# Atomic Rate Equations

This is a continuation from the previous tutorial - specialty single-mode fibers.

Applying a sinusoidal signal to a collection of atoms, with the frequency $$\omega$$ tuned near one of the atomic transition frequencies $$\omega_a$$, will produce a coherent induced polarization $$p(t)$$ or $$\tilde{P}(\omega)$$ in the collection of atoms. The strength of this induced response will be proportional to the instantaneous population difference $$\Delta{N}$$ on that particular transition.

At the same time, however, this applied signal field will also cause the populations $$N_1(t)$$ and $$N_2(t)$$ in the collection of atoms to begin changing slowly because of stimulated transitions between the two levels $$E_1$$ and $$E_2$$.

The rates of change of the populations are given by atomic rate equations, which contain both stimulated terms and relaxation or energy-decay terms (and possibly also other kinds of pumping terms). Deriving the quantum form for these stimulated and relaxation terms is the primary objective of this tutorial.

These atomic rate equations are of great value in analyzing pumping and population inversion in laser systems. Solutions of the rate equations for strong applied signals also lead to population saturation effects, which are of very great importance in understanding the large-signal saturation behavior of laser amplifiers and the power output of laser oscillators. Solving the atomic rate equations and understanding these solutions for some simple cases will therefore be the principal objective of a later tutorial.

## 1. Power Transfer from Signals to Atoms

We will derive the stimulated transition rates for an atomic transition in this tutorial by examining the power flow or the energy transfer between an applied optical signal and an atomic transition.

To get started on this, let us learn something about the rate at which power is transferred from an applied signal to any material medium, including a collection of resonant oscillators or atoms.

### Power Transfer to a Collection of Oscillators: Mechanical Derivation

When an electric field $$\mathcal{E}_x(t)$$ acts on a moving charge, it delivers power to (or perhaps receives power from) that moving charge. In a single classical oscillator (as in Figure 4.1), a purely mechanical argument says that the amount of work $$dU$$ done by a force $$f_x$$ acting on the electron, when the electron moves through a distance $$dx$$ is

$\tag{1}dU=f_xdx=-e\mathcal{E}_xdx$

The instantaneous rate at which power is delivered by the field to the classical oscillator is then

$\tag{2}\frac{dU(t)}{dt}=-e\mathcal{E}_x(t)\frac{dx(t)}{dt}=\mathcal{E}_x(t)\frac{d\mu_x(t)}{dt}$

where $$\mu_x(t)$$ is, of course, the dipole moment of the oscillator.

If we sum this power flow over all the oscillators or atoms in a small unit volume $$V$$, this result says that the average power per unit volume, $$dU_a/dt$$, delivered by the field to the atoms or oscillators is

$\tag{3}\frac{dU_a}{dt}=V^{-1}\mathcal{E}_x(t)\sum_{i=1}^{NV}\frac{d\mu_{xi}(t)}{dt}=\mathcal{E}_x(t)\frac{dp_x(t)}{dt}$

This equation, although derived from a mechanical argument, is a very general electromagnetic or even quantum-mechanical result. That is, this equation still holds true whether $$p_x(t)$$ represents the sum of a large number of classical oscillator dipoles with a number density $$N$$, or whether $$p_x(t)$$ represents the effect of a large number of quantum dipole expectation values proportional to a population difference $$\Delta{N}$$.

### Time-Averaged Power Flow

To obtain the time-averaged power delivered to a collection of atoms by a sinusoidal signal field, we can write the applied signal and the resulting polarization in phasor form as

$\tag{4}\mathcal{E}_x(t)=\frac{1}{2}[\tilde{E}(\omega)e^{j\omega{t}}+\tilde{E}^*(\omega)e^{-j\omega{t}}]$

and

$\tag{5}p_x(t)=\frac{1}{2}[\tilde{P}(\omega)e^{j\omega{t}}+\tilde{P}^*(\omega)e^{-j\omega{t}}]$

The steady-state sinusoidal polarization $$\tilde{P}(\omega)$$ on an atomic transition will then be related to the applied field by

$\tag{6}\tilde{P}(\omega)=\tilde{\chi}(\omega)\epsilon\tilde{E}(\omega)=[\chi'(\omega)+j\chi^"(\omega)]\epsilon\tilde{E}(\omega)$

(Remember that we use the host dielectric constant $$\epsilon$$ and not $$\epsilon_0$$ in this relation.)

If we substitute these phasor forms into Equation 4.3 and take the time average (by dropping the $$e^{\pm2j\omega{t}}$$ terms) we obtain a useful result for the average power absorbed from the fields, by the atoms, per unit volume, namely,

$\tag{7}\left.\frac{dU_a}{dt}\right|_{\text{av}}=\frac{j\omega}{4}\left(\tilde{E}^*\tilde{P}-\tilde{E}\tilde{P}^*\right)=-\frac{1}{2}\epsilon\omega\chi^"(\omega)|\tilde{E}(\omega)|^2$

The most important point here is that the power absorption (or emission) by the atoms depends only on the $$\chi^"(\omega)$$ part of the complex susceptibility $$\tilde{\chi}(\omega)$$. This is the "resistive" or lossy part of $$\tilde{\chi}(\omega)$$, whereas $$\chi'(\omega)$$ is the purely reactive part.

The minus sign in the final term of Equation 4.7 merely means that if we use the definition $$\tilde{\chi}\equiv\chi'+j\chi^"$$, then the quantity $$\chi^"$$ for an absorbing medium will turn out to be a negative number, as indeed we have already found for the classical electron oscillator. (Some authors, attempting to avoid this minus sign, use instead the definition that $$\tilde{\chi}\equiv\chi'-j\chi^"$$. )

### Poynting Derivation of Energy Transfer

The results for power transfer obtained above are in fact general electromagnetic results, having nothing directly to do with the particular atomic or quantum process that creates the polarization $$p_x(t)$$. To verify this, let us carry through a standard electromagnetic derivation of this same result, starting by writing Maxwell's equations

\tag{8}\begin{align}\nabla\times\boldsymbol{\mathcal{E}}&=-\partial{\pmb{b}}/\partial{t},\quad\nabla\times\pmb{h}=\pmb{j}+\partial{\pmb{d}}/\partial{t}\\\pmb{d}&=\epsilon_0\boldsymbol{\mathcal{E}}+\pmb{p},\quad\pmb{b}=\mu_0(\pmb{h}+\pmb{m})\end{align}

and then substituting them into the vector identity

$\tag{9}\pmb{h}\cdot(\nabla\times\boldsymbol{\mathcal{E}})-\boldsymbol{\mathcal{E}}\cdot(\nabla\times\pmb{h})\equiv\nabla\cdot(\boldsymbol{\mathcal{E}}\times\pmb{h})$

Note that all the vector quantities here, for example, $$\boldsymbol{\mathcal{E}}(\pmb{r},t)$$, are general vector functions of space and time at this point.

Equation 4.9 can then be integrated over an arbitrary volume $$V$$, bounded by a closed surface $$S$$ as in Figure 4.2, using the additional vector identity that

$\tag{10}\int_V\nabla\cdot(\boldsymbol{\mathcal{E}}\times\pmb{h})dV=-\int_S(\boldsymbol{\mathcal{E}}\times\pmb{h})\cdot{d}\pmb{S}$

where $$d\pmb{S}$$ is an inward unit vector normal to the surface $$S$$. Rearranging terms then gives as a general formula

\tag{11}\begin{align}\int_S(\boldsymbol{\mathcal{E}}\times\pmb{h})\cdot{d}\pmb{S}&=\frac{d}{dt}\int_V\left(\frac{1}{2}\epsilon_0|\boldsymbol{\mathcal{E}}|^2+\frac{1}{2}\mu_0|\pmb{h}|^2\right)dV\\&\quad+\int_V(\boldsymbol{\mathcal{E}}\cdot\pmb{j})dV\\&\quad+\int_V\left(\boldsymbol{\mathcal{E}}\cdot\frac{d\pmb{p}}{dt}+\mu_0\pmb{h}\cdot\frac{d\pmb{m}}{dt}\right)dV\end{align}

We can give a physical interpretation to each term in this equation.

The surface integral on the left-hand side of this equation gives the integral over the closed surface $$S$$ of the inwardly directed instantaneous Poynting vector $$\boldsymbol{\mathcal{E}}\times\pmb{h}$$. According to the standard interpretation of electromagnetic theory, this Poynting integral gives the total electromagnetic power being carried by fields $$\boldsymbol{\mathcal{E}}$$ and $$\pmb{h}$$ and flowing into the volume $$V$$ at any instant.

The terms on the right-hand side of Equation 4.11 tell where this power is going. The volume integral on the right-hand side of the first line is a purely reactive or energy-storage term. It gives the instantaneous rate of increase or decrease in the stored electromagnetic field energy terms $$\frac{1}{2}\epsilon_0\boldsymbol{\mathcal{E}}^2$$ and $$\frac{1}{2}\mu_0\pmb{h}^2$$ in the volume $$V$$. (These are vacuum energy density terms—that is, they do not include any energy going into atomic polarizations $$\pmb{p}(t)$$ or $$\pmb{m}(t)$$ in the volume $$V$$.)

The integral on the right-hand side of the second line gives the instantaneous power per unit volume being delivered by the $$\boldsymbol{\mathcal{E}}$$ field to any currents $$\pmb{j}$$, whether these currents come from ohmic losses ($$\pmb{j}=\sigma\boldsymbol{\mathcal{E}}$$) or any other real currents $$\pmb{j}(\pmb{r},t)$$ that may be present in the volume.

The integral on the right-hand side of the final line of Equation 4.11 then accounts for the instantaneous powers per unit volume $$\boldsymbol{\mathcal{E}}\cdot(d\pmb{p}/dt)$$ and $$\mu_0\pmb{h}\cdot(d\pmb{m}/dt)$$ that are being delivered by these fields to any electric and magnetic polarizations $$\pmb{p}(\pmb{r},t)$$ and $$\pmb{m}(\pmb{r},t)$$ that may be present, as a consequence of any kind of atomic or material medium. The $$\boldsymbol{\mathcal{E}}\cdot(d\pmb{p}/dt)$$ term represents in particular the vector generalization of the simple mechanical derivation we gave at the beginning of this section.

### Reactive Versus Resistive Power Flow

Note that power transfer from the signal fields to these atomic polarization terms does not necessarily mean this power is being dissipated in the atoms. If the medium has a purely reactive susceptibility, with $$\chi^"=0$$, then there can be no time-averaged power transfer, because $$\boldsymbol{\mathcal{E}}(t)$$ and $$\pmb{p}(t)$$ will be 90° out of time-phase, and the time-averaged value of the $$\boldsymbol{\mathcal{E}}\cdot{d}\pmb{p}/dt$$ term will be zero: energy will flow from the signal into the medium during one quarter cycle, and back out during the following quarter cycle.

The energy transfer into the polarization in this situation is basically reactive stored energy, which flows into the atoms during one half-cycle and back out during the following half-cycle. This reactive energy flow could be combined with the first integral on the right-hand side of Equation 4.11. If this were done, the expanded first term would become the time derivative of the more familiar expressions $$\frac{1}{2}\epsilon|\boldsymbol{\mathcal{E}}|^2$$ and $$\frac{1}{2}\mu|\pmb{h}|^2$$, which give the total electromagnetic energy stored in a medium rather than in vacuum.

We can also see that the rate of change of polarization $$d\pmb{p}/dt$$ in the $$\boldsymbol{\mathcal{E}}\cdot{d}\pmb{p}/dt$$ term plays the same role as the current density $$\pmb{j}$$ in the $$\boldsymbol{\mathcal{E}}\cdot\pmb{j}$$ term. It is sometimes convenient to define a "polarization current density" $$\pmb{j}_p$$ by

$\tag{12}\pmb{j}_p\equiv{d}\pmb{p}/dt$

which can be added to the real current density $$\pmb{j}$$ in Maxwell's equations. We realize that this polarization current simply represents the sloshing back and forth of the bound but oscillating atomic charge clouds that lead to the oscillating dipole moments $$\boldsymbol{\mu}(t)$$ in each atom and to the macroscopic polarization $$\pmb{p}(t)$$ in the collection of atoms.

To the extent that this current is in phase with the $$\boldsymbol{\mathcal{E}}(t)$$ term [that is, comes from the $$\chi^"(\omega)$$ part of the susceptibility], it represents additional resistive loss or dissipation in the medium; to the extent that it is 90° out of phase [the $$\chi'(\omega)$$ part], it represents reactive energy storage.

