Mølmer–Sørensen gate

Given that the Mølmer-Sørenson gate only uses a single motional mode of the trapped ions and two of the electronic states to form the qubit, the Hamiltonian of the system for two ions can be expressed as:[1]

${\displaystyle H_{0}=\hbar \nu (a^{\dagger }a+1/2)+{\frac {\hbar \omega _{eg}}{2}}\left(\sigma _{z}^{(1)}+\sigma _{z}^{(2)}\right)}$

The time evolution of the state of a two qubit implementation of the Mølmer-Sørensen gate over two gate times from the ions starting in the ground state. States are labelled first by the qubit of each ion and then by the phonon occupancy of the motional mode. The grey circle represents a probability of 1/2 of that state.

where ${\displaystyle a^{\dagger }}$ and ${\displaystyle a}$ are the creation and annihilation operators of phonons in the ions collective motional mode, ${\displaystyle \hbar \nu }$ is the energy of those phonons, and ${\displaystyle \sigma _{z}^{(i)}}$ is the Pauli z matrix on the ${\displaystyle i}$th qubit. The Hamiltonian of the system in the interaction picture and with the biochromatic light, after applying the rotating wave approximation, is given by:[4]

${\displaystyle H(t)=\hbar \Omega (e^{-i\delta t}+e^{-\delta t})e^{i\eta (ae^{-i\nu t}+a^{\dagger }e^{i\nu t})}(\sigma _{+}^{(1)}+\sigma _{+}^{(2)})+h.c.}$

where ${\displaystyle \Omega }$ is the Rabi frequency on the qubit carrier transition and ${\displaystyle \eta }$ is the Lambe Dicke parameter. In the Lamde Dicke regime this Hamiltonian can be approximated and exactly integrated to yield the propagator ${\displaystyle U(t)}$:[4]

${\displaystyle H(t)\approx -\hbar \eta \Omega (a^{\dagger }e^{i\epsilon t}+ae^{-i\epsilon t})S_{y}}$

${\displaystyle U(t)={\hat {D}}\left({\frac {\eta \Omega }{\epsilon }}(e^{i\epsilon t}-1)S_{y}\right)exp\left[i{\frac {\eta ^{2}\Omega ^{2}}{\epsilon }}\left(t-{\frac {\sin {\epsilon t}}{\epsilon }}\right)S_{y}^{2}\right]}$

with ${\displaystyle \epsilon =\nu -\delta }$, ${\displaystyle S_{y}=\sigma _{1y}+\sigma _{2y}}$ and ${\displaystyle {\hat {D}}(\alpha )=e^{\alpha a^{\dagger }-\alpha ^{*}a}}$ is a displacement operator. So for ${\displaystyle t_{gate}=2\pi /\epsilon }$ the displacement operator vanishes and if ${\displaystyle \Omega =\epsilon /(4\eta )}$ the gate will be maximally entangle the ions.