Theory - the "M" part of MOT

This week we will be learning about the “M” part of MOT - magnetism. We will start with a discussion of some more atomic physics, namely the hyperfine structure of atoms. We will be studying these higher resolution states because our Rubidium atom states that we care about are hyperfine states. Following this, we will discuss Zeeman Splitting once again in the context of these hyperfine states. These are the principles that allow us to use magnetic fields to trap and cool atoms. From a practical perspective, you will learn about Helmholtz coils and creating the proper magnetic field you will need for trapping the Rubidium atoms. Alongside this comes the study of magnetometry.

Goals

• Develop a high level understanding of where the hyperfine states come from.

• Recognize which quantum number(s) are attached to the Zeeman affect in the hyperfine basis.

• Understand the effect of the magnetic fields to the total energy.

• Know the principles of magnetic trapping and what magnetic coil configuration creates our desired field.

Deliverables

• Create a program or a drawing of the magnetic field generated by the coils.

Why do we need this resolution?

We previously discussed some basic atomic physics. As you have already seen by the treatment of the fine structure and Zeeman effect, atoms are far from perfect Bohr model atoms defined solely by the principal quantum number. Although there are hundreds of levels we can get by adding more and more corrections, the reality is that we often only care about a few of these levels.

Remember our goal: we want to trap and cool Rubidium. Since the Rubidium is going to start as a vapor cloud, each individual atom will have a lot of momentum! There’s really no way for us to force the Rubidium to lose an excitation and immediately hit a low temperature. Thus, in real experiments, we isolate a cooling cycle*in which the atoms will jump between a few levels. It will gain a photon, release the photon to negate some momentum, then absorb another photon and repeat along this cycle. Finding these cooling cycles are actually an active area of research, and is part of the reason why only a few atomic species have been used in AMO experiments.

We will also be using a magnetic trap. Thus, we need to resolve our states to look at the levels that are trapped by the magnetic field. With the motivation established, let’s look at the hyperfine structure of atoms.

The Hyperfine Structure

Up until now, we have treated the nucleus like a stable blob that considers nothing. However, just like how the electrons orbiting the nucleus had a magnetic moment, so does the nucleus. The nuclear spin should then contribute some change in energy as it interacts with a magnetic field. But what provides this magnetic field?

Recall from E&M that a current carrying wire produces a magnetic field. It then shouldn’t be surprising that the electrons orbiting the nucleus creates a magnetic field as well (though it isn’t exactly the perfectly classical picture you saw in E&M).

Remember that $\pmb{J}$ is our total angular momentum for the electron. We will use $\pmb{I}$ to describe our nuclear spin. Thus, we should expect that the hyperfine contribution involves some coupling between $\pmb{J}$, the electron angular momentum, and $\pmb{I}$, the quantity relating to the nuclear magnetic moment. Thus, we will use a basis that uses the total angular momentum,

$$\pmb{F} = \pmb{J} + \pmb{I}$$

The corresponding quantum numbers will be $f$ for the total angular momentum and $m_f$, the angular momentum quantized in some direction (often the $z$-direction). In total, the basis we will use for the Hyperfine structure is $| n, I, , j, f, m_f\rangle$.

If you want a more formal treatment, the Hamiltonian causing the hyperfine structure takes the form

$$\hat{H}_{HFS} = A \pmb{I} \cdot \pmb {J}$$

which is very similar to that of Spin-Orbit coupling. External readings and references to learn more about this are provided at the end of the module.

The Zeeman effect on Hyperfine states

Now that we understand the hyperfine states, let’s revisit the Zeeman effect. Recall that the Zeeman effect appeared when our atoms were subject to a magnetic field. The different magnetic dipole moments, controlled by the $m_l, m_s$, or $m_j$ quantum numbers, directly correlated to the shift in energy from the Zeeman effect.

In our hyperfine basis, we will see the same thing. If our atoms are subject to some magnetic field, the total magnetic dipole moment will interact with a magnetic field to lead to an energy shift

$$E_{\text{Zeeman}} = g_F \mu_B m_f B$$

Where $B$ $b$is the magnitude of the magnetic field, $\mu_B$ is again the Bohr magneton, $m_f$ is the magnetic quantum number, and $g_F$is the Landé $g$ factor which scales the energy shift depending on the type of magnetic dipole moment being coupled.

