Cfrgtkky

What is the Hamiltonian of a single ion in an ion trap? How to derive that formula? Also, how exactly can we create superposition by sending laser beams? I am interested in knowing the exact solutions of the Schrodinger's equation in this scenario. It will be helpful if someone can answer them or at least point to a book where I can find the answer.

The general Hamiltonian of a trapped ion is simply its kinetic, vibrational energy along with the potential energy interaction (Coulombic) with other ions (if in the neighborhood):
(For a single trapped ion)
begin{equation}
H=frac{|p|^2}{2m}+frac{m}{2}(omega_x^2x^2+omega_y^2y^2+omega_z^2z^2)
end{equation}
where usually $omega_x,omega_yll omega_z$ (nearly a 1D vibrational mode).
But this not via which the quantum information processing is achieved. In principle, the aim is to minimize the effect of this Hamiltonian and cool the ion via laser cooling to minimize the splitting levels via this Hamiltonian.

The key to quantum computation lies in the atomic structure (as @DaftWullie pointed out) of the ion, which is coupled to a laser. We usually isolate the atomic levels and tune the laser with the transition frequency. The effective way is to extract out a $mathbb{C}^2$ Hilbert space associated with the hyperfine structure of the atom where nuclear and electronic spin both interact.

The basic idea is to isolate two internal states in an approximately harmonic potential, which is used for multiple ions (qubits). For single qubit operations, for example, apply a EM field with a sharp frequency $omega_0$, the transition frequency of internal states, it turns on an internal Hamiltonian

begin{equation}
H_{int}=frac{hbar omega_0}{2}(S_+ e^{iphi}+S_-e^{-iphi})
end{equation}
which will give rise to a superpositon of the two internal states. $S_pm$ are the spin operators governing the internal total spin states and $phi$ is the phase associated (rotation). From this by tuning the parameters involved, we can form single qubit gates.

The procedure is similar for multi-qubit gates where we couple both the ions by a known mechanism, like the laser coupling and we formulate the Hamiltonian associated with the given transitions. Gates like the Controlled-unitary are somewhat hard to achieve because of their non-local nature. All these Hamiltonians can be easily solved analytically in the matric form (writing the Hamiltonian in the basis of the possible ket states involved with the given isolated levels).

In experiment it is very difficult to achieve a particular transition without other couplings involved and hence the ion is cooled to the order of $mK$ and tuned at very fine frequencies. The scalability is very hard to achieve in experiment.

To my knowledge, the best place to start it is the Nielsen and Chuang book, Physical Realization chapter from pg.309 onwards.
Also refer to this webpage by Oxford (who have a great group in Ion Trap experiments): https://www2.physics.ox.ac.uk/research/ion-trap-quantum-computing-group/intro-to-ion-trap-qc

Watch this talk: https://www.youtube.com/watch?v=WOQ_jWe62EA

Read via the wikipage: https://en.wikipedia.org/wiki/Trapped_ion_quantum_computer and few references therein.

And if time permits, refer to https://www.sciencedirect.com/science/article/pii/S0370157308003463 which was one of the foundational work in this area.

Hope this helps.