I am constructing the interference of two Gaussian wave packets onto which I apply a complex phase, attempting to obtain the coherence measure C (how well they overlap).
The $\Delta x(T)$ term in the argument represents the spatial splitting of the packets (as a real phase) and the complex exponential one between the two is my main concern, as, usually, it leads to a simple real value of C:
However, for a special case when the two $\Delta x(T)$ terms are taken to be different, it leads to a complex phase outside the integral:
where the complete formula for C now becomes a super long one with the terms inside the integral leading to a real term (as any complex term resulting from the complex exponential takes the form of a sine which cancels-out in the even integral above).
The point is that I do not understand the meaning of a complex or even a negative coherence measure C (which is meant to represent the overlap of the packets). As seen in the picture below, the overlap is easy to interpret and a negative value of C simply implies reflecting the graphs on the x-axis. But my question is how can I interpret a negative value of C?
And just as importantly, what about a complex value of C?
I apologize for the lengthy and perhaps odd question, but I really found it versing and had to explain it well.