### Quality Factor

The absorptive susceptibility $$\chi^"$$ in an atomic medium can be interpreted as a kind of inverse $$Q$$ or quality factor for the ratio of signal energy stored in a volume to signal power dissipated in that volume, in just the same fashion as the $$Q$$ factor is defined for a mechanical system or an electrical circuit. That is, the time-averaged stored signal energy per unit volume associated with a sinusoidal signal field in a host medium of dielectric constant $$\epsilon$$ can be written as

$\tag{13}U_\text{sig}=\frac{1}{2}\epsilon|\pmb{E}|^2$

The inverse $$Q$$ factor for this little volume can then be defined as

$\tag{14}\frac{1}{Q}\equiv\frac{\text{energy dissipated}}{\omega\times\text{energy stored}}=\frac{1}{\omega{U}_\text{sig}}\frac{dU_a}{dt}=-\chi^"$

The dimensionless atomic susceptibility $$\chi^"$$, as we have defined it, is thus essentially an inverse $$Q$$ factor describing the average power absorption per unit volume, by the atoms, from the signal. Of course for an amplifying transition, this $$Q$$ becomes a negative number.

For real laser transitions this $$Q$$ is always very high, since in all practical laser situations $$|\chi^"|\ll1$$. We usually think of a high $$Q$$ value in a system as being in some sense "good". Here, however, a high $$Q$$ means a weak susceptibility, and hence a small gain in an amplifying laser medium, which is generally not what we would like to have.

### Tensor Formulation of Power Flow

Real laser transitions may have a linear but anisotropic response, in which the induced polarization must be described by a tensor susceptibility. To describe the power transfer properly in this case we must employ a more sophisticated form for the analysis in terms of the hermitian and antihermitian parts of this susceptibility tensor.

To do this, we note that the full vector formula for instantaneous power delivered per unit volume is

$\tag{15}\frac{dU_a(t)}{dt}=\boldsymbol{\mathcal{E}}\cdot{d}\pmb{p}/dt$

The time-averaged power flow in an atomic medium with a tensor atomic susceptibility $$\boldsymbol{\chi}$$ is then given by

$\tag{16}\left.\frac{dU_a}{dt}\right|_\text{av}=\frac{j\omega\epsilon}{4}[\pmb{E}^*\cdot\boldsymbol{\chi}\pmb{E}-\pmb{E}\cdot\boldsymbol{\chi}^*\pmb{E}^*]=\frac{j\omega\epsilon}{4}\sum_{i=1}^3\sum_{j=3}^3\tilde{E}_i^*(\tilde{\chi}_{ij}-\tilde{\chi}_{ji}^*)\tilde{E}_j$

where $$i$$ and $$j$$ are both summed over the three directions $$x$$, $$y$$, $$z$$. If $$\boldsymbol{\chi}$$ happens to be an isotropic or even a diagonal tensor, then these sums reduce directly to our previous scalar results.

For a general anisotropic tensor susceptibility, however, we must separate the complex tensor $$\boldsymbol{\chi}$$ not into its real and imaginary parts, but into its hermitian and antihermitian parts, as given by

$\tag{17}\boldsymbol{\chi}\equiv\boldsymbol{\chi}_h+j\boldsymbol{\chi}_{ah}$

where $$\boldsymbol{\chi}_h$$ and $$\boldsymbol{\chi}_{ah}$$ are defined by

$\tag{18}\boldsymbol{\chi}_h\equiv(1/2)(\boldsymbol{\chi}^\dagger+\boldsymbol{\chi})\qquad\text{and}\qquad\boldsymbol{\chi}_{ah}\equiv(j/2)(\boldsymbol{\chi}^\dagger-\boldsymbol{\chi})$

with $$\boldsymbol{\chi}^\dagger$$ being the hermitian conjugate of $$\boldsymbol{\chi}$$. Note that $$\boldsymbol{\chi}_h$$ and $$\boldsymbol{\chi}_{ah}$$ are not necessarily the same as the real and imaginary parts of $$\boldsymbol{\chi}$$, since $$\boldsymbol{\chi}^\dagger$$ and $$\boldsymbol{\chi}$$ are in general not simply the complex conjugates of each other.

It can then be shown that the time-averaged power transfer is given by

$\tag{19}\left.\frac{dU_a}{dt}\right|_\text{av}=-\frac{1}{2}\omega\epsilon\pmb{E}^*(\omega)\boldsymbol{\chi}_{ah}(\omega)\pmb{E}(\omega)$

In the general tensor case it is the antihermitian part $$j\boldsymbol{\chi}_{ah}$$ of the susceptibility tensor, and not just the imaginary part $$\chi^"$$, that is the resistive or power-absorbing part.

## 2. Stimulated-Transition Probability

We will next use these results to derive a stimulated-transition probability, which gives the stimulated-transition rate at which atoms make transitions back and forth between quantum energy levels under the influence of an applied signal.

We will do this by considering the rate at which an applied signal will deliver energy to a collection of real quantum atoms, and the manner in which these atoms can accept this energy.

### Energy Transfer From Signal To Atoms

We have learned that when a sinusoidal signal field $$\tilde{E}(\omega)$$ produces a steady-state polarization $$\tilde{P}(\omega)$$ on a transition in a collection of atoms, the time-averaged power transfer per unit volume from the signal to the atoms must be given by

$\tag{20}\frac{dU_a}{dt}=-\frac{1}{2}\omega\chi^"(\omega)\epsilon|\tilde{E}|^2$

where the susceptibility for a homogeneous lorentzian atomic transition is given by

$\tag{21}\tilde{\chi}^"(\omega)=-\frac{3^*}{4\pi^2}\frac{\Delta{N}\gamma_\text{rad}\lambda^3}{\Delta\omega_a}\frac{1}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}$

(For an inhomogeneous gaussian transition exactly the same expression as Equation 4.21 would apply, except that the homogeneous linewidth $$\Delta\omega_a$$ would be replaced by the inhomogeneous linewidth $$\Delta\omega_d$$; the lorentzian frequency dependence would be replaced by a gaussian; and an additional factor of $$\sqrt{\pi\ln2}\approx1.48$$ would appear in front.)

The rate of energy transfer from the signal to the atoms can thus be written in the general form

$\tag{22}\frac{dU_a}{dt}=\left[\frac{3^*}{8\pi^2}\frac{\gamma_\text{rad}}{\Delta\omega_a}\frac{\omega\epsilon|\tilde{E}|^2\lambda^3}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\right](N_1-N_2)$

Note that this energy absorption is directly proportional to the population difference $$\Delta{N}\equiv{N_1}-N_2$$.

### Energy Storage by the Atoms

Now, what will the atoms do with this energy, or how can they accept this energy from the applied fields? From a quantum viewpoint, the total oscillation energy stored in a collection of atoms is given by the number of atoms $$N_j$$ in each quantum energy level, times the energy eigenvalue $$E_j$$ associated with that level, summed over all the energy levels $$E_j$$, as in Figure 4.3.

In a collection of identical two-level atoms, for example, the total oscillation energy density $$U_a$$ (energy per unit volume) is given by

$\tag{23}U_a(t)=N_1(t)E_1+N_2(t)E_2$

where $$N_1$$ and $$N_2$$ are the populations in levels $$E_1$$ and $$E_2$$. More generally, the total energy in a collection of multilevel atoms is the sum over all levels

$\tag{24}U_a(t)=\sum_{j=1}^MN_j(t)E_j$

Since the energy eigenvalues $$E_j$$ are fixed quantities, if the collection of atoms is to accept energy the level populations $$N_j$$ must change, with atoms flowing from a lower energy level to a higher energy level.

In the classical oscillator model the energy of each oscillator was associated with the internal oscillatory motion $$|x(t)|^2$$. In a quantum description, however, the internal energy of each atom must be calculated from the level populations and energy eigenvalues as above. These two descriptions are not unrelated—for example, we noted earlier that an atom in a mixture of populations at levels $$E_1$$ and $$E_2$$ has an internal oscillating dipole moment $$\boldsymbol{\mu}(t)$$ at the transition frequency $$\omega_{21}$$ that is associated with that mixture of populations.

In any event, if energy is to be delivered from a signal to a collection of atoms, the only way in which the atoms can accept this energy and change their total internal energy $$U_a(t)$$ is by changing the populations $$N_j(t)$$ in the collection of atoms.

The signal field induces a dipole moment in each atom, and thus produces a coherent macroscopic polarization proportional to the population difference $$\Delta{N}\equiv{N_1}-N_2$$. But, it must also cause the quantum state of each individual atom to begin to change in such a way that the populations $$N_1(t)$$ and $$N_2(t)$$ in the collection of atoms also begin to change.

### Stimulated Transition Probabilities

We can emphasize this point by rewriting Equation 4.22 in the alternative form

$\tag{25}\frac{dU_a}{dt}=W_{12}N_1\hbar\omega_a-W_{21}N_2\hbar\omega_a$

where either of the quantities $$W_{12}\bar\omega_a=W_{21}\bar\omega_a$$ corresponds to the collection of factors inside the set of square brackets in Equation 4.22.

By rewriting Equation 4.22 in this alternative form, however, we make the energy flow from the signal to the atoms seem to be produced by two flows of atoms, one upward from level 1 to level 2 at an upward stimulated-transition rate given by $$W_{12}N_1$$ (units of atoms per second), as shown by the upward arrow in Figure 4.4; plus an opposite flow of atoms downward from level 2 to level 1 at a downward stimulated-transition rate given by $$W_{21}N_2$$.

In other words, the energy transfer from the signal to the atoms, as given by Equation 4.25, can be accounted for by a net flow rate of atoms across the gap, upward minus downward, given by

$\tag{26}\left.\frac{dN_2}{dt}\right|_\text{stim}=-\left.\frac{dN_1}{dt}\right|_\text{stim}=W_{12}N_1-W_{21}N_2$

as illustrated in Figure 4.4.

Both of these flow rates are expressed in units of atoms per second. To get the net energy flow, each of these rates must be multiplied by the transition energy or photon energy $$\hbar\omega_a$$, since each transit of one atom across the energy gap represents a net absorption or emission of one quantum of energy by the atoms.

The quantities $$W_{12}$$ and $$W_{21}$$ are then referred to as the upward and downward stimulated-transition probabilities, per atom and per unit time, produced by the applied signal acting on the lower-level and upper-level atoms, respectively.

By equating Equations 4.22 and 4.25, we can see that these stimulated-transition probabilities are given by

$\tag{27}W_{12}\equiv{W_{21}}=\frac{3^*}{8\pi^2}\frac{\gamma_\text{rad}}{\hbar\Delta\omega_a}\frac{\epsilon|\tilde{E}|^2\lambda^3}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}$

With this interpretation we may say that the applied signal gives each atom in the lower level $$E_1$$ a probability $$W_{12}$$ per unit time of making a stimulated transition to the upper level, absorbing a quantum of energy in the process; similarly, the applied signal gives each atom in the upper level an equal probability $$W_{21}$$ per unit time of making a transition downward to the lower level, giving up one quantum of energy to the signal in the process.

Equations 4.25 through 4.27 provide an important and general result, sometimes referred to as "Fermi's Golden Rule", which is usually derived only with the aid of quantum theory.

We have obtained the correct quantum answer, however, from a simple energy argument, thus further illustrating the power of the classical oscillator arguments.

You might reasonably ask, however, since $$W_{12}=W_{21}$$, why didn't we describe them both by a single symbol? The answer is, first of all, that later on in writing multilevel rate equations it may help to keep various terms straight if we use $$W_{ij}$$ to mean a transition probability from level $$i$$ to level $$j$$, and $$W_{ji}$$ to mean the same transition probability in the reverse direction. In addition, there are some slight additional complications for the rate equations between degenerate energy levels, as we will see in a moment.

### Quantum Description of Stimulated Transitions

These signal-stimulated transition rates for atoms between two energy levels are often described in simplified fashion as a process in which the applied signal causes individual atoms to make discrete jumps back and forth between the two levels under the influence of the applied signal, exchanging one photon with each jump. This is the "billiard-ball" or photon model of laser amplification.

A much more correct description, however, even in quantum theory, is to say that each individual atom in the collection of atoms, rather than making a discrete "jump" or transition from one level to the other, in fact really only changes its quantum state by a small amount in response to the applied signal. We have pointed out earlier that the quantum state of an atom involved in a transition between two levels $$E_1$$ and $$E_2$$ can be written in the general form

$\tag{28}\psi(\pmb{r},t)=\tilde{a}_1(t)e^{-iE_1t/\hbar}\psi_1(\pmb{r})+\tilde{a}_2(t)e^{-iE_2t/\hbar}\psi_2(\pmb{r})$

where the expansion coefficients $$\tilde{a}_1(t)$$ and $$\tilde{a}_2(t)$$ are constant (stationary) in the absence of an applied signal. The time evolution of this quantum state in each individual atom in the presence of an applied signal must then be calculated in a proper quantum analysis by solving the Schrödinger equation of motion for the atom in the presence of the signal field.

The net result of such a calculation will be that, under the influence of an applied signal, the expansion coefficients $$\tilde{a}_1(t)$$ and $$\tilde{a}_2(t)$$ of each individual atom will begin to change slowly but continuously with time. In other words, the quantum state makeup of each individual atom will begin to shift by a small but continuous amount from quantum state $$\psi_1$$ towards quantum state $$\psi_2$$ or vice versa.

