I am studying the concept of a "gain medium" inside a laser cavity. The active atoms are taken to have the two levels $a$ and $b$, separated by energy $\hbar \omega$ and represented by a density matrix $\rho$. The atoms are stationary.
The equation of motion of the density matrix is
$$\dot{\rho} = -i[H, \rho] – \dfrac{1}{2}\left(\Gamma \rho + \rho \Gamma \right) + \lambda,$$
where
$$\rho = \begin{bmatrix} \rho_{aa} & \rho_{ab} \\ \rho_{ba} & \rho_{bb} \end{bmatrix}, \ \ \ \ \ H = \begin{bmatrix} W_{a} & V \\ V & W_{b} \end{bmatrix}, \\ \Gamma = \begin{bmatrix} \gamma_{a} & 0 \\ 0 & \gamma_{b} \end{bmatrix}, \ \ \ \ \ \lambda = \begin{bmatrix} \lambda_{a} & 0 \\ 0 & \lambda_{b} \end{bmatrix}$$
The perturbation Hamiltonian is $\hbar V$, and the unperturbed energies of the levels are $\hbar W_a$ and $\hbar W_b$. Furthermore, the two levels decay with damping constants $\gamma_a$ and $\gamma_b$, and are populated by pumping at rates $\lambda_a$ and $\lambda_b$.
Therefore, using the Fourier expansion of the electric field $E(z, t) = \sum\limits_n A_n(t) u_n(z)$, where $u_n(z) = \sin(k_n z)$ and $k_n = \dfrac{n \pi}{L}$, my notes claim that the electric dipole approximation for the perturbation becomes
$$V(t) = -A(t) \wp u(z) / \hbar$$
Note that we are now considering a single cavity mode, so we drop the subscript $n$.
[See this related question.]
We now write the time dependence of the electric field $A(t)$ in terms of the amplitude $E(t)$ and phase $\phi(t)$, which vary slowly in an optical period $2\pi/\nu$:
$$A(t) = E(t) \cos[\nu t + \phi(t)]$$
[See this related question.]
Now take
$$\delta(t) = E(t) e^{-i[\nu t + \phi(t)]}$$
to be the positive frequency part of the electric field.
[See this related question.]
So we can now write the off-diagonal terms in the density-matrix equation as
$$\dot{\rho}_{ab} = -(i\omega + \gamma)\rho_{ab} – i(\rho_{aa} – \rho_{bb})A(t) \wp u(z)/\hbar,$$
where $\gamma = \dfrac{1}{2} (\gamma_a + \gamma_b)$
[See this related question.]
The steady-state solution of this equation gives
$$\rho_{ab} = -\dfrac{1}{2} \dfrac{i(\rho_{aa} – \rho_{bb}) \delta(t) \wp u(z)}{\hbar[\gamma + i(\omega – \nu)]},$$
where it is assumed that the population inversion varies slowly compared with $\rho_{ab}$, and the "rotating-wave approximation" has been used.
We can now combine $V(t) = -A(t) \wp u(z) / \hbar$ and $\rho_{ab} = -\dfrac{1}{2} \dfrac{i(\rho_{aa} – \rho_{bb}) \delta(t) \wp u(z)}{\hbar[\gamma + i(\omega – \nu)]}$ to get
$$iV(t)(\rho_{ab} – \rho_{ba}) = -\dfrac{1}{2} \left( \dfrac{\wp}{\hbar} \right)^2 (\rho_{aa} – \rho_{bb}) A(t) \times \left( \dfrac{\delta(t)}{\gamma + i(\omega – \nu)} + \dfrac{\delta^*(t)}{\gamma – i(\omega – \nu)} \right) u^2(z)$$
Now, if we only keep the "slowly varying terms" in this expression, then the diagonal terms in the equation $\dot{\rho} = -i[H, \rho] – \dfrac{1}{2}\left(\Gamma \rho + \rho \Gamma \right) + \lambda$ become rate equations as follows:
$$\dot{\rho}_{aa} = -\gamma_a \rho_{aa} + \lambda_a + R(\rho_{bb} – \rho_{aa}), \\ \dot{\rho}_{bb} = -\gamma_b \rho_{bb} + \lambda_b – R(\rho_{bb} – \rho_{aa})$$
The rate $R$ of transitions between $a$ and $b$ is said to be
$$R = \left[ \dfrac{\wp^2 E(t)^2}{2\gamma \hbar^2} \right] \mathscr{L}(\omega – \nu)u^2(z) = IR_s \mathscr{L}(\omega – \nu)u^2(z),$$
where $I = \dfrac{\wp^2 E(t)^2}{\hbar^2 \gamma_a \gamma_b}$ is the dimensionless intensity of the field in the cavity, $R_s = \dfrac{1}{2} \dfrac{\gamma_a \gamma_b}{\gamma}$, $u(z) = \sin(kz)$ is the single cavity mode under consideration, and the Lorentzian factor $\mathscr{L}(\omega – \nu) = \dfrac{\gamma^2}{\gamma^2 + (\omega – \nu)^2}$ describes the effect of detuning the laser frequency from the atomic frequency.
What exactly is the point of having / how is it useful to have a "dimensionless intensity"? Isn't dimensionality/units of "intensity" one of the main things that makes the concept useful/meaningful? So how does it even make sense / how is it even meaningful to speak about "intensity" without dimensionality? Or am I confusing "dimensionality" with "units" here?
I remember learning about "dimensionality reduction" and "dimensionless quantities" in a partial differential equations / mathematical modelling class, but, from what I remember, it was moreso used as a mathematical tool/trick for simplification, rather than explained in any meaningful or physical way.
Best Answer
It's quite common to spot that there is some sort of natural size or unit for a physical quantity, and then manipulate the equations so that some terms are expressed in terms of these natural sizes. For example, when treating angular momentum in quantum physics it is very common to express it in units of $\hbar$ and in relativity it is very common to express speed in units of $c$. One introduces ${\bf J} = \tilde{\bf J}/\hbar$ and $\beta = v/c$ and then one says $\bf J$ 'is' the angular momentum, although strictly it is a dimensionless quantity which when multiplied by $\hbar$ gives the angular momentum. Similar one might say that $\beta$ 'is' the speed, but really it is a dimensionless quantity which when multiplied by $c$ gives the speed.
In your example there arises a natural intensity scale, in terms of the interaction strength compared to decay rates, and it is useful to notice this. The quantity $I$ here is the ratio of the intensity of the light to the unit of intensity which naturally arises in the physics of atom--light interaction. That intensity unit is often called the saturation intensity.