0 - Introduction to Atomic Physics

This is meant to serve as a quick introduction to atomic physics and critical concepts/motivations for the experiment.

0 - Introduction to Atomic Physics

Brief History

The history of atomic physics has been intertwined with the development of quantum mechanics ever since the development of the first models of the hydrogen atom by Bohr. Significant work since then has been spent on both theoretical pursuits of increasingly accurate models for more complex atoms and molecules, and experimental realisations to cool down (i.e., remove thermal energy/velocity), trap and study exciting fundamental, even new physics with these atoms and molecules. Thanks to the fantastic development over the past few decades, a wide variety of atomic gases (and molecular) are mastered to quantum degeneracy by many experimental teams around the world with extreme precision via a number of magnetic, optical and/or electrical field techniques, and are leveraged to study a variety of theoretical models/systems relevant in other fields of physics.

One of the main goals of this course/curriculum is to expose you to one of the basic building blocks of any ultracold atomic experiment, which is the magneto-optical trap (Nobel Prize 1997), as well as the necessary components to build one.

Magneto-Optical Trap (MOT) Basics

A magneto-optical trap represents the first step for any atomic physics experiment where we can demonstrate control of the internal dynamics (atomic energy states) and the center-of-mass motion (kinetic energy) of atoms. In order to trap atoms, we will need two types of forces: (1) a position-dependent force (provided by a magnetic field gradient), (2) and a velocity-dependent force (provided by optical light). A combination of both is required to create a MOT. We will cover a few basic principles in this section on atom-magnetic field and atom-light interaction. For a more in depth treatment, refer to excellent resources here.

Zeeman shift / Magnetic trapping

The tendency for magnetic moments and magnetic fields to align implies the internal energy level of an atom is influenced by the strength of the magnetic field it is placed in, also known as the Zeeman effect. We can write the magnetic potential experienced by an alkali atom as $U_B=-\vec{\mu}(B)\cdot \vec{B}=m_F g_F \mu(B) B$ where,

• the direction of the magnetic field strength $\vec{B}$ defines the $z$ direction

• $g_F$ is the Landé g-factor for total (nucleus + electrons) angular momentum $\vec{F}$

• $m_F$ is the (integer or half-integer) magnetic quantum number for the same total angular momentum $\vec{F}$

• $\mu(B)$ is the magnetic moment obtained from the Breit-Rabi equation for alkali atoms.

At low magnetic fields, the non-linear magnetic-field dependence of the magnetic moment can be dropped simplifying $\mu(B)\approx \mu_B$, with $\mu_B$ the Bohr magneton. In this case, the magnetic potential experienced by the atom reduces to $U_B=m_F g_F \mu_B B$.

This magnetic potential may be used to form traps for some $m_F$ states, depending on the overall polarity of $U_B$. A magnetic trap generated by an anti-Helmholtz coil pair generates a center with a vanishing magnetic field strength, thus weak field seeking states (atomic states that have minimum $U_B$ when $B=0$ are trappable in this configuration.

Scattering Rate

Consider a simplified atom with 2 energy levels, a ground state $\ket{g}$and an excited state $\ket{e}$. Assuming the atom has zero velocity, and is exposed to monochromatic light with frequency $\omega_L$ will scatter photons at a rate of $\Gamma_\text{scat}$:

$$\Gamma_{scat}=\frac{\Gamma}{2}\times \frac{s}{1+s+(2\delta/\Gamma)^2}$$

where

• $\Gamma$ is the natural linewidth of the transition

• $s=2\Omega^2/\Gamma^2=I/I_{sat}$ is the saturation parameter, where $I$ is the laser intensity and $I_{sat}$ is the saturation intensity of the transition

• $\Omega$ is the Rabi frequency which describes the coupling strength between the two states

• $\delta=\omega_L-\omega_0$ is the detuning of the laser frequency from the atomic resonance frequency $\omega_0$

Radiation pressure

Every time an atom absorbs a photon, there is momentum transfer from the laser given to the atom equivalent to the photon momentum $\hbar \vec{k}_L$. The atom can then spontaneously emit a photon. Spontaneous emission happens in a random direction, thus its effect on the atomic momentum averages out after many cycles. This means the result of the absorption followed by spontaneous emission over many cycles gives a net force along the laser light direction

$$F=\hbar k_L \Gamma_\text{scat}$$

This is the so-called radiation pressure, describing the pressure exerted on the atoms during the scattering process.

Doppler cooling

Consider the one-dimensional motion of an atom in the presence of counter-propagating laser beams with equal frequencies and with wavevectors $k_1=k$ and $k_2=-k$ respectively. Summing the radiation pressure from the two beams, we obtain the overall force on the atom as

$$F=F_1+F_2=\hbar k \frac{\Gamma}{2} s \left( \frac{1}{1+s+\left(\frac{2(\delta - kv)}{\Gamma}\right)^2} - \frac{1}{1+s+\left(\frac{2(\delta + kv)}{\Gamma}\right)^2}\right)$$

Expanding this equation to first order in the velocity, we obtain $F=-\beta v$. For $\beta > 0$, the radiation pressure provides a damping force to the atoms, slowing them down.

Question: Under what frequency detuning condition do we satisfy the requirement for a damping force?

< Expand $\Rightarrow \beta \sim -\delta = \omega_0-\omega_L.$ Red detuning, tune laser frequency below the resonance>

Why Rubidium?

In atomic physics, the scientific community has been working hard to create ultracold samples across the entire periodic table, but we have mainly obtained the most success with elements found in the Group-I and Group-II columns with some exceptions. The elements most explored by research groups around the world are largely determined by a few factors.

• The simplicity of the internal energy level structure (allowing for a closed energy level cycle for doppler cooling)

• The elements natural properties (Feshbach resonances, vapor pressure at room temperature, isotopes, magnetic and electric field sensitivity)

• Accessibility to (cheap and stable) lasers with resonant wavelengths

• and more recently, properties when combined with another atom (e.g., electric field polarisability)

We will be largely working with Rubidium, primarily for the first 3 reasons.

Note: A large area of active research is in exploring laser cooling of more complex atoms and molecules

Atomic energy levels

Rubidium is composed primarily of two stable isotopes ${}^{85}\text{Rb}$ and ${}^{87}\text{Rb}$. In this experiment, we create a MOT for ${}^{87}\text{Rb}$. Here goes the ${}^{87}\text{Rb}$ D2 energy level diagram of interest.

We will be focusing on two particular transitions:

• $5^2S_{1/2} \quad F=2 \to 5^2P_{3/2} \quad F=3$ transition (cooling transition)

• $5^2S_{1/2} \quad F=1 \to 5^2P_{3/2} \quad F=2$ transition (repump transition)

As per the figure above, these transitions are dominated by the $5^2S_{1/2} \to 5^2P_{3/2}$ energy split of ~780.5nm and can be driven by the absorption of corresponding resonant light.

References

Atomic Physics by Christopher Foot

Empirically measured properties of Rubidium-87 and Rubidium-85.