# Electric dipole interaction of atoms with radiated electromagnetic waves

16 Feb 2024 - tsp
Last update 16 Feb 2024
22 mins

This blog article is just a short summary of some theory basics - a quick look at a generic two level quantum system, the general amplitude rate equations that work for any two level system, the coupling of an classical electrical field to a dipole and the arising Rabi oscillations as well as the rotating frame approximation for monochromatic waves. In the end there will be a short look at the concept of $\pi$ and $\frac{\pi}{2}$ pulses as well as Bloch vectors and Bloch spheres. References that go much more into detail can be found at the end of the article.

## The two level system

Lets take a look at a simple two level atom and it’s interaction with electromagnetic radiation in a semi-classical way. It’s assumed that the radiation is a classical (non quantum) electromagnetic wave and the two level electric system (for example an Alkali atom) will be modeled quantum mechanically.

[ i \hbar \frac{\partial}{\partial t} \psi = \hat{H} \psi ]

Let’s assume we can split the Hamiltonian into an unperturbed part $\hat{H_0}$ and a time dependent perturbation or interaction part $\hat{H_I}$. The solution for Eigenvalues and Eigenvectors of the unperturbed Hamiltonian are the same as usual:

[ \begin{aligned} \psi_n(\vec{r}, t) &= \psi_n(\vec{r}) e^{-i \frac{E_n t}{\hbar}} \\ \mid n(t) > &= \mid n > e^{-i \omega_n t} \end{aligned} ]

Assume that the Eigenstates and Eigenvalues for the unperturbed systems are known as $E_n$ and $\mid n >$:

[ \hat{H_0} \mid n > = E_1 \mid n > ]

For a simple two level system as it will be used later in this article this would imply:

[ \begin{aligned} \hat{H_0} \psi_1(\vec{r}) &= E_1 \psi_1(\vec{r}) \\ \hat{H_0} \psi_2(\vec{r}) &= E_2 \psi_2(\vec{r}) \end{aligned} ]

Now we know those states form a complete Eigenbasis of our unperturbed system. Since the perturbed system exists in the same space but the unperturbed wavefunctions won’t be Eigenstates of the complete perturbed or interacting system one can assume that the full wavefunction is composed of a mixture of all components:

[ \begin{aligned} \psi(\vec{r}, t) &= \sum_{n} c_n(t) \psi_n(\vec{r}) e^{-i \frac{E_n}{\hbar} t} \\ \mid \psi_0 > &= \sum_n c_n \mid n > e^{-i \frac{E_n}{\hbar} t} \\ \mid \psi (t) > &= \sum_n c_n(t) \mid n > e^{-i \frac{E_n}{\hbar} t} \end{aligned} ]

## General amplitude rate equations

Let’s first take a look into some generic statement about the prefactors $c_n(t)$ or a perturbed system. As one can see the time dependent Schrödinger equation already describes the time evolution of such a simple perturbed system in form of a set of coupled differential equations:

[ \begin{aligned} i \hbar \frac{\partial}{\partial t} \mid \psi(t) > &= \hat{H} \mid \psi(t) > \\ i \hbar \frac{\partial}{\partial t} \mid \psi(t) > &= \left(\hat{H_0} + \hat{H_I}(t) \right) \mid \psi(t) > \\ \to 0 &= \left(\hat{H_0} + \hat{H_I}(t) - i \hbar \frac{\partial}{\partial t} \right) \mid \psi(t) > \\ 0 &= \sum_n \left( c_n(t) \hat{H_0} \mid n > + c_n(t) \hat{H_I} \mid n > - i \hbar \frac{\partial}{\partial t} \left(c_n(t) \mid n > \right) \right) e^{-i \omega_n t} \\ 0 &= \sum_n \left( \underbrace{c_n(t) \hat{H_0} \mid n >}_{c_n(t) E_n \mid n >} + c_n(t) \hat{H_I} \mid n > - i \hbar \left( \frac{\partial}{\partial t} c_n(t \right) \mid n > - \underbrace{i \hbar c_n(t) \left(\frac{\partial}{\partial t} \mid n > \right)}_{c_n(t) E_n \mid n >} \right) e^{-i \omega_n t} \\ 0 &= \sum_n \left( c_n(t) \hat{H_I} \mid n > - i \hbar \frac{\partial c_n(t)}{\partial t} \mid n > \right) e^{i \omega_n t} \end{aligned} ]

