# I Mandel-Q Parameter and Phonon Statistics

1. May 21, 2016

### jamie.j1989

Hi, i'm struggling to understand how the Mandel-Q parameter (MQ) can be used to evaluate the quantum dynamics of a single trapped ion. A trapped ion has a quantised degree of motional freedom so can be discussed in terms of the phonon.

Im studying the dynamics of a trapped ion which is subject to a detuning causing it to couple to the n±2 phonon states, . What I'm struggling with is how the Mandel-Q parameter is observed to be positive for an average phonon number which is less one, $\left<n\right> < 1$, where n is the phonon number, at the steady state. My understanding of the Mandel-Q parameter is that for an MQ<0 is said to be deviating from classical statistics, so more quantum, and vice versa for MQ>0.

My calculation of the MQ is

$$Q=\frac{\langle n^2 \rangle-\langle n\rangle ^2}{\langle n\rangle}-1$$

I would have thought that for a system with $\left<n\right> < 1$, is very quantised as the distribution of the occupation of phonon states is confined to only a few of the lowest phonon numbers. So should give and MQ<0.

My current reasoning as to why I'm observing MQ>0 is due to the coupling of n±2 phonon states, when a transition to the n±2 phonon state occurs we count a bunching of the of the phonons so the MQ becomes more positive. But then I'm not sure how a 'counting' of phonon transitions would occur as I'm using the master equation approach which is continuous in its evolution.

Any help with understanding this is greatly appreciated.

Thanks

(I can provide additional information if more of the physics is needed to help explain)

2. May 26, 2016