The probability of finding each atom in one level or the other begins to change by a small amount; and when these probabilities for individual atoms are averaged over the entire collection, it appears as if the population in one level has decreased and in the other has increased.

For many purposes, however, it is acceptable to summarize the results of this calculation by simply saying that, in the presence of an applied signal, atoms begin to make stimulated transitions or jumps back and forth between the two levels $$E_1$$ and $$E_2$$, thus changing $$N_1(t)$$ and $$N_2(t)$$.

The final result averaged over the collection of atoms is basically the same whether we think of each individual atom changing its quantum state by a small amount (which is what really happens), or whether we think of a small fraction of the atoms making discrete "jumps" from one level to the other (which is how the situation is often described).

### General Atomic Transition With Degeneracy

To take care of the more general case in which we have transition rates between two arbitrary energy levels $$E_i$$ and $$E_j\gt{E}_i$$, where these levels may have degeneracy factors $$g_i$$ and $$g_j$$, we must note that the complex susceptibility on such a transition is given by

$\tag{29}\tilde{\chi}_{ij}^"(\omega)=-\frac{3^*}{4\pi^2}\frac{\gamma_{\text{rad},ji}\lambda_{ij}^3}{\Delta\omega_a}\frac{[(g_j/g_i)N_i-N_j]}{1+[2(\omega-\omega_{ji})/\Delta\omega_{a,ij}]^2}$

Hence, the power transfer from signal to atoms can be written, first in electromagnetic form, and then in rate-equation form, as

\tag{30}\begin{align}\frac{dU_a}{dt}&=\left[\frac{3^*}{8\pi^2}\frac{\gamma_{\text{rad},ji}}{\Delta\omega_a}\frac{\omega\epsilon|\tilde{E}_{ij}|^2\lambda_{ij}^3}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\right]\left(\frac{g_j}{g_i}N_i-N_j\right)\\&=W_{ij}N_i\hbar\omega_{ji}-W_{ji}N_j\hbar\omega_{ji}\end{align}

By equating these two forms, we see that the general expression for the stimulated transition probabilities in this case becomes

$\tag{31}W_{ji}\equiv\frac{g_i}{g_j}W_{ij}=\frac{3^*}{8\pi^2}\frac{\gamma_{\text{rad},ji}}{\hbar\Delta\omega_{a,ij}}\frac{\epsilon|\tilde{E}_{ij}|^2\lambda_{ij}^3}{1+[2(\omega-\omega_{ji})/\Delta\omega_{a,ij}]^2}$

where $$\tilde{E}_{ij}$$ is the electric field of the applied signal on the $$i-j$$ transition.

Again the flow of atoms upward from level $$E_i$$ to level $$E_j$$ is given by the number of atoms in the lower level $$N_i$$ times an upward stimulated-transition probability $$W_{ij}$$, and the quantity $$W_{ji}$$ is similarly the probability of an upper level atom being stimulated to make a downward transition. The stimulated-transition probabilities $$W_{ij}$$ and $$W_{ji}$$ in the two directions are, however, related in general by

$\tag{32}g_iW_{ij}=g_jW_{ji}$

A very fundamental point is that the stimulated-transition probabilities in the two directions are still identical, except for the minor complication of the degeneracy factors $$g_i$$ and $$g_j$$.

### Fundamental Properties of the Stimulated-Transition Probabilities

From Equations 4.27 or 4.31, the important physical parameters involved in these signal-stimulated transition probabilities $$W_{ij}$$ and $$W_{ji}$$ are evidently

• The applied signal strength, or the signal energy per wavelength cubed, as measured by $$\epsilon|\tilde{E}|^2\lambda^3$$.
• The relative strength of the atomic transition, measured by its radiative decay rate or Einstein A coefficient, $$\gamma_\text{rad}$$.
• The inverse atomic linewidth, $$1/\Delta\omega_a$$.
• The frequency of the applied signal $$\omega$$ relative to the atomic transition frequency $$\omega_a$$, as measured by the atomic lineshape. Applied signals tuned away from line center are less effective in producing stimulated transitions.
• And, finally, the tensor alignment between the applied field and the atoms, as measured by the factor $$0\le3^*\le3$$.

For an inhomogeneous gaussian transition the formulas in Equations 4.27 or 4.31 must be modified by replacing $$1/\Delta\omega_a$$ by $$1/\Delta\omega_d$$; replacing the lorentzian frequency dependence by a gaussian; and adding a factor of $$\sqrt{\pi\ln2}\approx1.48$$ in front.

Note also that in the preceding analysis we speak of the upward and downward stimulated rates $$W_{12}N_1$$ and $$W_{21}N_2$$ as if they were separate and distinct processes.

It is, however, only the net transition rate between levels, $$W_{12}N_1-W_{21}N_2$$, that really counts. There is no way to "turn off" one of these rates and produce only the other one. They are physically identical or at least physically inseparable.

The transition rates discussed in this section are called stimulated transition rates because they are caused by applied signals producing changes in the populations $$N_1(t)$$ and $$N_2(t)$$.

Populations of atomic levels also change with time because of pumping effects, and because of energy decay or relaxation transitions between the levels. These relaxation processes produce additional terms in the rate equations, which we must describe in subsequent sections. The stimulated and relaxation terms must be added together in the total rate equations to describe how the populations change with time.

The next objective in this tutorial must be to understand how thermal fluctuations, or blackbody radiation fields, can also cause stimulated transitions between atomic energy levels.

We will then go on to show how these "noise-stimulated" transitions are related to the spontaneous emission or radiative decay processes we have discussed earlier, and how they provide a very important part of the relaxation processes in an atomic system.

One of the most basic conclusions of thermodynamics is that any volume of space that is in thermal equilibrium with its surroundings must contain a blackbody radiation energy density, made up of noise-like blackbody radiation fields.

Furthermore, the magnitude of these fields depends only on the temperature of the region and of its immediate surroundings and not at all on the shape or construction of the volume (provided only that the volume is large compared to a wavelength of the radiation involved).

The electromagnetic fields that make up this blackbody radiation energy are real, measurable, broadband, noise-like $$E$$ and $$H$$ fields, with random amplitudes, phases, and polarization, which are present everywhere in the volume.

The amount of blackbody radiation energy per unit volume that is present within a region at temperature $$T_\text{rad}$$, at frequencies within a narrow frequency range $$d\omega$$ about $$\omega$$, is given in fact by the blackbody radiation density

$\tag{33}dU_\text{bbr}=\frac{8\pi}{\lambda^3}\frac{\hbar{d\omega}}{\exp(\hbar\omega/kT_\text{rad})-1}$

In more precise terms, $$dU_\text{bbr}/d\omega$$ is the spectral density of the blackbody radiation energy, i.e., the amount of energy per unit volume and per unit frequency range centered at frequency $$\omega$$.

We write the temperature as $$T_\text{rad}$$ in this expression to indicate that it is the temperature of the "radiative surroundings" of this region—that is, the temperature of the nearest electromagnetically absorbing walls or boundaries—that determines the blackbody radiation energy density in the region.

The energy density dUbbr in any narrow frequency range du will have associated with it a mean-square electric field intensity $$d|E_\text{bbr}|^2$$ given by

$\tag{34}dU_\text{bbr}=\frac{\epsilon}{2}d|\tilde{E}_\text{bbr}|^2$

That is, there will be real measurable electric fields of noise-like character associated with the blackbody energy within the frequency range $$d\omega$$, and these fields will have a root-mean-square phasor amplitude $$\tilde{E}_\text{bbr}$$ given by

$\tag{35}d|\tilde{E}_\text{bbr}|^2=\frac{16\pi}{\lambda^3}\frac{\hbar{d}\omega}{\exp(\hbar\omega/kT_\text{rad})-1}$

With a sensitive enough antenna or probe and a receiver with a low enough noise figure (Figure 4.5), these noise-like fields can be detected and measured as a function of center frequency $$\omega$$ and temperature $$T_\text{rad}$$ inside the enclosure.

### Blackbody-Stimulated Atomic Transitions

Any atoms that may be present in the region under consideration are then exposed to these entirely real though noise-like $$\tilde{E}_\text{bbr}$$ fields. These $$E$$ fields will in fact act on the atoms just like signal fields, and will cause stimulated transitions and power absorption at exactly the same rate as would be caused by any other applied signal fields with the same mean-square amplitude. We can calculate the stimulated-transition rates that will be caused by these blackbody radiation fields by means of the following argument.

Figure 4.6 illustrates how the broadband continuum spectral distribution of the blackbody noise fields will overlap with the narrow atomic lineshape of a typical atomic transition.

The amount of stimulated-transition probability $$dW_{12}$$ that will be caused in a two-level atomic system by those blackbody radiation components lying within a small frequency bandwidth $$d\omega$$ centered at $$\omega$$ within the atomic linewidth will then be given (for a lorentzian transition) by exactly the same stimulated-transition probability expression as was derived in the previous section, namely,

$\tag{36}dW_{12,\text{bbr}}=dW_{21,\text{bbr}}=\frac{3^*}{8\pi^2}\frac{\gamma_\text{rad}}{\hbar\Delta\omega_a}\frac{\epsilon{d}|\tilde{E}_\text{bbr}|^2\lambda^3}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}$

with $$d|\tilde{E}_\text{bbr}|^2$$ given by Equation 4.35. Because these blackbody $$\tilde{E}$$ fields will be randomly polarized, the $$3^*$$ factor will have an averaged value of unity; so we will drop it from here on.

The total transition rate on the $$1\rightarrow2$$ transition due to blackbody radiation fields at all frequencies is then easily calculated by integrating the contribution from each narrow range $$d\omega$$, as given by Equation 4.36, over all the blackbody signals that are present at all frequencies, in the form

\tag{37}\begin{align}W_{12,\text{bbr}}&=W_{21,\text{bbr}}=\int{d}W_{12,\text{bbr}}\\&=\int_{-\infty}^{\infty}\left[\begin{split}\text{radiation density}\\\text{at frequency }\omega\end{split}\right]\times\left[\begin{split}\text{transition response}\\\text{at frequency }\omega\end{split}\right]d\omega\end{align}

For any reasonable atomic transition, the atomic linewidth will always be very much narrower than the blackbody spectral distribution, as in Figure 4.6.

It is then an entirely valid approximation to give the blackbody distribution function its value at the line center, $$\omega=\omega_a$$, and take it outside the integral over $$d\omega$$. The integral of Equation 4.36 over the lineshape then reduces to the simple form

$\tag{38}W_{12,\text{bbr}}=W_{21,\text{bbr}}=\frac{\gamma_\text{rad}}{\exp(\hbar\omega_a/kT_\text{rad})}\int_{-\infty}^{\infty}\frac{2}{\pi\Delta\omega_a}\frac{d\omega}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}$

But the integral on the right-hand side of this equation has unity area independent of its linewidth $$\Delta\omega_a$$; hence we obtain the very simple and fundamental result that

$\tag{39}W_{12,\text{bbr}}=W_{21,\text{bbr}}=\frac{\gamma_\text{rad}}{\exp(\hbar\omega_a/kT_\text{rad})-1}$

These blackbody-stimulated transition rates turn out to be independent of any properties of the atomic transition except its radiative decay rate $$\gamma_\text{rad}$$.

The very basic result that we obtain here is thus that the stimulated transition rate between any two atomic levels caused by blackbody fields depends only on the radiative decay rate for that transition, and on the Boltzmann factor at the temperature of the radiation, and on nothing else. In particular this result does not depend at all on the linewidth, or even the lineshape, of the transition.

### Power Absorption from the Surroundings?

This argument says that even without any externally applied signals, thermal-noise-stimulated transitions or "jumps" will be continually taking place in both directions between any two energy levels $$E_1$$ and $$E_2$$, with stimulated-transition probabilities $$W_{12,\text{bbr}}$$ and $$W_{21,\text{bbr}}$$ given by Equation 4.39.

More precisely, for two energy levels $$E_i$$ and $$E_j\gt{E_i}$$ having level degeneracies $$g_i$$ and $$g_j$$, respectively, these thermally stimulated transition rates will be given by

$\tag{40}W_{ji,\text{bbr}}=\frac{g_i}{g_j}W_{ij,\text{bbr}}=\frac{\gamma_{\text{rad},ji}}{\exp(\hbar\omega_{ji}/kT_\text{rad})-1}$

These transitions are caused entirely by the unavoidable blackbody radiation fields in which the atoms are always immersed (unless the electromagnetic surroundings can be cooled all the way to absolute zero).

But this in turn implies that there will necessarily be net power absorption, proportional to the atomic population difference $$\Delta{N}=N_1-N_2$$, from the blackbody fields to the atoms.

In other words, the blackbody fields will be continuously delivering energy, or heat, to the atoms through these stimulated transitions. But this in turn raises serious questions about thermal equilibrium between the atoms and the surroundings.

