In this blog-post, we present our path and thoughts towards using ultra-cold atom experiments for quantum computation. They are the result of a two month internship where we studied the feasibility of such an undertaking in our group. Many associate only universal devices, especially qubit devices, to be valid quantum computers. We show how we think of our ultra-cold atoms in terms of quantum circuits and implement first steps in the software framework PennyLane.

Dynamical gauge fields are a fundamental concept of high-energy physics. However, learning about them typically takes enormous amounts of time and effort. As such, they are typically a bit mystical to students (including me) of other fields of physics like condensed-matter or AMO. Here, we will give a simple introduction into some of the concepts that might allow for the quantum simulation of these theories with ultracold atomic gases.The reader should know about second quantization and the basics of quantum mechanics as the arguments are based on this formalism.

In this lecture we will discuss some basic properties of the Helium atom. We will introduce first some useful notations for the specific Hamiltonian at hand. Then we will focus on the important consequences played by the electron-electron interaction on the spin structure and the level scheme of the system. Finally, we will introduce the variational method for the estimation of the ground state energy.

In this lecture we will discuss a systematic approach to build up molecules from more complex atoms.

In this lecture we will first discuss the diagonalization of the harmonic oscillator and then discuss the main properties of the hydrogen atom.

We will finish our discussion of the Helium atom. Most importantly, we will dive into the strong separation between singlet and triplet states.

In this lecture we will simply summarize the most important messages from the previous lectures.

In this lecture we will simply summarize the most important messages from the previous lectures.

In the last lecture we have seen that we can typically treat complex atomic systems within the central field approximation. &&=& {2} ^2_{_i} + V_ (r_i) \right)}_{_0} + \left( {r_{ij}} - S(r_i) \right)}_{_1} So we we can can treat atoms through the shell structure known from the atom, but the screening lifts the l degeneracy. For a single outer electron, we have even seen how this screening can be described by the quantum defect. We would now like to go beyond this simple picture and discuss the following questions: - How should the residual term $_1$ be taken into account? - How do we properly take into account the Pauli principle ? - How can we treat the fine-splitting ? ON THE RESIDUAL COUPLING If we ignore the residual coupling, we obtain a spherically symmetric problem, which implies that the angular momentum $_i$ of each electron is conserved. This conservation will be broken by $_{1}$. However, these forces are internal, which implies that the total angular momentum $ = \sum_i _i$ is conserved. So we should label the states in the complex Hamiltonian by $$. The total angular momentum will then set the symmetry of the spatial wavefunction. As already discussed in some detail for the He atom, this has wide-reaching consquence on the spin degree of freedom through exchange interaction. THE PAULI PRINCIPLE AND SPIN - According to the Pauli principle, each single-particle state can be occupied only by one electron. After distributing all electrons over different single-particle eigenstates (“orbitals”), the resulting state needs to be fully antisymmetrized (Slater determinant). - There is a simplification for atoms with many electrons: The angular momenta and spins of a complete subshell with n,  l,  {m−l, ⋯, ml} add to zero and can be ignored in the further considerations (“shell structure”). Note that this is often broken in molecular binding! - Alkali atoms are the simplest atoms with shell structure: All but one _valence_ electron add to L = 0, S = 0. The ground state thus has L = 0, S = 1/2. - For more complex atoms, the valence electrons couple to a total orbital angular momentum L with a given symmetry according to particle exchange. Let us have a look at two examples for light atoms, starting with : - 1s² → L = 0, S = 0. The corresponding term is ¹S - 1s2s → L = 0, {S = 0, S = 1}. The corresponding terms are ¹S and ³S. . The electronic configuration of is: _{L=0,\,S=0} 3p^2 Per valence electron we have l = 1 and s = 1/2. So we get L = 0, 1, 2 and S = 0, 1. Here S = 1 means symmetry and S = 0 antisymmetry with respect to particle exchange. In principle we can form the following terms: ^1S,\,^3S,\,^1P,\,^3P,\,^1D,\,^3D Which of these terms can be fully antisymmetrized? Here, only the terms ¹S, ³P and ¹D fulfill Pauli’s principle. In general the exchange interaction (seen in the discussion of He), will then lower the energy of the states with high spins. Optional: Symmetry of the L states We can construct the following L-states for them: &= ^{l_1},^{m_{l_1}};^{l_2},^{m_{l_2}}} &= {} ( ^{m_{l_1}},^{m_{l_2}}} - ) &= {}(^{m_{l_1}},^{m_{l_2}}} - + ) The states and are symmetric and the state is antisymmetric with respect to particle exchange.