Projecting onto an arbitrary basis vector $< m \mid$ in Hilbert state space yields:

[ \begin{aligned} 0 &= \sum_n \left( c_n < m \mid \hat{H_I} \mid n > - i \hbar \frac{\partial c_n}{\partial t} < m \mid n > \right) e^{-i \omega_n t} \\ 0 &= \left( \sum_n c_n < m \mid \hat{H_I} \mid n > e^{-i \omega_n t} \right) - i \hbar \frac{\partial c_m}{\partial t} \underbrace{< m \mid m >}_{1} e^{-i \omega_m t} \\ \to i \hbar \frac{\partial c_m}{\partial t} \underbrace{< m \mid m >}_{1} e^{-i \omega_m t} &= \sum_n c_n < m \mid \hat{H_I} \mid n > e^{-i \omega_n t} \\ \to i \hbar \frac{\partial c_m}{\partial t} &= \sum_n c_n < m \mid \hat{H_I} \mid n > e^{i \left(\omega_m - \omega_n \right) t} \end{aligned} ]

Assuming now that the interacting Hamiltonian has only off diagonal elements (no diagonal elements) - since it will only provide information about state changes by projecting from one state into another but no preserved state information, this is contained in $\hat{H_0}$ - one can directly describe two coupled differential equations for a simple two level system:

[ \begin{aligned} i \hbar \frac{\partial}{\partial t} c_1 = c_2 < 1 \mid \hat{H_I} \mid 2 > e^{i (\omega_1 - \omega_2) t} \\ i \hbar \frac{\partial}{\partial t} c_2 = c_1 < 2 \mid \hat{H_I} \mid 1 > e^{i (\omega_2 - \omega_1) t} \end{aligned} ]

## Perturbation of an electric dipole by an oscillating electric field - Rabi oscillations

This description was generic for any perturbed quantum system that can be described via a diagonal unperturbed Hamiltonian $\hat{H_0}$ and a perturbation $\hat{H_I}$. Now let’s take a look at a specific simple two level system formed by an electric dipole that is perturbed by an oscillating electric field - this models for example Alkali atoms in presence of an external driving microwave field. The dipole - assuming the position of one charge carrier (for example the nucleus) is fixed at $\vec{0}$ by arbitrary choice - can be described by it’s charge $q$ and the distance $\vec{r}$ via $- q \vec{r}$. For an electron this leads to $-e \vec{r}$. Coupling to an external electric field is described as $-e \vec{r} \vec{E(t)}$ (this is also called the Stark effect):

[ \begin{aligned} \vec{E}(t) &= \vec{E_0} \cos(\omega t) \\ H_I(t) &= e * \vec{r} * \vec{E}(t) \\ H_I(t) &= e * \vec{r} * \vec{E_0} \cos(\omega t) \end{aligned} ]

Inserting the interaction Hamiltonian $\hat{H_I}$ into the general amplitude rate equations:

[ \begin{aligned} i \hbar \frac{\partial}{\partial t} c_1 = c_2 < 1 \mid e \vec{r} \vec{E_0} \cos(\omega t) \mid 2 > e^{i(\omega_1 - \omega_2) t} \\ i \hbar \frac{\partial}{\partial t} c_2 = c_1 < 2 \mid e \vec{r} \vec{E_0} \cos(\omega t) \mid 1 > e^{i(\omega_2 - \omega_1) t} \\ \end{aligned} ]

Rearranging and using the fact that $<m \mid n > = \left(<n \mid m>\right)^{*}$ yields:

[ \begin{aligned} i \frac{\partial}{\partial t} c_1 = \overbrace{\frac{< 1 \mid e \vec{r} \vec{E_0} \mid 2 >}{\hbar}}^{\Omega} e^{i(\overbrace{\omega_1 - \omega_2}^{-\omega_0}) t} \cos(\omega t) c_2 \\ i \frac{\partial}{\partial t} c_2 = \underbrace{\frac{< 2 \mid e \vec{r} \vec{E_0} \mid 1 >}{\hbar}}_{\left(\frac{1}{\hbar} < 1 \mid e \vec{r} \vec{E_0} \mid 2 >\right)^{*} = \Omega^{*}} e^{i(\overbrace{\omega_2 - \omega_1}^{\omega_0}) t} \cos(\omega t) c_1 \\ \end{aligned} ]