How can a collection of atoms, which are nominally in thermal equilibrium, remain in equilibrium if they are continually absorbing energy from their surroundings? Even more serious, how can a collection of atoms which are supposedly at an atomic temperature $$T_a$$ (defined by the Boltzmann ratio) continually absorb energy from surroundings that might be at a different thermodynamic temperature $$T_\text{rad}$$—especially if the surrounding temperature $$T_\text{rad}$$ might in some cases be colder than the atomic temperature $$T_a$$?

### Power Emission to the Surroundings

The answer to these questions comes in remembering that there will also be in any atomic system purely spontaneous and entirely downward transitions, due to the spontaneous emission or radiative decay from the upper-level atoms; and these spontaneous transitions or purely radiative decays will transfer power from the atoms back to the electromagnetic surroundings, with a spontaneous decay rate given by $$\gamma_\text{rad}$$.

These spontaneous downward transitions in the atoms are to be viewed as genuinely "spontaneous" and not as "noise-induced" transitions, at least in the approach we are taking here, since they simply occur spontaneously in a manner explainable only by quantum theory.

However, we will see that these spontaneous-emission transitions from the atoms to the surroundings can and do exactly balance the noise-stimulated absorption from the surroundings to the atoms, when the atoms are in thermal equilibrium with their electromagnetic surroundings.

(Some people find it helpful to describe the spontaneous downward transitions as being "one-way stimulated transitions" which are stimulated by quantum zero-point fluctuations in the electromagnetic field; but we will not get involved in that argument here.)

### Thermal Balance with the Electromagnetic Surroundings

Figure 4.7 shows schematically the overall transfer of energy that takes place in both directions between a collection of atoms and their "electromagnetic surroundings," through stimulated absorption and emission of blackbody radiation, plus spontaneous emission of energy from the atoms to the surroundings.

Each arrow in Figure 4.7 indicates the direction and magnitude of an energy flow. The ratio of energy flow from the atoms into the surroundings caused by blackbody-stimulated plus spontaneous emission, compared to energy flow in the reverse direction due to blackbody-stimulated absorption, is given by

\tag{41}\begin{align}\frac{\text{energy flow out of atoms}}{\text{energy flow into atoms}}&=\frac{(W_{21,\text{bbr}}+\gamma_\text{rad})N_2}{W_{12,\text{bbr}}N_1}\\&=\frac{W_{21,\text{bbr}}+\gamma_\text{rad}}{W_{12,\text{bbr}}}\times\frac{N_2}{N_1}\end{align}

Now, the population ratio in a collection of two-level atoms can be described at any instant by an "atomic temperature" $$T_a$$, in the sense that the Boltzmann ratio between the energy-level populations is given by

$\tag{42}\frac{N_2}{N_1}=\exp\left(-\frac{\hbar\omega_a}{kT_a}\right)$

At the same time, by using Equation 4.39 the ratio of spontaneous and noise-stimulated emission rates to noise-stimulated absorption rates is related to the temperature $$T_\text{rad}$$ of the electromagnetic surroundings by

$\tag{43}\frac{W_{21,\text{bbr}}+\gamma_\text{rad}}{W_{12,\text{bbr}}}=\exp\left(\frac{\hbar\omega_a}{kT_\text{rad}}\right)$

The ratio of the energy flow rates in the two directions is thus given, in terms of the temperatures of the atoms and the surroundings, by

$\tag{44}\frac{\text{energy flow out of atoms}}{\text{energy flow into atoms}}=\exp\left(\frac{\hbar\omega_a}{kT_\text{rad}}-\frac{\hbar\omega_a}{kT_a}\right)$

These rates will be equal and opposite if and only if the atomic temperature $$T_a$$ exactly equals the surrounding electromagnetic temperature $$T_\text{rad}$$.

The net energy received by the atoms from the blackbody fields will thus, at thermal equilibrium, exactly equal the energy radiated back to the surroundings by the atoms. There will be no net flow of atoms between levels $$E_1$$ and $$E_2$$, and no net power transfer between atoms and surroundings—as should certainly be the case at thermal equilibrium.

### Discussion: Thermal Equilibrium

There are several very fundamental conclusions that can be drawn from the preceding analytical results.

First, the existence of a spontaneous, purely downward emission in any collection of atoms appears to be essential, if for no other reason than to maintain energy balance with the atomic surroundings at thermal equilibrium.

A collection of atoms in thermal equilibrium at any finite temperature will always have a net power absorption on its atomic transitions; and the volume containing these atoms will always have finite blackbody signals to be absorbed by the atoms (unless the surroundings are at absolute zero).

The atoms will therefore always absorb energy from the blackbody fields, producing a net flow of atoms into the upper energy levels.

These upper-level atoms must then spontaneously drop down and radiate away energy at a rate given by $$\gamma_\text{rad}$$ times the number of atoms in the upper level. This energy reradiation will exactly equal the energy that the same atoms inevitably absorb from the blackbody radiation fields in which they are immersed, if the atoms and the surroundings are at the same temperature.

In the more general situation, the atomic temperature $$T_a$$ of a collection of atoms and the electromagnetic temperature $$T_\text{rad}$$ of their surroundings might be different, at least on a temporary basis.

That is, the atoms might be in internal thermal equilibrium at a temperature $$T_a$$, in the sense that all the phases of individual atomic oscillations are fully dephased or randomized, and all level populations satisfy the Boltzmann ratios with this temperature value.

This temperature might, for example, be relatively hot because the atoms have been immersed in a hot environment. These atoms might then be suddenly moved into an enclosure which has walls at a substantially colder (or hotter) temperature $$T_\text{rad}$$.

The atoms will now form one thermal reservoir at temperature $$T_a$$, and the walls and the blackbody radiation will form another reservoir at $$T_\text{rad}\ne{T_a}$$. Whichever is hotter, energy will flow from the hotter system to the colder.

The total system will eventually come to a thermal equilibrium at some temperature in between the initial temperatures, depending on the relative heat capacities of the two systems. This kind of "atomic transition calorimetry" can in fact be carried out experimentally, on nuclear magnetic transitions, for example.

### Detailed Balance

Overall thermal equilibrium requires, in fact, that the blackbody absorption and spontaneous emission rates be in exact equilibrium transition by transition, for each one of the $$E_i\rightarrow{E_j}$$ pairs in a collection of multilevel atoms.

This necessity for the net absorption and spontaneous emission to be in balance on each individual transition at thermal equilibrium is sometimes referred to as "detailed balance."

Detailed balance applies, in fact, not just transition by transition, but also frequency component by frequency component within any single transition: the net absorption rate by the atoms at any frequency $$\omega$$ and the spontaneous emission in a very narrow range $$d\omega$$ about that same $$\omega$$ must also balance.

An atomic transition must, therefore, by fundamental thermodynamic arguments, have exactly the same atomic lineshape for spontaneous emission as it does for stimulated absorption, whether this lineshape be lorentzian, gaussian, or whatever.

The simple relationship derived in Equation 4.39 between $$W_{ij,\text{bbr}}$$ and $$\gamma_{\text{rad},ji}$$ is therefore hardly accidental. This relation is rather a basic and necessary condition for thermal equilibrium to ensue.

The same relation between $$W_{ij,\text{bbr}}$$ and $$\gamma_{\text{rad},ji}$$ must hold generally for any kind of stimulated transition, with any lineshape or form of tensor response, and any order of electric or magnetic dipole or multipole character.

The direct proportionality we noted earlier between the stimulated response $$\tilde{\chi}(\omega)$$ and the spontaneous emission rate $$\gamma_\text{rad}$$ for an atomic transition is also a necessary consequence of the balance between net blackbody absorption and spontaneous emission that is required to reach thermal equilibrium.

The logical arguments we have developed here might be represented by Figure 4.8. The stimulated-transition circle indicates the processes of noise-stimulated absorption and emission in a collection of atoms.

These noise-stimulated processes can be derived by a semiclassical derivation—that is, a derivation in which the atoms are quantized but the electromagnetic fields are not.

The blackbody-radiation and spontaneous-emission circles then indicate the existence of these two phenomena, either of which can be derived independently of the other, but only by employing a full quantum electrodynamic calculation in which the electromagnetic field itself is quantized.

The connecting arrows then indicate that we can use the existence of any two of these phenomena, plus the criterion of thermal equilibrium, to derive the existence and magnitude of the third.

It is a matter of choice, for example, whether we begin with the existence of blackbody radiation, and then use this to imply the necessity for spontaneous emission; or whether we take some other direction around the circle. Any two of these processes imply the third.

The total energy-decay rates for quantum energy levels in atoms can involve both radiative and nonradiative transfer of energy from atoms to their surroundings.

In a broader viewpoint, therefore, we must really be concerned with the total atomic relaxation processes that result from interactions between the atoms and their thermal surroundings, both through electromagnetic or "radiative" interactions and through nonelectromagnetic or "nonradiative" interactions.

In this section we will try to make clear how an atomic transition interacts with both its electromagnetic and its nonelectromagnetic surroundings; how these interactions lead to both radiative and nonradiative decay; and how these in turn lead to two different but similar kinds of relaxation transitions associated with these two mechanisms.

### Radiative Relaxation Rates and Transition Probabilities

In the preceding section we obtained the remarkable and very fundamental result that blackbody radiation from the "electromagnetic surroundings" of a nondegenerate two-level atom will cause "blackbody stimulated transitions" with upward and downward transition probabilities given by

$\tag{45}W_{12,\text{bbr}}=W_{21,\text{bbr}}=\frac{\gamma_\text{rad}}{\exp(\hbar\omega_a/kT_\text{rad})-1}$

where $$T_\text{rad}$$ is the temperature of the electromagnetic surroundings. This is a very fundamental relationship. We can view it as being imposed by the necessity for thermal equilibrium between the rate at which an atom spontaneously radiates energy and the rate at which it absorbs energy from blackbody fields.

The transition rates $$W_{12,\text{bbr}}$$ and $$W_{21,\text{bbr}}$$ are thus from one viewpoint stimulated transitions caused by the real (if weak), random, omnipresent blackbody radiation fields. The existence of these fields depends only on the temperature of the surroundings, however, and on nothing else.

There is nothing we can do in practice to control or modify these blackbody fields (short of cooling everything in the vicinity down toward absolute zero). Hence we may just as well think of the blackbody-stimulated transition rates as being part of the relaxation mechanisms which are always present among the atomic-level populations, independent of anything that we ourselves do.

In earlier tutorials we spoke for simplicity only of energy decay, i.e., only of spontaneous downward relaxation from upper levels to lower levels.

The possibility of "upward relaxation," caused by energy coming back from the thermal surroundings to the atoms, was not mentioned. We are now seeing that, in a complete and accurate description, when an atom is coupled to external surroundings it can do more than just relax downward and give energy to those surroundings, as we said earlier.

It can also (but with inherently lower probability) receive energy from its thermal surroundings and be lifted or relaxed upward in energy. This is directly related to the fact that in thermal equilibrium there are always some numbers of atoms, given by the Boltzmann ratios, in upper energy levels (though these may be very small numbers).

At any temperature greater than absolute zero, the atoms never relax completely into the lowest energy level, as would always happen if only downward relaxation occurred.

Of course, for optical-frequency transitions at room temperature, the Boltzmann ratio is enormously small ($$\approx10^{-36}$$). Both the upper-level populations and the upward relaxation rates are truly negligible, and only downward relaxation need be considered.

For lower frequencies and more closely spaced levels, however, Boltzmann ratios and upward relaxation rates do need to be taken into account, and therefore we do need to understand the full situation described here.

For microwave and lower-frequency transitions, in fact, the Boltzmann ratio becomes nearly unity, and upward and downward relaxation rates become very nearly equal.

### Nonradiative Relaxation Rates and Transition Probabilities

Blackbody relaxation and energy-exchange mechanisms represent, however, only the interactions of the atoms with their electromagnetic surroundings, acting through the blackbody radiation and the radiative decay rate. These interactions are shown in a schematic form in the top part of Figure 4.9.

We must recognize, however, that real atoms will usually also be in thermal contact with what we will refer to, in general terms, as "other surroundings" or "nonradiative surroundings," as shown schematically in Figure 4.9(b).

These nonradiative surroundings, to which the atoms can also be coupled, can include a crystal lattice in which the laser atoms are imbedded; or a surrounding liquid medium in which the laser molecules are dissolved; or other atoms or walls with which the atoms of interest are colliding in a gas.

The atoms may then exchange energy with these "nonradiative surroundings" by means of the nonradiative decay processes that are included in the nonradiative decay rate $$\gamma_\text{nr}$$, in essentially the same way as the atoms exchange energy with the "electromagnetic surroundings" through the purely radiative processes that are involved in $$\gamma_\text{rad}$$.

But this necessarily implies, from the same kind of thermodynamic reasoning we employed earlier, that these "nonradiative surroundings" must also be able to cause "nonradiatively stimulated transitions" between the atomic levels, with stimulated-transition probabilities $$W_{12,\text{nr}}$$ and $$W_{21,\text{nr}}$$ in a manner exactly analogous to the blackbody transitions $$W_{12,\text{bbr}}$$ and $$W_{21,\text{bbr}}$$ described earlier.

