My Honors Thesis- Into the world of Ion Coulomb Crystals

A week ago I graduated from the University of Colorado Boulder with my B.S. in Engineering Physics, which was an amazing experience where I learned more than I ever could imagine about natural science, engineering, and developed skills to conduct research. One of the best things I got to do as part of my degree program was write an honors thesis- a piece of independent research that at CU includes a defense and can earn you Latin honors. My thesis, Improving Molecular Dynamics Simulations of Ion Coulomb Crystals, is now available on CU’s public honors thesis repository. I was able to graduate magna cum laude thanks to this work, and I am quite proud of it! The details are all there in that 79 page document- good bedtime reading if I do say so myself. If you’re not inclined, I wanted to give a big picture summary of what exactly I did.

My thesis is centered on improving molecular dynamics simulations of ion Coulomb crystals. What are ion Coulomb crystals?

An ion Coulomb crystal composed of hundreds of Calcium 40 ions

Ion coulomb crystals are three-dimensional objects composed of charged particles. They are one-component plasmas- the one in the above image is composed of calcium ions. They are governed by the classical forces you likely learned about in physics class- Coulomb repulsion between the charged particles in a field provided by an ion trap force them to arrange themselves in these ordered structures. The one in the above image is football-shaped due to the trapping field it is exposed to. This trapping field can come from Paul and Penning traps, and therefore from electric and magnetic fields.

This doesn’t really give you a reason why we care about them though- and that is where the story gets interesting. These humble little crystals (the ions are about 10 microns apart – that’s micrometers) are thought to be found in extreme environments such as the surface of Neutron stars, and they are being used today for simulating chemistry in the interstellar medium as well as for quantum information experiments, paving the way towards quantum computing! These crystals have the potential to become fantastic high-tech engineering tools for building advanced computers or even helping to pave the way for putting together designer molecules like LEGOs. The crystals we make in the lab are very cold- to form crystals in the lab we use ultra-high vacuum chambers (single water molecules coming and reacting with your calcium will ruin your day) and Doppler laser cooling to cool the ions to below 10 millikelvin- extremely cold, just above absolute zero. Lasers are often thought to make objects warmer, but here we are exploiting quantum mechanics to cause the ions to lose energy. Recall high school chemistry when you learned about electron energy levels. When you kick an electron up an energy level it will cascade back down and emit a photon. This is great because these photons can be captured by a camera, creating beautiful pictures like the one above. As an added benefit, the ions over time will lose their momentum and eventually come nearly to rest, at which points they form the neatly ordered structures in these crystals. This cold temperature makes them advantageous for studying chemistry in extreme environments, which is something we know very little about. It also means that we can keep these ions confined and stored for long periods of time, making them advantageous for quantum computing research.

The lab I worked in was trying to characterize reactions. In order to do this, they needed analysis tools to help them out. This is where the work in my thesis comes in. I extended their existing molecular dynamics suite to help them produce accurate simulated images of ion Coulomb crystals that could be compared to the pictures they could take experimentally of these crystals. Why would we do this? We need to know the temperature of the ions, as well as well as confirm our understanding of how the system works. Molecular dynamics simulations use computers to evolve simulated ions over time until they form crystals using the same parameters as those used in the experiment- voltages on the ion trap electrodes, ion masses, number of ions, etc. With the crystals extremely isolated in the center of an oscillating trapping field a way was needed to figure out what the temperature of those ions is, which comes from the calculated energy of the ions. By using the Mathieu equations, which are a set of differential equations that can be used to model the motion of the ions in an ion trap, the program evolves the ions in time until they arrive in their final positions, where an image can be generated.

The work I undertook extended these simulations in a number of key ways. First, the original images rarely matched with experimental images for the same parameters and often looked blurry or had ions in non-physical positions. Second, the accuracy of the positions of the ions were improved by writing a new fourth-order integrator, which is the part of the computer program that actually evolves these ions in time by solving the equations of motion over and over (usually using a time step on the order of nanoseconds) so that the ion positions could be predicted with great accuracy. This had the added benefit of helping to fix some uncertainties in the ion energy we had, which is important when one of the main goals of the project is to extract temperature data for experimentally characterizing reaction rates. Finally I fixed the laser cooling model in the program- when Doppler laser cooling is being applied the atom will be hit with photons from the laser and absorb a photon if it corresponds with the right transition. The electron hops up an energy level and cascades back down, emitting a photon. This is called scattering. This happens in a completely random direction- there is no way to predict with certainty what direction this photon is going to go. This process heats the ions slightly. In the original program, a random direction would get picked but then the magnitude of the momentum kick applied to the ion from scattering this photon would be picked from a normal distribution. This means that most of the time the kick was extremely small because the distribution was centered by zero, and low values had the highest probability. The new model corrected this, and as a result we were able to find that the heating of the ions was now symmetric, making the images of the crystals look much more realistic.

An actual ion Coulomb crystal versus a simulated ion Coulomb crystal with the same experimental parameters.

As one can see from the above image, the simulations match the experimental images quite well. The ion positions are turned into images and then returned to the user. This also means that the ion counts can be extracted from the images by matching simulations of varying number of ions to actual crystals where you don’t know how many ions are present, say if you did not dump the crystal into the mass spectrometer to count them. This makes the simulations a nice check on the experiment and they work together to analyze data as well as to predict interesting parameters to use experimentally to produce crystals that are adventageous for measuring reaction rates in extreme environments.

More details and some sample code can be found in my thesis, as linked above. The simulations were written in C++ and I also developed a handy GUI to make using the simulations in the lab easier. This was a lot of fun and I learned a lot. It was also fantastic to be able to do some work that I knew would contribute to science and would help make the work done by the researchers in the lab easier. It was a great way to wrap up nearly four years of work at JILA at CU Boulder ahead of starting a new engineering position at Lockheed Martin this June.

If you have any questions, feel free to drop a comment or send me an email on my contact page!

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