Using the definitions of

• The resonance frequency $\omega_0 = \omega_2 - \omega_1 = \frac{E_2 - E_1}{\hbar}$
• The Rabi frequency $\Omega = \frac{1}{\hbar} <1 \mid e \vec{r} \vec{E_0} \mid 2 >$

those equations can be expressed very compact:

[ \begin{aligned} \to i \frac{\partial}{\partial t} c_1 = \Omega e^{-i \omega_0 t} \cos(\omega t) c_2 \\ i \frac{\partial}{\partial t} c_2 = \Omega^{*} e^{i \omega_0 t} \cos(\omega t) c_1 \end{aligned} ]

Assuming that the wavelength of the external oscillating field is way larger than the spatial extension of the dipole (i.e. $\lambda » a_0$) one can use the dipole approximation. This means one assumes that the amplitude of the electric field is constant over the whole spatial area of the dipole and thus one can move it out of the transition amplitude:

[ \begin{aligned} \Omega_{\text{dipole}} &= \frac{1}{\hbar} e \vec{E_0} <1 \mid \vec{r} \mid 2 > \end{aligned} ]

In case one assumes that the radiation is polarized along the $x$ axis of the coordinate system this simplifies further to:

[ \begin{aligned} \Omega_{\text{dipole}} &= \frac{e \mid E_0 \mid}{\hbar} <1 \mid \underbrace{\hat{e_x} \vec{r}}_{x} \mid 2 > \end{aligned} ]

## Rotating wave approximation for monochromatic waves

Starting from the differential amplitude rate equation:

[ \begin{aligned} i \frac{\partial}{\partial t} c_1 &= \Omega e^{-i \omega_0 t} \cos(\omega t) c_2 \\ \to i \frac{\partial}{\partial t} c_1 &= \Omega e^{-i \omega_0 t} \frac{e^{i \omega t} + e^{-i \omega t}}{2} c_2 \\ \to i \frac{\partial}{\partial t} c_1 &= \frac{\Omega}{2} \left(e^{i(\omega - \omega_0)t} + e^{-i(\omega + \omega_0)t}\right) c_2 \\ \end{aligned} ]

Now one can do the first approximation - the so called rotating wave approximation. The term $\omega + \omega_0$ is oscillating very fast in comparison to any reasonable interaction time. Thus one can assume it averages out in the integral over the interaction Hamiltonian:

[ \begin{aligned} i \frac{\partial}{\partial t} c_1 &= \frac{\Omega}{2} e^{i (\omega - \omega_0) t} c_2 \\ i \frac{\partial}{\partial t} c_2 &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} c_1 \\ \end{aligned} ]

Those equations can be combined into a very simple second order differential equation by taking another time derivative of the second equation:

[ \begin{aligned} i \frac{\partial^2}{\partial t^2} c_2 &= -i \frac{\Omega^{*}}{2} (\omega - \omega_0) e^{-i(\omega - \omega_0)t} c_1 + \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0)t} \frac{\partial c_1}{\partial t} \\ &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} \left(-i(\omega - \omega_0) c_1 + \frac{\partial c_1}{\partial t}\right) \\ &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} \left(-i(\omega - \omega_0) c_1 + \frac{\Omega}{2i} e^{i (\omega - \omega_0) t} c_2\right) \end{aligned} ]

Using the fact that

[ c_1 = i \frac{\partial c_2}{\partial t} \frac{2}{\Omega^{*}} e^{i (\omega - \omega_0) t} ]

One finally arrives at

[ \begin{aligned} i \frac{\partial^2}{\partial t^2} c_2 &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} \left(-i(\omega - \omega_0) i \frac{\partial c_2}{\partial t} \frac{2}{\Omega^{*}} e^{i (\omega - \omega_0) t} + \frac{\Omega}{2i} e^{i (\omega - \omega_0) t} c_2\right) \\ &= (\omega - \omega_0) \frac{\partial c_2}{\partial t} + \frac{\Omega \Omega^{*}}{4i} c_2 \\ \to 0 &= - i \frac{\partial^2}{\partial t^2} c_2 + (\omega - \omega_0) \frac{\partial c_2}{\partial t} + \frac{\Omega \Omega^{*}}{4i} c_2 \\ 0 &= \frac{\partial^2}{\partial t^2} c_2 + i (\omega - \omega_0) \frac{\partial c_2}{\partial t} + \mid \frac{\Omega }{2} \mid^2 c_2 \\ \end{aligned} ]