These additional transitions we will refer to generally as nonradiative relaxation transitions. The basic physics involved in the nonradiative interaction of a collection of atoms with their "nonradiative surroundings" will then be the same in every important aspect as that of the radiative interaction of these same atoms with their electromagnetic surroundings.

### Example: Phonon Interactions in Crystal Lattices

As a specific example of this, let us consider the interaction between a collection of laser or maser atoms and the lattice vibrations in a surrounding host crystal lattice, since this is one important type of "nonradiative surroundings."

A crystal lattice containing laser atoms can propagate acoustic waves, often referred to as phonons, at many different frequencies and in many different directions, just as a vacuum or dielectric medium can propagate electromagnetic waves, or photons.

Moreover, like electromagnetic waves, these acoustic waves can interact with atomic transitions of atoms contained in the crystal lattice, and can produce stimulated transitions and induced atomic responses.

That is, there will generally be some weak coupling or interaction between the quantum wave function of an atom imbedded in a crystal lattice and the acoustic vibrations in the surrounding crystal lattice.

This coupling is very analogous to the weak electric-dipole coupling between the atomic wave functions and the electromagnetic vibrations (fields) in the surrounding electromagnetic "ether."

The basic physical principles that apply to electromagnetic interactions with atoms therefore apply in almost exactly the same way to what we may call generalized acoustic interactions with the atoms.

For example, a coherent acoustic signal in the form of a lattice vibrational wave at a frequency $$\omega$$ near an atomic transition frequency $$\omega_a$$ can be absorbed or amplified through its interaction with the atoms, just like an electromagnetic wave; and this absorption or amplification of the acoustic wave will depend on the atomic population difference (and the atomic linewidth and lineshape) exactly like an electromagnetic wave interaction.

It is entirely possible to use an inverted atomic population to amplify acoustic waves and to produce acoustic-wave oscillation in a crystal at the atomic transition frequency.

Such "acoustic lasers" or "acoustic masers" have been experimentally demonstrated at microwave frequencies, using some of the same pumping methods and maser materials used to produce electromagnetic maser oscillation at the same frequencies on the same transitions.

### Acoustic Transition Rates

Of more importance to us here is the fact that at any finite temperature such a crystal lattice will have thermal lattice vibrations, or "blackbody acoustic radiation," which is exactly analogous in character to blackbody electromagnetic fields (although the appropriate energy density formulas are somewhat different).

These thermally induced vibrations represent the heat content of the crystal lattice, and as such can be characterized by a lattice temperature which we will label more generally as $$T_\text{nr}$$, with the subscripts standing for "nonradiative surroundings". The lattice vibrations of course go to zero only if the lattice temperature $$T_\text{nr}$$ itself goes to absolute zero.

A critically important point is that the atoms will then be affected by these thermal lattice vibrations in the surrounding crystal, in basically the same way that they are affected by the blackbody radiation in the electromagnetic surroundings.

To describe this interaction we must use exactly the same arguments as for the electromagnetic surroundings, but now we refer to interactions with the "nonradiative" or lattice surroundings rather than with the "electromagnetic surroundings."

In fact, by invoking the necessity for detailed thermal balance in the energy transfer processes between the atoms and the lattice acoustic modes, we can argue that these acoustically stimulated transition rates $$W_{12,\text{nr}}$$ and $$W_{21,\text{nr}}$$ must be related to the nonradiative decay rate $$\gamma_\text{nr}$$ by exactly the same fundamental relationship as Equation 4.45 for the radiative case, namely,

$\tag{46}W_{12,\text{nr}}=W_{21,\text{nr}}=\frac{\gamma_\text{nr}}{\exp(\hbar\omega_a/kT_\text{nr})-1}$

Only if these relations hold will the power delivered to the atoms by the surrounding lattice through the $$W_{12,\text{nr}}$$ and $$W_{21,\text{nr}}$$ transitions always be exactly balanced, under thermal equilibrium conditions, by the power delivered from the atoms back to the nonradiative lattice surroundings through $$\gamma_\text{nr}$$.

This equation applies, in fact, in a completely general fashion, not just to the interaction of atoms with acoustic lattice surroundings in crystals, but also to the nonradiative interactions of a collection of quantum atoms with any kind of nonradiative thermal surroundings.

That is, suppose the upper-level atoms in a collection of atoms do in fact lose some of their excitation energy by transferring energy into any kind of "nonradiative surroundings," whether to a surrounding crystal lattice or cell walls, or by collisions with other atoms in a gas mixture.

Suppose this energy loss rate is described by a nonradiative decay rate $$\gamma_\text{nr}$$ times the upper-level population, and that the surroundings which receive this energy are describable by a temperature $$T_\text{nr}$$.

These other surroundings must then necessarily produce upward and downward thermally stimulated transitions on the same transition in the collection of atoms, with thermally stimulated transition probabilities $$W_{12,\text{nr}}$$ and $$W_{21,\text{nr}}$$ exactly as given by Equation 4.46.

We use the notations $$W_\text{nr}$$ and $$T_\text{nr}$$ in this equation, and in Figure 4.9(b), to emphasize that the net interaction with any "nonradiative thermal surroundings" is completely analogous to the interaction of the same atoms with the blackbody radiation surroundings, even though electromagnetic radiation and blackbody radiation fields in the usual sense are not involved.

The nonradiative decay rate $$\gamma_\text{nr}$$ thus plays the same role in interacting with any kind of "nonradiative surroundings" as the radiative decay rate $$\gamma_\text{rad}$$ plays in interacting with the radiative or electromagnetic surroundings.

The generalization of Equation 4.46 to degenerate transitions is also the same as for the electromagnetic case, namely,

$\tag{47}W_{ji,\text{nr}}=\frac{g_i}{g_j}W_{ij,\text{nr}}=\frac{\gamma_{\text{nr},ji}}{\exp(\hbar\omega_{ji}/kT_\text{nr})-1}$

The combined influence of radiative and nonradiative interactions for any collection of atoms (actually for any single transition in a collection of atoms) can then be illustrated by an expanded diagram like Figure 4.9(b), in which we indicate separately the interactions and the relaxation transition rates for the radiative and the nonradiative surroundings.

The only significant parameters in these interactions are the two relaxation rates $$\gamma_\text{rad}$$ and $$\gamma_\text{nr}$$, and the associated temperatures of the surroundings $$T_\text{rad}$$ and $$T_\text{nr}$$, respectively.

These two temperatures $$T_\text{rad}$$ and $$T_\text{nr}$$ will usually have the same value; but in special cases the temperature $$T_\text{nr}$$ of the "nonradiative surroundings" could be different from the temperature $$T_\text{rad}$$ of the electromagnetic surroundings.

Suppose the crystal lattice of an atomic medium is essentially lossless and transparent to electromagnetic radiation at all frequencies of interest, so that the lattice itself is not part of the electromagnetic surroundings.

The temperature $$T_\text{nr}$$ characteristic of the lattice vibrations when the crystal is cooled, for example, in a liquid helium bath, may be much colder than the temperature Trad of the warmer electromagnetic surroundings seen by the atoms through the windows of the helium dewar.

### Another Nonradiative Example: Inelastic Collisions in Gases

As another example of nonradiative interactions, suppose that excited atoms of type A in a mixture of two different gases can lose some of their excitation energy through inelastic collisions with atoms of type B, with this energy going into heating up the kinetic motion of the type B atoms.

This is a form of nonradiative decay for the excited atoms of type A, which can be accounted for by a nonradiative decay rate $$\gamma_\text{nr}$$ (which will probably be directly proportional to the pressure or density of the atoms of type B).

From the same arguments.as before, these same collisions must then also produced collision-stimulated transitions in both directions between the levels of the type A atoms, with transition rates $$W_{12,\text{nr}}$$ and $$W_{21,\text{nr}}$$ given by Equation 4.47, and with $$T_\text{nr}$$ given by the kinetic temperature of the type B atoms.

The physical details of how the kinetic motion of the type B atoms can react back to produce collision-stimulated upward and downward transitions in the type A atoms may not be particularly obvious; and it is certainly not at all clear how we might use a population inversion in the type A atoms to "amplify" the type B kinetic motion.

The general rule is, however, that if a collection of excited atoms can deliver energy in any fashion to some part of their nonradiative surroundings, then they are in some way coupled to those surroundings.

As a result, these "other surroundings" are necessarily coupled back to the atoms, and thermal fluctuations in these "other surroundings" can cause upward and downward thermally stimulated transition rates in the atomic system by acting through the same nonradiative interaction mechanisms.

Note in this instance that collisions between atoms in a gas may contribute to the homogeneous line broadening of transitions in these atoms in either of two distinct ways.

Elastic collisions between atoms cause dephasing effects, and thus give a homogeneous line-broadening contribution $$2/T_2$$ which is directly proportional to the collision frequency and thus to the gas pressure.

Inelastic collisions may cause both additional dephasing and an additional nonradiative energy decay term $$\gamma_\text{nr}$$, which will in turn give an additional pressure-dependent lifetime broadening contribution.

### Total Relaxation Transition Rates

It is important to understand how there can be separate but essentially similar relaxation effects produced by both the radiative and the nonradiative surroundings, as illustrated in Figure 4.9(b).

Once we understand the underlying physics, however, it is then much simpler to combine these two effects (including the spontaneous relaxation effects) into a single pair of thermally stimulated relaxation transition probabilities, which we will henceforth denote by $$w_{12}$$ and $$w_{21}$$, and which are defined as follows.

Let the transition rate or flow rate (in atoms/second) in the downward direction due to all these interactions be written in the form

\tag{48}\begin{align}\left|\frac{dN_2}{dt}\right|_{\begin{split}\text{downward}\\\text{relaxation}\end{split}}&=(W_{21,\text{bbr}}+\gamma_\text{rad}+W_{21,\text{nr}}+\gamma_\text{nr})N_2\\&=w_{21}N_2\end{align}

and let the corresponding flow rate in the upward direction be written as

\tag{49}\begin{align}\left|\frac{dN_1}{dt}\right|_{\begin{split}\text{upward}\\\text{relaxation}\end{split}}&=(W_{12,\text{bbr}}+W_{12,\text{nr}})N_1\\&=w_{12}N_1\end{align}

Obviously we then have

$\tag{50}w_{21}\equiv{W}_{21,\text{bbr}}+W_{21,\text{nr}}+\gamma_\text{rad}+\gamma_\text{nr}$

in the downward direction, and

$\tag{51}w_{12}\equiv{W}_{12,\text{bbr}}+W_{12,\text{nr}}$

in the upward direction.

The downward relaxation transition probability $$w_{21}$$ includes both the thermally stimulated downward transitions and the spontaneous emission transitions from both radiative and nonradiative mechanisms, whereas the upward transition probability $$w_{12}$$ represents the thermally stimulated upward transitions due to both mechanisms.

Figure 4.10 illustrates these net relaxation rates between any pair of atomic levels. For an arbitrary pair of levels $$E_i$$ and $$E_j\gt{E_i}$$, the downward relaxation probability must be written as

$\tag{52}w_{ji}\equiv{W}_{ji,\text{bbr}}+W_{ji,\text{nr}}+\gamma_{\text{rad},ji}+\gamma_{\text{nr},ji}$

and the upward relaxation probability on the same transition is written as

$\tag{53}w_{ij}\equiv{W}_{ij,\text{bbr}}+W_{ij,\text{nr}}$

We will from here on use these lowercase notations $$w_{12}$$ and $$w_{21}$$, or more generally $$w_{ij}$$ and $$w_{ji}$$, as defined above, to indicate the total relaxation transition probabilities (per atom and per unit time) in the upward and downward directions between any two levels $$i$$ and $$j$$, due to all the purely thermal interactions plus energy decay processes connecting the atoms to their surroundings.

Also, from now on we will restrict the uppercase symbols $$W_{12}$$ and $$W_{21}$$, or more generally $$W_{ij}$$ and $$W_{ji}$$, to indicate signal-stimulated transition probabilities that are produced by external signals or pumping mechanisms that we either deliberately apply to the atoms, or that we allow to build up in a laser cavity, as shown schematically in Figure 4.9(c).

That is, from here on the uppercase $$W_{ij}$$'s signify deliberately induced transition probabilities that we can turn off or suppress; the lowercase $$w_{ij}$$'s are relaxation transition probabilities that we can in essence do nothing about (except possibly by cooling the surroundings).

### Boltzmann Relaxation Ratios

Note that if the surroundings of an atom, radiative and nonradiative, are both at the same equilibrium temperature $$T_\text{nr}=T_\text{rad}=T$$, then the preceding expressions show that the ratio between upward and downward relaxation probabilities is always given by the Boltzmann ratio

$\tag{54}\frac{w_{12}(\uparrow)}{w_{21}(\downarrow)}=e^{-\hbar\omega_a/kT}$

or, more generally,

$\tag{55}\frac{w_{ij}}{w_{ji}}=\frac{g_j}{g_i}\exp\left(-\frac{E_j-E_i}{kT}\right)$

where $$T$$ is the temperature of the thermal surroundings. The upward thermally induced relaxation rate is always smaller (and on optical-frequency transitions usually much smaller) than the combination of downward thermally induced relaxation plus energy decay.

