Mandel-Q Parameter and Phonon Statistics

[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)

 

[eljose79]

Thank you for your question about the use of the Mandel-Q parameter in evaluating the quantum dynamics of a single trapped ion. I can understand your confusion about the observed positive value of MQ for an average phonon number less than one.

First, let me clarify that the Mandel-Q parameter is a measure of the deviation of the photon number distribution from a classical distribution. It is not directly related to the quantization of the system. Therefore, even for a system with a quantized degree of motion, the MQ can still be positive or negative.

Now, for a system with an average phonon number less than one, it is indeed highly quantized as you mentioned. This means that the majority of the occupation is concentrated in the lower energy levels, and there is a low probability for transitions to higher energy levels. However, this does not necessarily mean that the phonon number distribution is classical. In fact, for a system with a highly quantized motion, the phonon number distribution is usually non-classical.

In your calculation of MQ, you have correctly considered the difference between the quadratic and linear moments of the photon number distribution. However, this difference is not directly related to the counting of phonon transitions. Instead, it reflects the fact that the distribution is non-classical, with a higher probability for having a certain number of phonons compared to a classical distribution.

In summary, the positive value of MQ for an average phonon number less than one indicates that the phonon number distribution is non-classical, which is expected for a highly quantized system. I hope this helps to clarify your understanding of the Mandel-Q parameter and its use in studying quantum dynamics. If you have any further questions, please don't hesitate to ask.