First one can solve the second order differential equation using the Ansatz $c_2(t) = \alpha e^{\beta t}$:

[ \begin{aligned} c_2(t) &= \alpha e^{i \beta t} \\ \frac{\partial}{\partial t} c_2(t) &= i \beta c_2(t) \\ \frac{\partial^2}{\partial t^2} c_2(t) &= - \beta^2 c_2(t) \end{aligned} ]

Inserting into the differential equation yields

[ \begin{aligned} 0 &= - \beta^2 c_2(t) - (\omega - \omega_0) \beta c_2(t) + \mid \frac{\Omega}{2} \mid^2 c_2(t) \\ \to 0 &= \beta^2 + (\omega - \omega_0) \beta - \mid \frac{\Omega}{2} \mid^2 \\ \to \beta_{1,2} &= - \frac{\omega - \omega_0}{2} \pm \sqrt{\left(\frac{\omega - \omega_0}{2}\right)^2 + \mid \frac{\Omega}{2} \mid^2} \\ \beta_{1,2} &= - \frac{\omega - \omega_0}{2} \pm \frac{\sqrt{(\omega - \omega_0)^2 + \Omega^2}}{2} \end{aligned} ]

One can now define $W^2 = \Omega^2 + (\omega - \omega_0)^2$

[ \begin{aligned} \beta_{1,2} = -\frac{\omega - \omega_0}{2} \pm \frac{\sqrt{W^2}}{2} \\ \beta_{1,2} = -\frac{\omega - \omega_0}{2} \pm \frac{W}{2} \end{aligned} ]

This yields the two solutions

[ \begin{aligned} c_{2,1} = \alpha e^{i \frac{\omega - \omega_0}{2} t} e^{-i \frac{W}{2} t} \\ c_{2,2} = \alpha e^{i \frac{\omega - \omega_0}{2} t} e^{i \frac{W}{2} t} \end{aligned} ]

The general solution can now be described as linear combination as usual:

[ c_2(t) = \alpha e^{i \frac{\omega - \omega_0}{2} t} * \left(A e^{-i \frac{W}{2} t} + B e^{i \frac{W}{2} t}\right) ]

Assuming that the second state is unpopulated at $t=0$, i.e. setting $c_2(t=0) = 0$ yields the relation between $A$ and $B$ and allows one to use the Euler formula to introduce a sine function:

[ \begin{aligned} c_2(t=0) &= \alpha e^{0} * (A e^0 + B e^0) \\ &= \alpha * (A + B) \\ &= 0 \\ \to A + B &= 0 \\ \to A &= -B \\ \to c_2(t) &= \alpha_2 e^{i \frac{\omega - \omega_0}{2} t} * (e^{i \frac{W}{2} t} - e^{-i \frac{W}{2} t}) \\ \to c_2(t) &= 2 i \alpha_2 e^{i \frac{\omega - \omega_0}{2} t} * \sin(\frac{W}{2}t) \\ \end{aligned} ]

To determine the prefactor $\alpha_3$ one can use the boundary condition of $c_1(t=0) = 1$ by inserting into the time evolution of $c_2(t)$:

[ \begin{aligned} i \frac{\partial}{\partial t} c_2 &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} c_1 \\ \frac{\partial}{\partial t} c_2(t) &= \underbrace{-i (\frac{\omega - \omega_0}{2} t) 2i \alpha_2 e^{i \frac{\omega - \omega_0}{2} t} \sin(\frac{W}{2}t)}_{t=0 \to 0} + \alpha_2 e^{i \frac{\omega - \omega_0}{2} t} 2i \cos(\frac{W}{2} t) \frac{W}{2} \\ \to \frac{\partial}{\partial t} c_2(t=0) &= i \alpha_2 W \\ \to i \alpha_2 W &= - i \frac{\Omega^{*}}{2} \underbrace{e^{-i (\omega - \omega_0) t} c_1(t=0)}_{1} \\ \alpha_2 &= - \frac{\Omega^{*}}{2 W} \end{aligned} ]

To determine the population of any state one projects the state vector onto the desired state as usual - for example in this case on the excited state $<2\mid$:

[ \begin{aligned} c_2(t) &= - i \frac{\Omega^{*}}{W} \sin(\frac{W}{2}t) e^{-i \frac{\omega - \omega_0}{2} t} \\ < 2(t) | \psi(t) > &= c_2^{*} e^{i \frac{E_2}{\hbar} t} < 2 \mid \left( c_1(t) e^{-i \frac{E_1}{\hbar} t} \mid 1 > + c_2(t) e^{-i \frac{E_2}{\hbar} t} \mid 2 >\right) \\ &= \underbrace{c_2^{*} e^{i \frac{E_2}{\hbar} t} c_1(t) e^{-i \frac{E_1}{\hbar} t} \underbrace{< 2 \mid 1 >}_{0}}_{0} + c_2^{*} c_2(t) \underbrace{e^{i \frac{E_2}{\hbar} t} e^{-i \frac{E_2}{\hbar} t}}_{1} \underbrace{< 2 \mid 2 >}_{1} \\ &= c_2^{*}(t) * c_2(t) \\ &= \mid c_2(t) \mid^2 \\ \to \mid c_2(t) \mid^2 &= \frac{\Omega^2}{W^2} \sin^2 \left(\frac{Wt}{2}\right) \end{aligned} ]

When one now moves on the resonance frequency of the system, i.e. setting $\omega = \omega_0$ and thus $W = \sqrt{\Omega^2 + (\omega - \omega_0)^2} = \Omega$ this leads to an on resonant population of the excited state of

[ \mid c_{2,\text{res}}(t) \mid^2 = \sin^2 \left(\frac{\Omega t}{2}\right) ]

As one can see a continuous excitation on the resonance frequency leads to an oscillation in the population of the excited state in contrast to the expected saturation from a classical point of view. Those oscillations are an indicator of the quantum nature of the system and are called Rabi oscillations.

#### $\pi$ pulse and $\frac{\pi}{2}$ pulse

The on resonant oscillation of the populated state leads to the concept of the so called $\pi$ and $\frac{\pi}{2}$ pulses. In case $\Omega t = \pi \to t = \frac{\pi}{\Omega}$ the population of the upper state $\mid 2 >$ is $c_2(t = \frac{\pi}{\Omega}) = 1$, i.e. the system is populated fully in the excited state when it was initially fully in it’s ground state. For a pulse duration of $\Omega t = 2 \pi$ on the other hand the population of the upper state will go to $c_2(t = \frac{2 \pi}{\Omega}) = 0$ again. This effect can be used to show the quantum behavior of a two level system - in a classical two level system excitation and decay would reach an equilibrium after some time - for a quantum system the effect of state inversion can be seen by observing the Rabi oscillations.

As one can also see a pulse of half the length - a so called $\frac{\pi}{2}$ pulse - puts the system in a superposition between lower and upper state with equal amplitudes:

[ \begin{aligned} \Omega t &= \frac{\pi}{2} \\ t &= \frac{\pi}{2 \Omega} \end{aligned} ] [ \begin{aligned} \mid c_{2,\text{res}}(t=\frac{\pi}{2 \Omega}) \mid^2 &= \sin^2 \left(\frac{\Omega}{2} \frac{\pi}{2 \Omega}\right) \\ \mid c_{2,\text{res}}(t=\frac{\pi}{2 \Omega}) \mid^2 &= \sin^2 \left(\frac{\pi}{4}\right) \\ \mid c_{2,\text{res}}(t=\frac{\pi}{2 \Omega}) \mid^2 &= \frac{1}{2} \\ \end{aligned} ]

The factor $c_1$ can be determined by normalization constraints very simple since

[ \begin{aligned} < \psi(t) \mid \psi(t) > &= 1 \\ \to 1 &= \sum_n \sum_m c_m^{*}(t) c_n(t) < m \mid n > e^{i \frac{E_m - E_n}{\hbar} t} \end{aligned} ]

Using again the orthonomality constraint on $< m \mid n>$, i.e. $<m \mid n> = 0 \forall m \neq n$ and $<n \mid n> = 1$ this reduces to

