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I am trying to understand this trapped ion paper.

More specifically, I am trying to understand what they are exactly varying, when they are varying 'the phase'to obtain the oscillation of the parity.

The parity is defined as $\pi = \sum_{n=0}^{N} (-1)^j p_j$ where $p_j$ is the probably of the jth spin pointing down.

It appears that parity oscillation amplitude is a way to measure the state preparation fidelity.

Here is another paper that shows a plot of the oscillating parity. Here, they define the parity as $p = p_{even} - p_{odd} $ with $p_{even} $or $p_{odd}$ corresponding to the probability of finding the state with an even or odd number of excitations. I think the two definitions are the same thing. Here, they are varying 'the phase.' They call it the phase of the analysis pulse. Not sure if this phase is the same as above.

The bottom line is, I am not sure what they are varying when they observe this parity oscillation.

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Short answer: they are varying the phase of the electromagnetic field (in particular, the laser field) that drives the ions.

Long answer: when they create their GHZ state, they want to try to characterize the fidelity of this state -- how close is the experimental state to the true GHZ state. Mathematically, this is characterized by the overlap of the experimental density matrix with the target state $|\psi\rangle = \frac{1}{\sqrt{2}}(|000...\rangle + |111...\rangle)$. The fidelity is given by $F = \langle \psi | \rho | \psi \rangle$.

The fidelity is dictated by two important terms: one term is the population in the correct states. If they prepare the state many times and then measure the ions each time, how often do they find all the ions to be in $|000...\rangle$ and how often do they find $|111...\rangle$? Ideally, they would obtain these two outcomes 50% of the time at random, but in practice the probabilities will be lower.

Assuming that indeed they measure nearly 50% of the time in $|000...\rangle$ and 50% of the time in $|111...\rangle$, there is another important piece of puzzle: were the ions truly in a superposition of these two states, or were they just randomly in one or the other each time (a so-called 'statistical mixture')? To assess the degree of coherence between the population in $|000...\rangle$ and in $|111...\rangle$, they measure these states in a different basis. Whereas a direct measurement could be thought of as measuring each spin in the $Z$-basis, by applying a short resonant laser pulse they can rotate the spins into the $X$ or $Y$ basis (or any combination of the two).

This final rotation into a different basis is sometimes referred to as an 'analysis pulse'. The phase of the analysis pulse determines what basis they measure in. A perfect GHZ state would display oscillations in the parity as a function of the analysis pulse phase (and the frequency of these oscillations would also grow with the number of entangled particles).

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