This Boltzmann relation does not depend on the nature or the strength of the radiative and/or nonradiative relaxation mechanisms that may be present; it will hold if they are all at the same temperature $$T$$.

If the radiative and nonradiative surroundings are somehow at different temperatures, however, each interaction must be considered separately, and this ratio becomes somewhat more complicated.

### Optical Frequency Approximation

A convenient rule of thumb for visible frequencies is that the equivalent temperature corresponding to $$\hbar\omega_a/k$$ is $$\approx$$ 25,000 K. For any reasonable temperature $$T$$ of the surroundings, therefore, the Boltzmann ratio at optical frequencies is always very small, on the order of

$\tag{56}\exp(-\hbar\omega_a/kT)\approx\exp(-25,000/300)\approx10^{-36}$

The thermally stimulated terms in the relaxation rates, either upward or downward, are then totally negligible compared to the spontaneous emission rates, and the relaxation transition probabilities in the two directions can be approximated by

$\tag{57}w_{ij}(\uparrow)\approx0\qquad(\text{upward direction})$

and

$\tag{58}w_{ji}(\downarrow)\approx\gamma_{ji}\equiv\gamma_{\text{rad},ji}+\gamma_{\text{nr},ji}\qquad(\text{downward direction})$

When we write out the rate equations for lower-frequency transitions, such as for magnetic resonance or microwave maser experiments, where the photon energy $$\hbar\omega$$ is $$\ll{kT}$$, then the relaxation terms in both upward and downward directions must be included; and we must use the more complete formulation involving the relaxation probabilities $$w_{ij}$$ and $$w_{ji}$$ in both upward and downward directions.

The simplified notation using only $$\gamma_{ji}$$ terms and including relaxation or energy decay in the downward direction only is more commonly employed in optical-frequency and laser analyses, where the optical-frequency approximation is almost always valid. Infrared and submillimeter laser transitions fall somewhere in between, and may require use of the more complete formulation on at least some of the transitions.

## 5. Two-Level Rate Equations and Saturation

The stimulated transition probabilities and relaxation transition probabilities derived in the preceding sections of this tutorial can now be used to write the general rate equations for any atomic system, taking into account both applied signals and relaxation processes.

In this section we will explore the rate-equation solutions for an ideal two-level system. This will allow us to introduce a number of useful concepts, particularly the idea of saturation of the population difference $$\Delta{N}$$ at high enough applied signal levels.

### Two-Level Rate Equation

In a simple two-level atomic system with an applied signal present, atoms flow from level 1 to level 2 at a rate $$(W_{12}+w_{12})N_1$$, and from level 2 to level 1 at a rate $$(W_{21}+w_{21})N_2$$, as illustrated in Figure 4.11.

The total rate equation for the level populations $$N_1$$ and $$N_2$$ in this system is

$\tag{59}\frac{dN_1(t)}{dt}=-\frac{dN_2(t)}{dt}=-[W_{12}+w_{12}]N_1(t)+[W_{21}+w_{21}]N_2(t)$

If the energy levels have no degeneracy, the stimulated-transition rates are related by $$W_{12}=W_{21}$$, and the relaxation rates are related by $$w_{12}/w_{21}=\exp(-\hbar\omega_a/kT)$$, where $$T$$ is the temperature of the surroundings of the atoms.

For a two-level system, however, it is usually more convenient to work with the total number of atoms $$N_1(t)+N_2(t)=N$$ and the population difference $$N_1(t)-N_2(t)=\Delta{N}(t)$$.

Since the thermal equilibrium populations $$N_{10}$$ and $$N_{20}$$ with no signal present are related by the Boltzmann ratio $$N_{20}/N_{10}=\exp(-\hbar\omega_a/kT)$$, the population difference $$\Delta{N_0}$$ on a nondegenerate two-level transition at thermal equilibrium, with no applied signal, can be written as

$\tag{60}\Delta{N_0}\equiv{N}_{10}-N_{20}=\frac{w_{21}-w_{12}}{w_{12}+w_{21}}N=N\tanh(\hbar\omega_a/2kT)$

For a simple system with just two levels and a fixed total population, only one rate equation for the population difference $$\Delta{N}(t)$$ is then really needed. The equations for $$dN_1(t)/dt$$ and $$dN_2(t)/dt$$ can be combined into a single rate equation in the form

$\tag{61}\frac{d}{dt}\Delta{N(t)}=-(W_{12}+W_{21})\Delta{N(t)}-(w_{12}+w_{21})\left(\Delta{N(t)}-\frac{w_{21}-w_{12}}{w_{12}+w_{21}}N\right)$

We can make this equation appear even simpler by using the fact that $$W_{12}=W_{21}$$ for the signal-stimulated transition probability, and by defining a two-level energy relaxation time or population recovery time $$T_1$$ by

$\tag{62}w_{12}+w_{21}\equiv{1/T_1}$

If we also recognize that the final term in Equation 4.61 is just the thermal-equilibrium population difference $$\Delta{N_0}$$ for the atoms in equilibrium with the surroundings at temperature $$T_\text{rad}$$, then this two-level rate equation takes on the particularly simple and yet very general form

$\tag{63}\frac{d}{dt}\Delta{N(t)}=-2W_{12}\Delta{N(t)}-\frac{\Delta{N(t)}-\Delta{N_0}}{T_1}$

This particularly simple form for the ideal two-level case with fixed total population turns out to be very useful and important for describing a great variety of laser and maser phenomena.

### Physical Interpretation: The Population Recovery Time $$T_1$$

Understanding this two-level rate equation is important for understanding many subsequent aspects of laser behavior.

For example, the relaxation term on the right-hand side of Equation 4.63, namely, $$-[\Delta{N(t)}-\Delta{N_0}]/T_1$$, obviously causes the population difference $$\Delta{N}(t)$$ to relax toward its thermal equilibrium value $$\Delta{N_0}$$ in the absence of an applied signal, with an exponential time constant $$T_1$$.

This time constant $$T_1$$ is therefore often called the population recovery time or the energy relaxation time of the system.

Suppose the two-level transition is an optical-frequency transition with $$\hbar\omega_a\gg{kT}$$. The upward relaxation probability $$w_{12}$$ is then essentially zero, whereas the downward relaxation probability $$w_{21}$$ is essentially the upper-level energy decay rate $$\gamma_{21}$$ as, we discussed earlier.

The definition of $$T_1$$ therefore becomes

$\tag{64}1/T_1\equiv{w}_{12}+w_{21}\approx\gamma_{21}\equiv1/\tau_{21}$

In the optical-frequency limit, the time constant $$T_1$$ is thus the same thing as the total lifetime or energy decay time $$\tau_{21}$$ of the upper energy level.

In contrast, the stimulated signal term $$-2W_{12}\Delta{N(t)}$$ on the right-hand side of Equation 4.63 obviously acts to drive the population difference $$\Delta{N(t)}$$ toward zero, that is, to saturate the population difference.

The stimulated-transition probability $$W_{12}$$ is, of course, proportional to the strength of the applied signal, and so the rate at which $$\Delta{N(t)}$$ is driven toward zero is proportional to the applied signal intensity.

Note that the factor of 2 appears in front of this stimulated term because the transition of a single atom from level 1 to level 2 both reduces $$N_1(t)$$ by one and increases $$N_2(t)$$ by one, and thus changes $$\Delta{N(t)}$$ by twice that much.

The steady-state behavior of the population difference $$\Delta{N}$$ in the presence of an applied signal $$W_{12}$$ must be a balance between these competing population-recovery and population-saturation effects. To obtain the steady-state solution, we can set the total time derivative in the rate equation equal to zero, i.e.,

$\tag{65}\frac{d}{dt}\Delta{N}=0=-2W_{12}\Delta{N}-\frac{\Delta{N}-\Delta{N_0}}{T_1}$

and obtain from this the steady-state population difference

$\tag{66}\Delta{N}=\Delta{N_\text{ss}}\equiv\Delta{N_0}\times\frac{1}{1+2W_{12}T_1}$

The ratio of the steady-state value $$\Delta{N_\text{ss}}$$ with signal present to the thermal-equilibrium value $$\Delta{N_0}$$ with no applied signal is plotted versus applied signal strength $$W_{12}$$ in Figure 4.12.

We see that as the applied signal strength or stimulated-transition rate $$W_{12}$$ increases, the steady-state population difference $$\Delta{N_\text{ss}}$$ is driven below the small-signal or thermal-equilibrium value $$\Delta{N_0}$$, and eventually is driven toward zero at large enough applied signal levels.

This steady-state value of the population difference results from a balance between the stimulated-transition term, which acts to transfer atoms from the more heavily populated level $$N_1$$ toward the less heavily populated level $$N_2$$, and thus tends to equalize the populations, and the relaxation term, which tends to pull $$\Delta{N}$$ back toward its thermal-equilibrium value $$\Delta{N_0}$$.

This reduction in the steady-state population difference with increasing signal strength has the general form

$\tag{67}\frac{\Delta{N_\text{ss}}}{\Delta{N_0}}=\frac{1}{1+W_{12}/W_\text{sat}}=\frac{1}{1+\text{const}\times\text{signal power}}$

where $$W_\text{sat}\equiv1/2T_1$$ is the value of the stimulated-transition probability at which the population difference is driven down to exactly half its initial or small-signal value.

This form of reduction in population difference with increasing signal strength is generally referred to as homogeneous saturation of the population difference on the two-level transition.

### Saturation in Real Laser Systems

This general type of saturation behavior is extremely important in laser theory.

Gain coefficients and loss coefficients in laser materials are directly proportional to the population difference on the laser transition. We will see later on that in a great many atomic systems the population difference on the atomic transition will very often saturate with increasing signal strength in the form given by Equation 4.67, even for initially inverted population differences produced by laser pumping.

As a result, either the attenuation coefficient or the gain coefficient $$\alpha_m$$ in an atomic medium will very often saturate with increasing signal intensity $$I$$ in the general fashion given by

$\tag{68}\alpha_m=\alpha_m(I)=\alpha_{m0}\times\frac{1}{1+I/I_\text{sat}}=\alpha_{m0}\times\frac{1}{1+\text{const}\times\text{signal power}}$

where $$\alpha_{m0}$$ is the small-signal (unsaturated) attenuation or gain coefficient; $$I$$ is the applied signal intensity (usually expressed as power per unit area); and $$I_\text{sat}$$ is a saturation intensity at which the gain or loss coefficient is saturated down to half its initial value $$\alpha_{m0}$$.

This form of saturation behavior is often referred to as homogeneous saturation, since it is characteristic of homogeneously broadened transitions. Inhomogeneously broadened transitions, such as doppler-broadened lines, exhibit a more complex saturation behavior, including "hole burning" effects, which we will describe in a later tutorial.

### Saturable Absorption and Saturable Gain

Materials specially chosen to operate as saturable absorbers are often used in laser experiments for $$Q$$-switching, mode-locking, and isolation from low-level leakage signals.

On the other hand, saturation of the inverted population difference and hence the gain in an amplifying laser medium is what determines a laser's power output.

When a laser oscillator begins to oscillate, the oscillation amplitude grows at first until the intensity inside the cavity is sufficient to saturate down the laser gain exactly as we have described.

Steady-state oscillation then occurs when the saturated laser gain becomes just equal to the total cavity losses, so that the net round-trip gain is exactly unity. Gain saturation is thus the primary mechanism that determines the power level at which a laser will oscillate.

Note that the reactive susceptibility $$\chi'(\omega)$$, and hence the phase shift on an atomic transition, is also directly proportional to the population difference $$\Delta{N}$$. An atomic transition will thus exhibit both saturable absorption or gain and saturable phase shift as the applied signal strength is increased.

### Transient Two-Level Solutions

Let us also look at the transient response of a two-level atomic system to an applied signal.

Suppose a two-level system has some initial population difference $$\Delta{N}(t_0)$$ at time $$t_0$$ (where this initial value may or may not be the same as the thermal-equilibrium value $$\Delta{N_0}$$); and assume that an applied signal with constant amplitude $$W_{12}$$ is then turned on at $$t=t_0$$.

The transient solution to the rate equation for $$t\gt{t_0}$$ is then

$\tag{69}\Delta{N(t)}=\Delta{N_\text{ss}}+[\Delta{N(t_0)}-\Delta{N_\text{ss}}]\exp[-(2W_{12}+1/T_1)(t-t_0)]$

where $$\Delta{N_\text{ss}}$$ is the steady-state or saturated value of $$\Delta{N}$$ given earlier. This transient response is plotted for a few typical cases in Figure 4.13.