[ \begin{aligned} < \psi(t) \mid \psi(t) > &= 1 \\ \to 1 &= \sum_n c_n^{*}(t) c_n(t) \underbrace{< n \mid n >}_{1} \underbrace{e^{i \frac{E_n - E_n}{\hbar} t}}_{1} \\ \to 1 &= \sum_n \mid c_n(t) \mid^2 \\ \to 1 &= \mid c_1(t) \mid^2 + \mid c_2(t) \mid^2 \\ \to c_1(t) &= \sqrt{1 - \mid c_2(t) \mid^2} \end{aligned} ]

So for the $\frac{\pi}{2}$ pulse one can see - assuming one starts with a fully populated ground state - a equally mixed state with $c_1 = c_2 = \frac{1}{2}$. When one looks at the equations more generally by simply inserting the $t = \frac{\pi}{2}$ into the differential equations for the prefactors one sees that the states are swapped by a $\frac{\pi}{2}$ pulse. This is a concept that is used heavily for NMR pulse sequences.

[ c_1 \mid 1 > + c_2 \mid 2 > \to -i (c_2 \mid 1 > + c_1 \mid 2> ) ]

## Bloch vectors and Bloch spheres for the electric dipole

Since they are often used for illustration and provide a very intuitive picture let’s take a short look at Bloch spheres and thus also Bloch vectors

Lets assume again the electric dipole - let’s also assume that the electric field is acting along the axis $\vec{e_x}$ to make evaluation of the interaction Hamiltonian easier. In this case we only have to look at the $x$ component of the dipole - it’s expectation value can then easily be calculated:

[ \begin{aligned} < \psi(t) \mid \psi(t) > &= - \int \psi^{t}(t) e \hat{x} \psi(t) d^3r \\ &= - e \underbrace{\int \psi^{t}(t) \hat{x} \psi(t) d^3r}_{D_x(t)} \\ &= -e D_x(t) \end{aligned} ]

Inserting the definition of the wavefunctions from before this yields

[ \begin{aligned} D_x(t) &= \int \left( c_1(t) e^{-i \omega_1 t} \psi_1 + c_2(t) e^{-i \omega_2 t} \right)^{*} \hat{x} \left( c_1(t) e^{-i \omega_1 t} \psi_1 + c_2(t) e^{-i \omega_2 t} \right) d^3r \end{aligned} ]

Above we have seen that the interaction Hamiltonian has only off diagonal entries so products on the diagonal $<1 \mid 1> = <2 \mid 2> = 0$ vanish:

[ \begin{aligned} D_x(t) &= c_2^{*}(t) c_1(t) e^{i (\omega_2 - \omega_1) t} <2 \mid \hat{x} \mid 1> + c_1^{*}(t) c_2(t) e^{-i (\omega_2 - \omega_1) t} <1 \mid \hat{x} \mid 2> \end{aligned} ]

Also dipole moments are real since $< 1 \mid \hat{x} \mid 2> = (< 2 \mid \hat{x} \mid 1>)^{*}$. For this problem it’s now helpful to use the density matrix description of the problem:

[ \begin{aligned} \mid \psi > < \psi \mid &= \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} \begin{pmatrix} c_1^* & c_2^* \end{pmatrix} \\ &= \begin{pmatrix} \mid c_1 \mid^2 & c_1 c_2^{*} \\ c_2 c_1^{*} & \mid c_2 \mid^2 \end{pmatrix} \\ &= \begin{pmatrix} \rho_{11} & \rho_{12} \\ \rho{21} & \rho{22} \end{pmatrix} \end{aligned} ]

The diagonal elements represent the population of the states $\mid 1 >$ and $\mid 2 >$, the off diagonal elements (the coherences) represent the response of the system while applying the external field. It’s not helpful to perform a basis transformation and look at the problem in terms of the detuning of the driving field. The detuning is defined as $\delta = \omega - \omega_0$:

[ \begin{aligned} \tilde{c_1} &= c_1 e^{-i \frac{\delta t}{2}} \\ \tilde{c_2} &= c_2 e^{i \frac{\delta t}{2}} \end{aligned} ]

As one can see this does not affect the population factors:

[ \begin{aligned} < 1 \mid 1 > \to c_1^{*} e^{i \frac{\delta t}{2}} c_1 e^{-i \frac{\delta t}{2}} = c_1^{*} c_1 = \mid c_1 \mid^2 \\ < 2 \mid 2 > \to c_2^{*} e^{-i \frac{\delta t}{2}} c_2 e^{i \frac{\delta t}{2}} = c_2^{*} c_1 = \mid c_2 \mid^2 \\ \end{aligned} ]

