Epole matrix element

[pathfinder49]

In calc_Epole() it states the following:
https://github.com/OxfordIonTrapGroup/ion_phys/blob/da3c5ef9fe868aa297426506801e9f02a77c2f67/ion_phys/common.py#L353-L362

I could work this through, but it seems prudent to verify the matrix element definition used. Happy to update this once I understand the conventions.

[RHanley1]

My pull request #10 aims to define the conventions etc used. I would suggest reading this, as I believe it to be consistent with the code.

[pathfinder49]

In your document, you define the matrix element as:

This relates to the spontaneous transition rates via:

This does not seem to match the ePole definition in the code. Instead the ePole contributions for a transition are calculated as np.sqrt(A*(2*Ju+1)) * <sum of transition wigber3js> .

Without checking the wigner3j maths, the usage looks like the ePole elements are defined such that:

abs(ePole)^2 = A_ji

This leaves the phase definition of the epole elements unclear.

[pathfinder49]

ePole is implemented as dtype=np.float64. (Entries are both +ve and -ve. )

• This is fine for coherent simulations. (Rabi fequencies between bound states are real).
• The sign of the ePole elements has no physical significance?

[RHanley1]

This relates to the spontaneous transition rates via:

This does not seem to match the ePole definition in the code. Instead the ePole contributions for a transition are calculated as np.sqrt(A*(2*Ju+1)) * <sum of transition wigber3js> .

The definition you quote is correct for a single two-level transition. As discussed in section 1.2.1, one typically has access to the manifold decay rate, which is related to the individual transitions by

. Now, this is only true for angular momentum eigenstates J. When the states is a superposition of J eigenstates, one must sum over the the superposition, as discussed in 1.4.1. In this case, the phase factor is relevant, and I believe the definition in the code to be correct. .

[pathfinder49]

Thank you for pointing out the relation to the fine structure decay rates. Though I'm more interested in the numerical definition of the Ion.ePole matrix elements and how they relate to more standard properties of the corresponding transition between hyperfine eigenstates. The relation to A_ij is useful for constructing the density matrix evolution.

I was working under the assumption that the of the code is correct for it's current purposes.

[pathfinder49]

The signs of Ion.ePole elements are subject to change, see #11

[hartytp]

This leaves the phase definition of the epole elements unclear.

It doesn't leave the phase definition totally unclear because at non-zero B-field you get state mixing so if you don't get the phases right you don't calculate the correct decay rates. However, this doesn't fix the phases for states in different levels (e.g. for processes involving multiple fine structure levels).

[pathfinder49]

What I mean is: I'm unclear on the definition of Ion.ePole phase within the ion_phys package.

Regarding the state mixing, Isn't this already accounted for by the conversion from Ion.ePole_hf to Ion.ePole?