With no applied signal present, so that $$2W_{12}T_1=0$$, the population $$\Delta{N}(t)$$ relaxes from the initial value $$\Delta{N}(t_0)$$ toward the thermal-equilibrium value $$\Delta{N_0}$$ with exponential time constant $$T_1$$.

When a constant applied signal is present, however, the population difference $$\Delta{N}(t)$$ relaxes—more accurately, is driven—toward the saturated steady-state value $$\Delta{N}_\text{ss}\lt\Delta{N_0}$$. Increasing the signal strength also speeds up the rate $$(2W_{12}+1/T_1)$$ at which the population difference approaches this saturated condition.

### Two-Level Systems With Degeneracy

The same simple results derived above can also be obtained, though with slightly more algebraic complexity, even if the two-level system has degeneracies $$g_1$$ and $$g_2$$ in its lower and upper energy levels.

To verify this we can recall that if degeneracy factors $$g_1$$ and $$g_2$$ are present, the stimulated transition rates are related by $$g_1W_{12}=g_2W_{21}$$, and the relaxation rates are related by $$w_{12}/w_{21}=(g_2/g_1)\exp[-(E_2-E_1)/kT]$$.

For the degenerate case, it also makes the most sense to define the population difference $$\Delta{N}$$ on the two-level transition in the form

$\tag{70}\Delta{N(t)}\equiv(g_2/g_1)N_1(t)-N_2(t)$

since this is the population difference that appears in the complex susceptibility $$\chi(\omega)$$, and hence in any absorption or gain expressions.

The population difference $$\Delta{N_0}$$ at thermal equilibrium must also now be written, using these definitions, in the slightly more complicated form

$\tag{71}\Delta{N_0}\equiv(g_2/g_1)N_{10}-N_{20}=N\frac{1-\exp[-\hbar\omega_a/kT]}{1+(g_1/g_2)\exp[-\hbar\omega_a/kT]}$

Here we must also define the effective signal-stimulated transition probability $$W_\text{eff}$$ by

$\tag{72}W_\text{eff}\equiv\frac{1}{2}(W_{12}+W_{21})$

and the energy relaxation time $$T_1$$ in the same fashion as above, namely, $$w_{12}+w_{21}\equiv1/T_1$$.

The two-level rate equation with degeneracy then takes on exactly the same simple form as in Equation 4.63, namely,

$\tag{73}\frac{d}{dt}\Delta{N(t)}=-2W_\text{eff}\Delta{N(t)}-\frac{\Delta{N(t)}-\Delta{N_0}}{T_1}$

A two-level system even with degeneracy thus behaves exactly like an ideal nondegenerate two-level system, and all the results we have just derived remain valid, provided that we use the special definitions of $$\Delta{N(t)}$$ and $$W_\text{eff}$$ that we have just introduced.

### Atomic Time Constants: $$T_1$$, $$T_2$$ and $$\tau$$

The notation $$T_1$$ that we have introduced for the two-level rate equation in this section is one example of several different notations for atomic time constants that will appear frequently in the rest of this text, as well as in many analyses of atomic behavior in the scientific literature. It is important to keep track of the physical meanings of these time constants, as well as the distinctions between them.

The symbol $$T_1$$ is used rather widely in the scientific literature as we have used it here, namely, to indicate in general the time constant with which a population $$N(t)$$ or a population difference $$\Delta{N}(t)$$ will return to its equilibrium value or—what is essentially the same thing—the time constant with which an atomic system will exchange energy with its surroundings.

The time constant $$T_1$$ is thus generally equivalent to the population recovery or energy decay times $$\tau$$ or $$\gamma^{-1}$$ often used in other analyses. For a two-level optical-frequency transition in particular, this time constant, is essentially the same as the upper-level lifetime or energy-decay lifetime $$\tau_{21}$$.

This same time constant $$T_1$$ is also, for reasons that we will learn later, sometimes referred to as the longitudinal relaxation time, especially in Bloch equation analyses, or the on-diagonal relaxation time in quantum analyses of atomic systems.

This time constant $$T_1$$ stands in contrast to the quite different time constant $$T_2$$ we introduced in an earlier tutorial to describe the elastic dephasing of the coherent macroscopic polarization $$p(t)$$.

The time constant $$T_2$$ is also widely used in the scientific literature, and is sometimes called the atomic dephasing time, the transverse relaxation time, or the off-diagonal relaxation time of the same atomic transition.

In most situations the energy decay or population recovery time $$T_1$$ is substantially longer than the dephasing time $$T_2$$ (although for highly isolated individual atoms, as in a very low-pressure gas cell, the usual dephasing mechanisms may be nearly eliminated, and then, in the usual notation, $$T_2\approx{T_1}$$).

The notations $$T_1$$ and $$T_2$$ are most commonly used to indicate these two different time constants in magnetic resonance and Bloch equation analyses, and for analyses on two-level atomic systems; the alternative notations $$\tau$$ or $$\gamma$$ and sometimes $$\Delta\omega_a=2/T_2$$ are also commonly used, especially in optical-wavelength and multilevel laser calculations.

We will jump back and forth between these alternative notations in different parts of this tutorial series, depending on what seems to match up best with the usual scientific literature.

## 6. Multilevel Rate Equations

A real atomic system will, of course, have a very large number of energy levels $$E_i$$, with different degeneracies $$g_i$$ and time-varying populations $$N_i(t)$$.

Signals may then be applied to this atomic system simultaneously at frequencies near several different transition frequencies $$\omega_{ji}=(E_j-E_i)/\hbar$$; and relaxation transitions will occur in general between all possible pairs of levels in the system.

We will now show how to write the complete rate equations applicable to such a multilevel, multisignal, multifrequency case.

### Multilevel Atomic Systems

Figure 4.14 shows a typical multienergy-level atomic system to which several different signals tuned near different transition frequencies may be simultaneously applied.

We assume for simplicity that all the transitions to which signals are applied have resonance frequencies $$\omega_{ji}$$ that differ from each other by at least a few atomic linewidths. This ensures that each applied signal is in resonance with (and thus affects) only the one transition to which it is tuned.

We also assume that all the applied signals will be weak enough that a rate equation approach is valid. This is a point that will be discussed in more detail in a later tutorial.

Consider first just the flow of atoms between some given level $$E_i$$ and some other higher-lying level $$E_j$$. If a signal is applied to this particular transition, the flow rate in the upward direction out of level $$E_i$$ will be $$(W_{ij}+w_{ij})N_i$$, and the flow rate in the downward direction into level $$E_i$$ will be $$(W_{ji}+w_{ji})N_j$$. The net flow rate between these two levels will thus be expressed by the rate equation terms

$\tag{74}\frac{dN_i}{dt}=-\frac{dN_j}{dt}=-W_{ij}N_i+W_{ji}N_j-w_{ij}N_i+w_{ji}N_j$

for any pair of $$i$$ and $$j$$ levels.

The stimulated transition rate produced by the applied signal on this particular transition will have the same general form as Equation 4.31, namely,

$\tag{75}W_{ji}=\frac{g_i}{g_j}W_{ij}=\frac{3^*}{8\pi^2}\frac{\gamma_{\text{rad},ji}}{\hbar\Delta\omega_{a,ij}}\frac{\epsilon|\tilde{E}_{ij}|^2\lambda_{ji}^3}{1+[2(\omega-\omega_{ji})/\Delta\omega_{a,ij}]^2}$

where all the quantities have values appropriate to that particular $$i\rightarrow{j}$$ transition.

This expression assumes a lorentzian homogeneous transition. An appropriately modified version must be substituted if the transition is a gaussian inhomogeneous transition. All atomic transition parameters such as $$\gamma_{\text{rad},ji}$$ and $$\Delta\omega_{a,ij}$$ as well as the applied signal field $$\tilde{E}_{ij}$$, will of course have different values for each transition in the system.

The relaxation transition probabilities $$w_{ij}$$ and $$w_{ji}$$ between an arbitrary pair of levels usually cannot be calculated in any simple fashion, and their numerical values are most often either measured or just guessed at.

These numerical values may differ (widely!) for different transitions in any given system, and may depend strongly on gas pressure, crystal-lattice temperature, or other properties of the atomic surroundings. These probabilities on any given transition will, however, be related, as always, by the Boltzmann ratios

$\tag{76}\frac{w_{ij}}{w_{ji}}=\frac{g_j}{g_i}\exp\left(-\frac{E_j-E_i}{kT}\right)$

for any pair of levels $$E_i$$ and $$E_j$$.

### Multilevel Rate Equations

When multiple signals are present simultaneously on several different transitions, each applied signal will produce stimulated transitions that affect only the populations $$N_i$$ and $$N_j$$ of the two levels involved in that particular transition.

The population changes produced by multiple signals are then taken into account by simply summing the stimulated transition rates produced by each applied signal on its particular transition, plus the relevant relaxation rates, as given by the preceding equations.

The signals on different transitions do not (to first order) interfere with each other, even if they happen to terminate on the same energy level. The relaxation transition rates between all the levels are similarly taken into account simply by adding their independent effects on each energy-level population.

Suppose as a general example that energy level $$E_i$$ is acted on by several applied signals each close to a different transition frequency $$|\omega_{ji}|=|E_j-E_i|/\hbar$$ as illustrated in Figure 4.15. (The other levels $$E_j$$ may be below or above level $$E_i$$.)

The general rate equation for the population $$N_i(t)$$ on this particular level is then

$\tag{77}\frac{dN_i}{dt}=\sum_{j\ne{i}}(-W_{ij}N_i+W_{ji}N_j)+\sum_{j\ne{i}}(-w_{ij}N_i+w_{ji}N_j)$

The first sum gives the stimulated transition rates to all other levels $$E_j$$ for which appropriate signals tuned at or near $$|\omega_{ji}|$$ are present. Each such signal will produce appropriate stimulated-transition probabilities related by $$g_iW_{ij}=g_jW_{ji}$$.

The second sum gives the relaxation rates to and from all the other levels $$E_j$$ of the atom, or at least all other levels $$E_j$$ for which the relaxation terms $$w_{ij}N_i$$ and $$w_{ji}N_j$$ have any appreciable magnitude.

In general, then, for an $$M$$-level atomic system we can write $$M$$ separate rate equations of the form in Equation 4.77, one for each level population $$N_i(t)$$ for $$i=1$$ to $$i=M$$.

The three rate equations for a 3-level atomic system, for example, have the form

\tag{78}\begin{align}dN_1/dt&=-(W_{12}+W_{13}+w_{12}+w_{13})N_1\\&\qquad+(W_{21}+w_{21})N_2+(W_{31}+w_{31})N_3,\\dN_2/dt&=-(W_{21}+W_{23}+w_{21}+w_{23})N_2\\&\qquad+(W_{12}+w_{12})N_1+(W_{32}+w_{32})N_3,\\dN_3/dt&=-(W_{31}+W_{32}+w_{31}+w_{32})N_3\\&\qquad+(W_{13}+w_{13})N_1+(W_{23}+w_{23})N_2,\\\end{align}

if we assume that applied signals may be present on all three possible transitions.

Note that we organized these equations by systematically writing the stimulated plus relaxation terms in each equation, first for the level $$N_i$$ itself, and then for all the other levels $$N_j$$ connected to this level. It would be equally possible to organize the terms in each equation in a pair-wise fashion, for example, for level $$N_2$$,

\tag{79}\begin{align}dN_2/dt&=-(W_{21}N_2-W_{12}N_1)-(W_{23}N_2-W_{32}N_3)\\&\quad-(w_{21}N_2-w_{12}N_1)-(w_{23}N_2-w_{32}N_3)\end{align}

where we first write all the stimulated terms and then all the relaxation terms, for each other level to which level $$E_2$$ is connected. The important point is obviously to include all the (necessary) terms, and then to organize them in a fashion which makes their solution the easiest.

### Conservation of Atoms

If the total number of atoms in all the energy levels is constant, however, there will also be a "conservation of atoms" equation, namely,

$\tag{80}\sum_{i=1}^MN_i=N_1+N_2+\ldots+N_M=N$

where $$M$$ is the total number of energy levels in the system.

But if this condition applies, then only $$M-1$$ of the $$M$$ rate equations will be linearly independent, since any one of the $$M$$ rate equations can be obtained as the negative sum of the other $$M-1$$ equations.

We will really have, therefore, $$M-1$$ rate equations for individual level populations, plus the conservation of atoms equation, to give a total of $$M$$ independent equations in the $$M$$ unknown populations $$N_i(t)$$, $$i=1$$ to $$M$$.

We will employ general multilevel rate equations of the type described here to analyze several different laser pumping and signal saturation processes in future tutorials.

In the remainder of this tutorial, however, let us look at some general characteristics of these multilevel equations and their solutions, without going into the details of any specific problems or specific examples.

Let us consider first the steady-state behavior of such a multilevel system when one or more signals of constant amplitude are applied to different transitions in the system.

In particular, how will the populations and population differences in a multilevel system saturate or move away from their thermal-equilibrium values in the presence of one or more strong applied signals, and how will this compare to the simple two-level saturation result?