On the other hand this transformation acts on the coherence factors:

[ \begin{aligned} D_x(t) &= c_2^{*} c_1 e^{- i \delta t} e^{i(\omega_2 - \omega_1)t} <2\mid\hat{x}\mid1> + c_1^{*}c_2 e^{i \delta t} e^{i(\omega_2 - \omega_1)t} <1\mid\hat{x}\mid2> \\ &= c_2^{*} c_1 e^{- i (\omega - \omega_0) t} e^{i \omega_0 t} <2\mid\hat{x}\mid1> + c_1^{*}c_2 e^{i (\omega - \omega_0) t} e^{i \omega_0 t} <1\mid\hat{x}\mid2>\\ &= c_2^{*} c_1 e^{- i \omega t} <2\mid\hat{x}\mid1> + c_1^{*}c_2 e^{i \omega t} <1\mid\hat{x}\mid2> \\ &= \tilde{c_2}^{*} \tilde{c_1} e^{i \omega t} <2\mid\hat{x}\mid1> + \tilde{c_1}^{*}\tilde{c_2} e^{-i \omega t} <1\mid\hat{x}\mid2> \\ \end{aligned} ]

Using the substitution

[ \begin{aligned} u &= \tilde{c_1}\tilde{c_2}^{*} + \tilde{c_2}\tilde{c_1}^{*} = \tilde{\rho}_{12} + \tilde{\rho}_{21} \\ v &= -i \left( \tilde{c_1}\tilde{c_2}^{*} - \tilde{c_2}\tilde{c_1}^{*} \right) = -i(\tilde{\rho}_{12} - \tilde{\rho}_{21}) \end{aligned} ]

one can express the dipole moment using $\sin$ and $\cos$ functions:

[ -e D_x(t) = - e < 1 \mid \hat{x} \mid 2 > (u \cos(\omega t) - v \sin(\omega t)) ]

The components $u$ and $v$ identify the in phase and quadrature components of a dipole in a frame rotating with frequency $\omega$. Now we can apply the rotating frame approximation and look at the time dependence of the population prefactors:

[ \begin{aligned} i \frac{\partial}{\partial t} c_1 &= \frac{\Omega}{2} e^{i (\omega - \omega_0) t} c_2 \\ i \frac{\partial}{\partial t} c_2 &= \frac{\Omega^{*}}{2} e^{-i (\omega - \omega_0) t} c_1 \\ \end{aligned} ]

Expressing them with the detuning $\delta = \omega - \omega_0$:

[ \begin{aligned} i \frac{\partial}{\partial t} c_1 &= \frac{\Omega}{2} e^{i \delta t} c_2 \\ i \frac{\partial}{\partial t} c_2 &= \frac{\Omega^{*}}{2} e^{-i \delta t} c_1 \\ \end{aligned} ]

one can now look at the time derivative of $\tilde{c_1} = c_1 e^{-i \frac{1}{2} \delta t}$ and $\tilde{c_2} = c_2 e^{i \frac{1}{2} \delta t}$

[ \begin{aligned} \tilde{c_1} &= c_1 e^{-i \frac{1}{2} \delta t} \\ \frac{\partial \tilde{c_1}}{\partial t} &= \frac{\partial c_1}{\partial t} e^{-\frac{1}{2} \delta t} - c_1 \frac{1}{2} \delta e^{-i \frac{1}{2} \delta t} \\ &= \frac{\partial c_1}{\partial t} e^{-\frac{1}{2} \delta t} - i \frac{c_1 \delta}{2} e^{-i \frac{1}{2} \delta t} \\ \to i \frac{\partial \tilde{c_1}}{\partial t} &= \underbrace{i \frac{\partial c_1}{\partial t}}_{c_2 e^{i \delta t} \frac{\Omega}{2}} e^{-\frac{1}{2} \delta t} + \frac{c_1 \delta}{2} e^{-i \frac{1}{2} \delta t} \\ i \frac{\partial \tilde{c_1}}{\partial t} &= c_2 e^{i \delta t} \frac{\Omega}{2} e^{-\frac{1}{2} \delta t} + c_1 \frac{\delta}{2} e^{-i \frac{1}{2} \delta t} \\ &= c_2 e^{i \frac{1}{2} \delta t} \frac{\Omega}{2} + c_1 \frac{\delta}{2} e^{-\frac{1}{2} \delta t} \\ \to i \frac{\partial \tilde{c_1}}{\partial t} &= \tilde{c_2} \frac{\Omega}{2} + i \tilde{c_1} \frac{\delta}{2} \\ &= \frac{1}{2} \left(\tilde{c_2} \Omega + \tilde{c_1} \delta\right) \end{aligned} ]

