Quantum Computation III Atomic Spectra Atomic spectra are defined as the spectrum of the electromagnetic radiation emitted or absorbed by an electron during transitions between different energy levels within an atom.

## Two-level atom

### Classical Model

• $N_1$ : Number of atoms in $\ket{g}$ per unit volume.

• $N_2$ : Number of atoms in $\ket{e}$ per unit volume.

• $\Gamma$ : Rate of spontaneous emission. The probability of transitioning of a single atom from $\ket{e}$ to $\ket{g}$ due to spontaneous emission per unit time.

• $I$ : Intensity. $I = \frac{1}{2} c n^2 \varepsilon_0 |E|^2$. Intensity has a dimension of $\mathrm{J\cdot m^{-2} \cdot s^{-1}}$.

• $\sigma(\omega)$ : Absorption cross-section, defined such that $\sigma(\omega) I$ is the energy absorbed by a single atom when irradiated by intensity $I$ with frequency $\omega$ per unit time.

#### Einstein coefficient

The change in number of the excited atoms $\frac{\mathrm{d}N_2}{\mathrm{d}t}$ (per unit volume) comes from three sources:

1. Spontaneous emission: In a unit time, there is a probability of $\Gamma$ for a single atom to transition to $\ket{g}$. For $N_2$ atoms, the total transition number is $N_2 \Gamma$ on average.
2. Stimulated absorption: According to the definition of absorption cross-section, $\sigma(\omega) I$ is the energy absorbed by a single atom per unit time. Since the incident beam has a frequency $\omega$, each time an atom absorb a photon $\hbar \omega$ , they will be excited into $\ket{e}$ (although the energy gap is $\hbar \omega_0$) . Therefore, in a unit time, $\sigma(\omega) I / \hbar \omega$ is the total number of atoms being excited into $\ket{g}$, which generates a term $\frac{\sigma(\omega) I}{\hbar \omega} N_1$.
3. Stimulated emission: Since stimulated absorption and stimulated emission have the equal probability, the mechanism is similar. We have a negative term $-\frac{\sigma(\omega) I}{\hbar \omega} N_2$ .

We write down

$\frac{\mathrm{d}N_2}{\mathrm{d}t} = -N_2 \Gamma -\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1)$

For steady state, spontaneous emission + stimulated emission = stimulated absorption, $\mathrm{d} N_2/\mathrm{d}t = 0$. We obtain the steady state equation

$-\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1) = N_2 \Gamma$

#### Intensity attenuation

When a laser beam with a frequency of $\omega$ and an intensity of $I$ is incident on the medium, we have

$\frac{\mathrm{d}I}{\mathrm{d} z} = -\alpha(\omega) I$

where $\frac{\mathrm{d}I}{\mathrm{d} z}$ is the dissipation (caused by pure absorption) of energy per unit volume per unit time. At the same time, $\sigma(\omega) I(N_2 - N_1)$ is also the dissipation of energy per unit volume per unit time, where $N_2 - N_1$ means stimulated emission minuses stimulated absorption, leaving the pure absorption. So

$\alpha(\omega) = -\sigma(\omega)(N_2 - N_1)$

Combining $N_2 + N_1 = N$ and the steady state equation, we could finally obtain

$\alpha(\omega) = \frac{\sigma(\omega) N}{1 + \dfrac{2\sigma(\omega)I}{\hbar \omega \Gamma}}$

#### Line shape function

We have found that for a fixed number of particles $N_1,N_2$ and a constant incident light intensity $I$, the energy absorbed by the medium per unit time varies depending on the frequency of the incident light $\omega$. This is reflected in the dependence of absorption cross-section $\sigma(\omega)$ on ω. Furthermore, we can extract a dimensionless line shape function $g(\omega)$ from it, describing the absorption efficiency of the medium for different frequencies of light. It also means the stimulated emission power spectrum.

### Quantum Model

• $\rho_{gg}$ : The probability of a single atom on the ground state.
• $\rho_{ee}$ : The probability of a single atom on the excited state.

By solving the Optical Bloch Equation, we can obtain

$\rho_{ee} (t\to \infty) = \frac{\Omega^2 / \Gamma^2}{1+\dfrac{4\Delta^2}{\Gamma^2}+\dfrac{2\Omega^2}{\Gamma^2}}$

Define a saturation parameter $S_0 = 2\Omega^2/\Gamma^2 = I/I_\text{sat}$, the density matrix element becomes

$\rho_{ee} (t\to \infty) = \frac{S_0/2}{1+S_0+\dfrac{4\Delta^2}{\Gamma^2}}$

When on-resonance and large intensity $S_0 \gg 1$, $1/2$ of the atoms occupy the excited state, reaches its maximum.

#### Resonance fluorescence

We will now consider the radiation due to a single, isolated atom driven by a monochromatic field. The radiation, viewed as scattered light, is called resonance fluorescence.

