= Trapped-ion internal and motional states
{title2=$|g,n\rangle=|g\rangle\otimes|n\rangle$}
An internal <qubit> uses two distinguishable electronic or <hyperfine structure> <quantum states>, denoted $|g\rangle,|e\rangle$. The <ion>'s centre-of-mass motion has a different <Hilbert space>. In one approximately harmonic direction, a <number state> $|n\rangle$ has <energy> $E_n=\hbar\nu(n+1/2)$, where $\nu$ denotes a motional <angular frequency>, not ordinary <frequency>. The <tensor product of quantum systems> gives $|g,n\rangle$. Cooling towards $n=0$ prepares motion; it does not automatically prepare the internal <qubit>. Nor does the <ground state> mean a stationary point particle: <zero-point energy> and position spread remain.
For several <ions>, <Coulomb forces> couple their displacements into collective <normal modes>. Each selected <normal mode> has its own <quantum harmonic oscillator> <number states>, while each <ion> retains internal <qubit> states. A pulse addressing one <ion> can change a shared mode. This supplies a route to correlations between distant internal <quantum states>; it also makes residual motional excitation and unwanted excitation of other <normal modes> relevant. A single-ion pulse calculation alone establishes no experimental multi-ion <quantum gate> performance.
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