To find out how the steady-state level populations $$N_{i,\text{ss}}$$ will vary in a multilevel system with a set of constant-intensity applied signals $$W_{ij}$$, we must solve the appropriate set of $$M-1$$ rate equations like those just described, with the time derivatives set equal to zero, plus the supplementary condition given by conservation of the total number of atoms.

This gives a set of $$M$$ coupled linear algebraic equations of the general form

\tag{81}\begin{align}\hat{W}_{11}N_{1,\text{ss}}&+\hat{W}_{12}N_{2,\text{ss}}&&+\ldots&&&+\hat{W}_{1M}N_{M,\text{ss}}&&&&=0\\\hat{W}_{21}N_{1,\text{ss}}&+\hat{W}_{22}N_{2,\text{ss}}&&+\ldots&&&+\hat{W}_{2M}N_{M,\text{ss}}&&&&=0\\\ldots&+\ldots&&+\ldots&&&+\ldots&&&&=0\\N_{1,\text{ss}}&+N_{2,\text{ss}}&&+\ldots&&&+N_{M,\text{ss}}&&&&=N\end{align}

where each of the $$\hat{W}_{ij}$$ elements is a linear combination of (constant) $$W_{ij}$$ and $$w_{ij}$$ factors. The first $$M-1$$ of these equations come from the $$M-1$$ rate equations, and the final one comes from the conservation of atoms. (These equations could, of course, be rearranged into any arbitrary order.)

The available methods for solving such a set of $$M$$ coupled algebraic equations in all their glory, in order to find the steady-state populations in an $$M$$-level system, are merely those used for solving any set of $$M$$ linear algebraic equations—and the calculations that are required, especially for $$M\ge3$$, become just as messy.

If you have not done this kind of calculation recently, try carrying out an explicit solution of the full-blown coupled equations for the 3-level system (Equations 4.78). It will rapidly become obvious how intractably messy the algebra becomes even for just three energy levels, let alone any case with $$Mgt3$$.

Real laser problems do sometimes involve, however, atomic or molecular systems having anywhere from four to several dozen simultaneous coupled rate equations of this type. Possible methods for attacking these $$M$$-level steady-state problems then include the following.

• Adopt some standard algebraic algorithm, such as Cramer's Rule, and keep tirelessly turning the crank until algebraic solutions emerge. (However, Cramer's Rule is well known to be a poor procedure for numerical computer calculations, because of round-off error in the repeated additions and subtractions that are involved.)
• Use a computer with a good packaged linear-equation routine other than Cramer's rule (such as gaussian elimination).
• Eliminate as many terms from the equations as possible by means of physical arguments (for example, eliminate levels you know have negligible populations or terms corresponding to negligible relaxation rates); and then be clever in substituting the remaining equations into each other.
• Give up and substitute some alternative attack.

The second of these methods is the only feasible one if you have a really big multilevel problem and it really has to be solved. The third approach is the only useful one for most other simple cases.

### Saturation in Multilevel Systems

If we do solve one of these multilevel systems for the steady-state populations as a function of relaxation rates and applied signal strengths, what will the saturation behavior on any particular transition look like?

We found in the previous section that the steady-state population difference $$\Delta{N}_{12}$$ in a two-level system saturates with increasing signal intensity $$W$$ in a simple homogeneous fashion.

We will also find quite generally in any multilevel system that increasing the signal strength $$W_{ij}$$ applied to any $$i\rightarrow{j}$$ transition will cause the population difference on that transition to saturate in essentially the same fashion, that is, in the form

$\tag{82}\Delta{N}_{\text{ss},ij}=\Delta{N}_{0,ij}\frac{1}{1+W_{ij}/W_{\text{sat},ij}}$

where $$W_{\text{sat},ij}$$ is a saturation value or saturation intensity for that particular transition under those particular circumstances.

The initial inversion $$\Delta{N}_{0,ij}$$ on this transition might be, for example, the small-signal population inversion on the lasing transition in a system which is being pumped on some other transition.

Both the initial inversion $$\Delta{N}_{0,ij}$$ and the saturation intensity $$W_{1j,\text{sat}}$$ will then depend in a complicated way on the relaxation rates of the system and the signals applied to any other transitions in the system.

So long as these are fixed in amplitude, however, turning on and increasing the strength of a signal $$W_{ij}$$ applied to the $$j\rightarrow{i}$$ transition will cause the population difference $$\Delta{N}_{\text{ss},ij}$$ on that particular transition to saturate in exactly the same homogeneous fashion as for the two-level system.

The saturation behavior on the inverted $$i\rightarrow{j}$$ transition will be formally identical to the saturation behavior of a two-level system, even though many other relaxation rates or other applied (fixed-intensity) signals may be present in the system.

### Proof of Saturation Behavior

The point that we have just made concerning saturation in a multilevel system is best illustrated by solving the multilevel rate equations for a few simple but still realistic practical cases, and examining the solutions, as we will do in later tutorials.

This point can also be proven in a slightly messy but quite general way, which we will outline here.

We first note that if a signal $$W_{ij}$$ is applied to a certain $$i\rightarrow{j}$$ transition, the rate equation for either of the two levels $$N_i$$ or $$N_j$$ involved in that transition may be rewritten at steady-state ($$d/dt\equiv0$$) in the form

$\tag{83}\frac{dN_i}{dt}=-W_{ij}N_i+W_{ji}N_j+f_i(N_k,w_{ik},W_{ik})=0$

where $$f_i(N_k,w_{ik},W_{ik})$$ is a complicated function of all the other level populations $$N_k$$, relaxation rates $$w_{ik}$$, and applied signals $$W_{ik}$$ on all the other levels in the system (but not $$W_{ij}$$ or $$W_{ji}$$).

Now, if the signal intensity on this one transition is turned up to a very large value, so that $$W_{ij}$$ and $$W_{ji}$$ approach $$\infty$$, the function $$f_i$$ must approach a finite limiting value, since all the factors in the $$f_i$$ expression remain finite.

Hence, to keep the first pair of terms in Equation 4.83 finite as $$W_{ij}$$ and $$W_{ji}\rightarrow\infty$$, the population difference on the transition must decrease in the limiting form

$\tag{84}\Delta{N}_{ij}\equiv\left(\frac{g_j}{g_i}N_i-N_j\right)\approx\frac{(g_j/g_i)f_i}{W_{ij}}\qquad(W_{ij},W_{ji}\rightarrow\infty)$

To examine this in more detail, we can note that since the rate equations are a linear coupled set, their steady-state solutions (for $$dN_i/dt=0$$) for any of the steady-state level populations $$N_k$$, expressed in terms of any one particular transition probability $$W_{ij}\equiv(g_j/g_i)W_{ji}$$, must be of the general form

$\tag{85}N_{\text{ss},k}=\frac{a_{kij}+b_{kij}W_{ij}}{c_{ij}+d_{ij}W_{ij}}$

where the constants $$a_{kij}$$, $$b_{kij}$$, etc., will in general be complicated mixtures of all the other relaxation probabilities $$w_{pq}$$ and transition probabilities $$W_{rs}$$ that are present in the system for all the $$pq$$ and $$rs$$ transitions other than $$W_{ij}$$ and $$W_{ji}$$.

(These coefficients might be found, for example, by expanding the steady-state solution of the rate equations using Cramer's rule; and noting that all the population expressions $$N_k$$ will have the same denominator $$c_{ij}+d_{ij}W_{ij}$$.)

As $$W_{ij}$$ and $$W_{ji}$$ become arbitrarily large, each of the level populations will approach some saturated steady-state value given by $$N_k\rightarrow{b}_{kij}/d_{ij}$$ as $$W_{ij}\rightarrow\infty$$.

Consider in particular the limit of the $$i\rightarrow{j}$$ population difference $$\Delta{N}_{ij}$$ as $$W_{ij}\rightarrow\infty$$. This can be gotten by subtracting two expressions like Equation 4.85 with $$k=i$$ and $$k=j$$.

But then the infinite signal limit implies that the factors $$(g_j/g_i)b_{iij}W_{ij}$$ and $$b_{jij}W_{ij}$$ involved in these two expressions must exactly cancel.

Hence $$\Delta{N}_{ij}$$ must have the form (for all values of $$W_{ij}$$)

$\tag{86}\Delta{N}_{\text{ss},ij}=\frac{(g_j/g_i)a_{iij}-a_{jij}}{c_{ij}+d_{ij}W_{ij}}$

Note again that all the constants in this expression, like $$a_{iij}$$ and $$c_{ij}$$, depend on all the other $$w_{pq}$$'s and $$W_{rs}$$'s, but not on $$W_{ij}$$ or $$W_{ji}$$.

The conclusion of the derivation just given is that the saturated population difference for a strong signal applied to any one single transition in a multilevel system can be written in the general form

$\tag{87}\Delta{N}_{\text{ss},ij}=\frac{\Delta{N}_{0,ij}}{1+(d_{ij}/c_{ij})W_{ij}}$

This is exactly like the saturation of a two-level system as derived in the preceding section.

### Transient Response of Multilevel Systems

We can also say some general things about the transient response of a multilevel atomic system, for example when an applied signal is suddenly turned on or turned off.

The $$M-1$$ coupled rate equations given earlier, plus the conservation of atoms equation, form a set of $$M$$ coupled linear differential equations for the level populations $$N_i(t)$$ versus $$t$$ (or at least these equations are linear so long as the applied signals $$W_{ij}$$ have either zero or constant values).

Linear coupled differential equations lead in general either to exponentially decaying or possibly to oscillating transient solutions. A two-level system exhibits, for example, a single exponential recovery with decay rate $$1/T_1$$ for no applied signal or $$(W_{12}+W_{21}+1/T_1)$$ for a constant applied signal (cf. Equation 4.69).

The transient solutions for an $$M$$-level atomic system will similarly exhibit $$M-1$$ transient terms with $$M-1$$ exponential time constants, very much like the transient response of a multiloop RC electrical circuit containing $$M-1$$ independent capacitances.

Each time constant of the multilevel system will in general be some complicated combination of all the $$w_{ij}$$'s and $$W_{ij}$$'s in the system. These time constants, or rather the corresponding decay rates, will be the multiple roots of a polynomial equation formed from the secular determinant for the coupled set of linear equations. Standard techniques such as Laplace transforms can be used to find these decay rates and transient solutions.

As a practical matter, however, in most multilevel systems one or two relaxation rates dominate in determining the transient behavior of any given level. Experimental results for the time behavior of any one level population $$N_i(t)$$ usually show either just one predominant time constant or perhaps in more complex cases a double-exponential type of transient behavior.

### Optical-Frequency Approximation

As we noted earlier, most laser systems at optical frequencies have $$\hbar\omega/kT\gg1$$ for all the transitions involved. To express this in still another way, the energy gap corresponding to visible-frequency radiation is $$\hbar\omega\approx2$$ eV, as compared to $$kT\approx25$$ meV at room temperature, so that $$\hbar\omega/kT\approx40$$.

In this limit, all upward relaxation probabilities $$w_{ij}$$ can be ignored, and all downward relaxation probabilities can be written as energy decay rates in the form $$w_{ji}\approx\gamma_{ji}$$. We can therefore use $$\gamma_{ji}$$ as an alternative notation for the downward relaxation probability from any level $$E_j$$ to any lower level $$E_i$$ in the rate equations, and there are no upward relaxation processes.

The relaxation terms for any given level $$E_i$$ in the general rate equations can then be simplified to the form

$\tag{88}\left.\frac{dN_i}{dt}\right|=-\sum_{k\lt{i}}\gamma_{ik}N_i+\sum_{k\gt{i}}\gamma_{ki}N_k$

where the first sum represents relaxation out of level $$E_i$$ into all lower levels $$E_k$$, and the second sum represents relaxation down into level $$E_i$$ from all higher levels $$E_k$$.

If we consider only the first term, the net energy-decay rate from level $$E_i$$ to all lower levels is

$\tag{89}\left.\frac{dN_i}{dt}\right|=-\sum_{k\lt{i}}\gamma_{ik}N_i=-\gamma_iN_i$

The total decay rate $$\gamma_i$$ and the net lifetime $$\tau_i$$ for level $$E_i$$ are thus given by summing over all the radiative and nonradiative decay rates from level $$E_i$$ to all lower levels $$E_k$$, i.e.,

$\tag{90}\gamma_i\equiv\frac{1}{\tau_i}=\sum_{k\lt{i}}\gamma_{ik}$

In the absence of pumping effects or relaxation from upper levels, an initial population $$N_i(t_0)$$ in level $$E_i$$ will decay as

$\tag{91}N_i(t)=N_i(t_0)e^{-\gamma_i(t-t_0)}=N_i(0)e^{-(t-t_0)/\tau_i}$

Note again that multiple decay processes acting in parallel combine by summing the decay rates, or summing the inverse lifetimes associated with each process.

The next tutorial introduces multidimensional optimized optical modulation formats.