The optical Wiener-Хи́нчин theorem writes

$I_\text{sc}(\boldsymbol{r},\omega) = \frac{1}{2\pi} c n^2 \varepsilon_0 \int_{-\infty}^{+\infty} \langle E^{(-)}(\boldsymbol{r},t) E^{(+)}(\boldsymbol{r},t+\tau) \rangle \, \mathrm{e}^{\mathrm{i}\omega\tau} \, \mathrm{d} \tau$

where $\hat{E}$ contains $\sigma, \sigma^\dagger$. This will become

$I_\text{sc}(\boldsymbol{r},\omega_\text{sc}) = \frac{\hbar \omega_0 \Gamma}{r^2} f(\theta,\phi) S(\omega_\text{sc})$

where $S(\omega_\text{sc})$ is the spectrum of the scattered light with a dimension of $\mathrm{Hz^{-1}}$, which describes the probability of the fluorescence having a frequency between $\omega_\text{sc}$ and $\omega_\text{sc} + \mathrm{d}\omega_\text{sc}$. However, $S(\omega_\text{sc})$ is not normalized. The total probability is

$\int_0^\infty S(\omega_\text{sc}) \, \mathrm{d}\omega_\text{sc} = \rho_{ee}(t\to \infty)$

This results in

$P_\text{sc} = \int \frac{\hbar \omega_0 \Gamma}{r^2} f(\theta,\phi) \,\mathrm{d} \Omega \int S(\omega_\text{sc}) \, \mathrm{d}\omega_\text{sc} = \Gamma \hbar \omega_0 \rho_{ee}$

Conclusion: In spite of the fact that atomic spectra are the superposition of waves at different frequencies, the total received power $P_\text{sc}$ is exactly equivalent to an atom in the excited state emitting a photon $\hbar \omega_0$ with a probability of $\rho_{ee}$ in a unit time, regardless of whether the external electric field is on-resonant $\omega = \omega_0$ or off-resonant $\omega \neq \omega_0$.

It’s worth mentioning that the spectrum of the scattered light is named by Mollow Triplet Spectrum

$S(\omega_\text{sc})=\frac{s}{(1+s)^2} \delta\left(\omega_\text{sc}-\omega\right)+\frac{s}{8 \pi(1+s)} \frac{\Gamma}{\left[(\omega_\text{sc}-\omega)^2+(\Gamma / 2)^2\right]} + \cdots$

where $s=2\Omega^2/\Gamma^2$ is the saturation parameter.

Therefore, by measuring the power of resonance fluorescence, which is actually measuring $\rho_{ee}$ since $P_\text{sc} \propto \rho_{ee}$ , we can draw a graph To explain the line shape, we recall the expression of $\rho_{ee}$ :

$\rho_{ee} (t\to \infty) = \frac{S_0/2}{1+S_0+\dfrac{4\Delta^2}{\Gamma^2}}$

Define a parameter $\Gamma^\prime = \Gamma \sqrt{1+S_0}$ where $S_0 = 2\Omega^2/\Gamma^2 = I/I_\text{sat}$ evaluates the intensity of the incident beam. The $\rho_{ee}$ is written

$\rho_{ee} = \frac{S_0}{2(1+S_0)} \frac{1}{1+\dfrac{4\Delta^2}{\Gamma^{\prime 2}}}$

The full width at half maximum (FWHM) is exactly $\Gamma^\prime$ (When $\Delta = \pm \Gamma^\prime/2$ we have $\frac{1}{1+\frac{4\Delta^2}{\Gamma^{\prime 2}}} = \frac{1}{2}$). Two mechanism of broadening could be found

• In the weak-field limit, $S_0 \simeq 0$, $\Gamma^\prime \simeq \Gamma$. This is called natural broadening.
• In the strong-field limit, $\Gamma^\prime = \Gamma \sqrt{1+S_0}$ . The FWHM of the transition is effectively larger due to the strong coupling to the field. This phenomenon is called power broadening of the transition.

#### Connection with classical model

Warning: Only self-consistent when on-resonance $\omega=\omega_0$.

Optical Bloch Equation writes

$\dot{\rho}_{ee} = -\Gamma \rho_{ee} - \frac{\Omega^2/\Gamma}{1 + \dfrac{4\Delta^2}{\Gamma^2}} (\rho_{ee}-\rho_{gg})$

Naturally $\rho_{ee} = N_2/N,\,\rho_{gg} = N_1 / N$. Recall that in classical model

$\frac{\mathrm{d}N_2}{\mathrm{d}t} = -N_2 \Gamma -\frac{\sigma(\omega) I}{\hbar \omega} (N_2 - N_1)$

thus

$\sigma(\omega) = \frac{\hbar \omega}{I} \frac{\Omega^2/\Gamma}{1 + \dfrac{4\Delta^2}{\Gamma^2}}$

Since

$\Omega^2 = \frac{|\langle g |\hat{d}|e\rangle|^2}{\hbar^2} |E|^2 \qquad \Gamma=\frac{\omega^3_0}{\pi \varepsilon_0 \hbar c^3} |\langle g |\hat{d}|e\rangle|^2 \qquad I = \frac{1}{2}c \varepsilon_0 |E|^2$

The absorption cross-section now becomes

$\sigma(\omega) = \frac{2 \pi c^2 \omega}{\omega_0^3} \frac{1}{1 + \dfrac{4\Delta^2}{\Gamma^2}} \approx \frac{\lambda^2_0}{4} \frac{\Gamma^2}{2\pi (\Gamma^2/4 +\Delta^2)}$

where $\lambda_0 = 2\pi c/\omega_0$. If we define a line shape function

$g(\omega) = \dfrac{1}{2\pi} \dfrac{\Gamma}{\Gamma^2/4 +\Delta^2}$

we find $\sigma(\omega) = \Gamma \frac{\lambda^2_0}{4} g(\omega)$. We can obtain the same result taking advantage of energy conservation:

$-\sigma(\omega) I (N_2 - N_1) = P_\text{sc} = N\Gamma \hbar \omega_0 \rho_{ee} \implies \sigma(\omega) = \Gamma \frac{\lambda^2_0}{4} g(\omega)$

which means, given the same incident intensity and the same population, the efficiency of energy absorption by atoms varies with the changing incident light frequency $\omega$, and reaches its maximum at resonance $\Delta = 0$.