphon off a portion of the population before it spontaneously returns to the ground state.
The existence of intermediate states in materials is essential to the technique of optical pumping of lasers (see below).
Creation A population inversion is required for laser operation, but cannot be achieved in the above theoretical group of atoms with two energy-levels when they are in thermal equilibrium. In fact, any method by which the atoms are directly and continuously excited from the ground state to the excited state (such as optical absorption) will eventually reach equilibrium with the de-exciting processes of spontaneous and stimulated emission. At best, an equal population of the two states, , can be achieved, resulting in optical transparency but no net optical gain. To achieve lasting non-equilibrium conditions, an indirect method of populating the excited state must be used.
In laser
In three-level laser
A three-level laser may achieve population inversion. It consists of a group of atoms, with each atom able to exist in any of three energy states, levels 1, 2 and 3, with energies , , and , and populations , , and , respectively. ; that is, the energy of level 2 lies between that of level 1 and level 3. The level 1 is also referred to as the ground state.
Initially, the system of atoms is at thermal equilibrium, and the majority of the atoms will be in the ground state, i.e., , . The atoms can be excited from level 1 to level 3 and this act is called pumping. It can happen when the atoms are subjected to light of a frequency
the process of optical absorption will excite electrons from level 1 to level 3. There are also methods not directly involving light absorption; such as electrical discharge or chemical reactions. The level 3 is sometimes referred to as the pump level or pump band, and the energy transition as the pump transition (labeled transition P in the diagram on the right).
Upon pumping the medium, an appreciable number of atoms will transition to level 3, such that . To have a medium suitable for laser operation, it is necessary that these excited atoms quickly decay to level 2. The energy released in this transition may be emitted as a photon (spontaneous emission). However, in practice, the transition is usually non-radiative (labeled transition R in the diagram), with the energy being released as phonons and transferred to vibrational motion (heat) of the host material surrounding the atoms, without the generation of a photon. This phenomenon is sometimes called collisional deactivation.