Field Gradients and Magnetic Trapping

Now we can talk about the “M” part of a MOT. For this lab, you will use a 6-beam MOT. As we get deeper into MOT physics, you’ll learn what all the letters really mean and what each of the beams are.

Magnetic Forces and Dipoles

Before we even get to the experimental setup, let’s talk about what forces are at play. Our goal is to create some magnetic field that will confine our desired atoms. The physics of this you may have seen or heard of before - the famous Stern-Gerlach Experiment that demonstrated that angular momentum in the form of spin was quantized relied on the interaction of a magnetic dipole and the magnetic field.

A magnetic dipole in the presence of a magnetic field has a potential energy

$$U = - \pmb{\mu} \cdot \pmb{B}$$

Where $\pmb{\mu}$ is the magnetic dipole moment and $\pmb{B}$ is the magnetic field. Since we are working with the hyperfine states of our atom, we can also evaluate this element in terms of our quantum numbers. The potential energy contribution takes the form (as discussed previously)

$$U = g_F \mu_B m_f B.$$

Note that for a given atom, everything above is constant except the magnetic field magnitude. The force that each of the atoms will feel will be

$$\pmb{F} = - \nabla U = - g_F \mu_B m_f (\nabla B)$$

We are going to use this force to our advantage! Using a quadrupole trap, we will create an energy minima at the axis, leading all of our “coldest” desired atoms to be “pushed” to the center.

Figure 10.1 from Foot “Atomic Physics.”

The easiest way for us to create our magnetic fields will be with Helmholtz coils. These are essentially big loops of wires carrying a current that will produce a (near) uniform magnetic field in our desired region. Our coils will be set up in an anti-Helmholtz configuration.

Concept Check: Use the Biot-Savart law to calculate the magnetic field produced by a loop of wire with a current passing through. Then, use this result to calculate the magnetic field everywhere in an anti-Helmholtz configuration. Once you’ve done this, use your favorite programming language to write a code that draws the magnetic field for you for a given set of parameters!

From Maxwell’s equations, we know that $\nabla \cdot \pmb{B} = 0$. If we have no field in the $z$ direction, we must have that

$$\frac{\partial B_x}{\partial x} + \frac{\partial B_y}{\partial y} = 0$$

The easiest solution to this problem would be to set

$$\frac{\partial B_x}{\partial x} = - \frac{\partial B_y}{\partial y} = b$$

Where $b$ is some constant. The resulting magnetic field is therefore

$$\pmb{B} = b (x \hat{e}_x - y \hat{e}_y) + \pmb{B}_0$$

where $\pmb{B}_0$ is some constant magnetic field. If the constant field is set to zero, the magnitude of this field is $|\pmb{B}| = br$ where $r$ is the radial distance from the center. This gives us the desired potential! The farther we are from the center, the stronger the magnetic force.

However, we run into a problem - the potential energy is sharp at $r=0$. This can be solved by adding a small bias field in the $z$-direction. Thankfully, this is already built into $\pmb{B}_0$. We can choose $\pmb{B}_0 = B_0 \hat{e}_z$.

Concept Check: Let’s assume the magnitude of the bias field is large and $r$$r$ is small ($br << B_0$). Show that the magnitude of the field is approximately

$$|\pmb{B}| = B_0 + \frac{b^2 r^2}{2 B_0}.$$

The quadrupole trap plus a bias field will work to radially confine our atoms.

Pre-lab questions and concept checks

• What leads to hyperfine splitting? Why is it often ignored in first year quantum mechanics?

• How does magnetic trapping work? What principles are in play?

• How are we generating the magnetic field for our MOT? Why is this configuration practical and useful?

• Create a code that will allow you to visualize the magnetic field as a vector field. How does changing the positions and size of your coils as well as the current in the coils affect your magnetic field?

References and Aditional Reading

• Sakurai: Chapter 5 (Zeeman Effect and Fine Structure)

• Foot: Chapter 1, 5 (Zeeman Effect and Fine Structure), 6 (Hyperfine Structure), 10 (Magnetic Coils and MOTs)

• Griffiths 3rd edition: Chapter 7 (Fine structure, Zeeman Effect, Hyperfine Structure)

• ChemLibreTexts on Fine and Hyperfine Structure

• MIT OCW 8.06 Lecture on the Relativistic Correction

• MIT OCW 8.06 Lecture on Zeeman Splitting and Fine Structure