Trapped Ions crystals

The birth of ion trapping

Trapped ions stand out as one of the most promising platforms for quantum computing and quantum simulations. The field traces its roots back to 1953 with the pioneering work of Wolfgang Paul^{1 2} who introduced the radio-frequency trap for ions (also know as Paul trap). The first experiments with trapped ions had broad applications in the field of mass spectroscopy. The use Paul traps for quantum computing began in 1995 when Cirac and Zoller ³ proposed an implementation of a trapped ion quantum computer that would satisfy the DiVincenzo criteria ⁴.

Various types of traps

There are several types of traps used to confine charged particles, each utilizing different physical principles.

The two most common designs in the field of quantum physics are:

• Paul traps : A Paul trap, or quadrupole ion trap, uses oscillating radiofrequency electric fields to confine charged particles in three dimensions. These traps are especially useful in quantum computing, where precise control over individual ions, which act as qubits, is essential.
• Penning traps : A Penning trap confines the motion of charged particles by using static electric and magnetic fields. The magnetic field forces the particles into a circular motion, while the electric field creates a potential well along the axis of the magnetic field, effectively trapping the particles in a stable configuration. Penning traps are widely employed in high-precision experiments, such as mass spectrometry, fundamental particle measurements, and antimatter research, due to their ability to hold particles in a controlled environment for extended periods.

Paul traps

Here we delve a bit more in detail on the operational principle of Paul traps. The force on a charged particle in an electric and magnetic field is given by:

where $q$ is the charge of the particle, $\textbf{v}$ is its speed and $\textbf{E}$ and $\textbf{B}$ are respectively the electric and magnetic field. Although, in practice most of experiments rely on an external magnetic field to lift the degeneracy of some internal states of the ions, these are usually small fields (few Gauss in strength) and can be ignored when considering the particle dynamics in a trap. This leaves us with only the first term and the force becomes $\textbf{F} = q\textbf{E}$. The force governing the dynamics of the ions then, is dependent on its mass, charge and the shape of electric field generated by the trap.

So far, we haven't discussed much about the shape of the electric field. It's useful to define the electric field in terms of its potential as $\textbf{E} = -\vec{\nabla}\Phi$, where $\vec{\nabla}$ represents the gradient of the field. According to Earnshaw's theorem, point-like charged particles cannot be held in a stable, stationary position using only electrostatic or magnetic fields. Mathematically, this is expressed by requiring that the Laplacian of the potential equals zero: $\Delta \Phi = 0$.

This means that static fields alone can't be used to trap ions. To trap them, we need to either combine static and magnetic fields (as in Penning traps) or use a mix of static and radio-frequency (RF) fields, like in a Paul trap.

The potential generated by a Paul trap is given by:

Here, $U^{\mathrm{DC}}$ is the strength of the static (DC) electric potential, while $U^{\mathrm{RF}}$ and $\Omega$ represent the strength and frequency of the radio-frequency (RF) potential. The parameters $\alpha^{\mathrm{RF/DC}}_i$ describe the curvature of the RF/DC fields in the $i$ direction. These parameters are dependent on the geometry of the trap.

The oscillating part of $\Phi_{\mathrm{PT}} \left(\mathbf{r},t\right)$ causes the field to switch polarity every half period ($\pi/\Omega$). The electric field generated at $t=0$ and $\pi/\Omega$ is shown in the figures below. When an ion is placed inside the trap, it experiences an oscillating force: anti-confining (red arrows) in one direction and confining (green arrows) in the opposite direction. If the polarity switches fast enough, this creates an effective trapping potential, keeping the ion confined.

$t = 0$

$t = \pi/\Omega$

 

To illustrate this, here's an animation made with p5js. It shows an ion trapped in the center of an oscillating potential. The surface represents the field $\Phi_{\mathrm{PT}} \left(\mathbf{r},t\right)$ in the $r_x$ ($x$) and $r_y$ ($y$) directions, and the blue projection below tracks the ion's position in the x and y directions. You can use the slider to adjust the speed of the ion's dynamics.

Note that in the animation, the ion appears to move on the surface, but this is more of a visual aid. The z-axis (blue) should be thought of as an energy scale, not a spatial direction. The ion’s actual motion, shown below the surface, involves slow elliptical movement with fast oscillations layered on top.

Following some math, one finds that the motion of the ion is governed by the following equations:

where,

Eq. 3 is known as Mathieu differential equation. The motion of an ion in a Paul trap can be described analytically by the solution of such differential equations. In the limit of $q_j\ll 1$ and $a_j\ll q_j$ one can expand to a lowest-order approximation and express the ion's motion in a simple form:

where $A_j$ is the amplitude of motion of the ion, $\phi_j$ a relative phase and

The motion described by Eq. 4 is shown in the figure below. This motion is a combination of slow harmonic motion at frequency $\omega_j$ and fast oscillations at frequency $\Omega$. In ion trapping, we refer to these as "secular motion" and "micromotion", respectively. The ion’s secular motion, without micromotion ($q=0$), is represented by the red dashed line, while the full motion (secular plus micromotion) is depicted by the thick blue line.

More ions

When adding more ions in the trap we have to consider for the Coulomb repulsion between the ions:

with $q_{i/j}$ the charge of ion $i/j$, $\epsilon_0$ the vacuum permittivity and $\mathbf{r}_{i,j} = \mathbf{r}_i - \mathbf{r}_j$ the displacement vector between ion $i$ and $j$.

Typically, all ions in the trap carry the same charge, so $q_{i} = q_{j}$. However, recent work has explored the use of a mix of different ion species or even a mix singly and highly charged ions ($q>2$); see, for example, ⁵. At low temperatures, when $q_{i} = q_{j}$, the ions arrange themselves into regular crystals—either one-dimensional (1D), two-dimensional (2D), or three-dimensional (3D)—depending on the trap configuration, which is determined by the a and q parameters in Eqs. 3 and 4.

For ions to crystallize, we need to include a cooling mechanism in the equations of motion. In trapped ion systems, cooling is usually achieved through laser cooling, and the range of techniques available for this would require a separate post (which is already in the works). For now, let's treat cooling as a linear drag force on the ions such that $\textbf{F}_{\rm{Cool}} = C_{\rm{Cool}} \textbf{v}_{i}$ with $\textbf{v}_{i}$ the velocity of ion $i$. Our simplified model is enough for determining quite precisely the shape and the equilibrium positions of a $N$ ion crystal depending on the experimental parameter. Below you can find another p5js animation where you can see the shapes of such crystal in a 2D plane. This simulation can be used to extract the equilibrium positions of the ions as in cystals with $N>3$ these have to be found numerically ⁶.