Applying exactly the same procedure yields the time derivative for $\tilde{c_2}$:

[ i \frac{\partial \tilde{c_2}}{\partial t} = \frac{1}{2} \left( \omega \tilde{c_1} - \delta \tilde{c_2} \right) ]

Using those expressions one can now describe the time derivative of the density matrix elements:

[ \begin{aligned} \frac{\partial \tilde{\rho}_{12}}{\partial t} = \left(\frac{\partial \tilde{\rho}_{21}}{\partial t} \right)^{*} &= -i \delta \tilde{\rho}_{12} + \frac{i \Omega}{2} \left(\rho_{11} - \rho_{22}\right) \\ \frac{\partial \rho_{22}}{\partial t} = -\frac{\partial \rho_{11}}{\partial t} &= \frac{i \Omega}{2} \left( \tilde{\rho}_{21} - \tilde{\rho}_{12} \right) \end{aligned} ]

The term $\rho_{11} - \rho_{22}$ is the population difference between excited and ground state. One can now look back at the expression for the dipole moment a few equations ago:

[ \begin{aligned} -e D_x(t) &= - e < 1 \mid \hat{x} \mid 2 > (u \cos(\omega t) - v \sin(\omega t)) \\ u &= \tilde{c_1}\tilde{c_2}^{*} + \tilde{c_2}\tilde{c_1}^{*} = \tilde{\rho}_{12} + \tilde{\rho}_{21} \\ v &= -i \left( \tilde{c_1}\tilde{c_2}^{*} - \tilde{c_2}\tilde{c_1}^{*} \right) = -i(\tilde{\rho}_{12} - \tilde{\rho}_{21}) \end{aligned} ]

The definition of $u$ and $v$ leads to the time derivatives for those expressions:

[ \begin{aligned} \frac{\partial v}{\partial t} &= - \delta u + \Omega\left(\rho_{11} - \rho{22}\right) \\ \frac{\partial u}{\partial t} &= \delta v \\ w &= (\rho_{11} - \rho{22}) \\ \frac{\partial w}{\partial t} = - \Omega v \end{aligned} ]

Those components make up the Bloch vector $\vec{R}$:

[ \vec{R} = u \hat{e}_1 + v \hat{e}_2 + w \hat{e}_3 ]

This definition allows to write a compact vector equation for the above time derivatives:

[ \begin{aligned} \frac{\partial \vec{R}}{\partial t} &= \vec{R} \times \left(\Omega \hat{e_1} + \delta \hat{e_3} \right) \\ &= \vec{R} \times \vec{W} \\ \vec{W} &= \Omega \hat{e_1} + \delta \hat{e_3} \end{aligned} ]

When one looks at the inner product of $\frac{\partial \vec{R}}{\partial t}$ and $\vec{R}$ one sees those vectors are orthogonal:

[ \frac{\partial \vec{R}}{\partial t} \vec{R} = 0 ]

This implies that the Bloch vectors length stays constant over time:

[ \mid \vec{R} \mid^2 = \text{const} ]

Since the length of the vector is $\mid R \mid^2 = \mid u \mid^2 + \mid v \mid^2 + \mid w \mid^2 = 1$ this means the Bloch vector is moving along the surface of a three dimensional sphere with unit radius. The time derivative of the Bloch vector $\frac{\partial \vec{R}}{\partial t}$ is also orthogonal to the vector $\vec{W}$ which means that excitations with fixed Rabi frequency and detuning follow a cone on the sphere with constant angle $\theta$ defined by $\vec{R} \vec{W} = R W \cos(\theta)$. The ratio of populations varies by

[ w = 1 - 2 \underbrace{\rho_{22}}_{\mid c_2 \mid^2} = 1 - \frac{2 \Omega^2}{W^2} \sin^2 \left(\frac{W t}{2} \right) ]