# Help required integrating this function over all time

## Main Question or Discussion Point

Hello,

I am trying to derive a function which describes the probability of two-photons interfering at a beamsplitter however I'm stuck on this particular integral. I need to solve:

$$P(\tau,\delta\tau,\Delta)=\frac{1}{4}\int|\zeta_{1}(t+\tau)\zeta_{2}(t)-\zeta_{1}(t)\zeta_{2}(t+\tau)|^2dt$$

The integral is over all t, i.e. t = -infinity to t = infinity

Where |...|^2 means the terms inside are multiplied by it's complex conjugate, and

$$$\begin{array}{l} \zeta _1 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t - \delta \tau /2) - i(\omega - \Delta /2)){\rm{ }}\forall {\rm{ }}t - \delta \tau /2 > 0 \\ \zeta _1 (t) = 0{\rm{ }}\forall {\rm{ }}t - \delta \tau /2 < 0 \\ \zeta _2 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t + \delta \tau /2) - i(\omega + \Delta /2)){\rm{ }}\forall {\rm{ }}t + \delta \tau /2 > 0 \\ \zeta _2 (t) = 0{\rm{ }}\forall {\rm{ }}t + \delta \tau /2 < 0 \\ \end{array}$$$

Here $$\tau, \delta\tau, \omega, \Delta$$ and t are real

The function that I expect to derive is:

$$$P(\tau ,\delta \tau ,\Delta ) = \frac{1}{2}\exp \left( { - 2\left| {\delta \tau - \tau } \right|} \right) + \frac{1}{2}\exp \left( { - 2\left| {\delta \tau + \tau } \right|} \right) - \cos \left( {\left| \tau \right|\Delta } \right)\frac{1}{2}\exp \left( { - 2\left| {\delta \tau } \right| + \left| \tau \right|} \right)$$$

|...| denotes absolute value.

In the integral above would I have to split it up into several contributions depending on which condition on "t" is satisfied. Also I don't see how integrating this function could lead to the absolute value terms in function P.

If anyone could offer any help at all I'd be very greatful... My research is stagnent because of this!

Thanks

Last edited:

anyone? :/

djeitnstine
Gold Member
I'd have to consult my physics professor D=

hehe, i don't think it should be that difficult btw in the eqs above it should be

$$$\begin{array}{l} \zeta _1 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t - \delta \tau /2) - i(\omega - \Delta /2)t){\rm{ }}\forall {\rm{ }}t - \delta \tau /2 > 0 \\ \zeta _1 (t) = 0{\rm{ }}\forall {\rm{ }}t - \delta \tau /2 < 0 \\ \zeta _2 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t + \delta \tau /2) - i(\omega + \Delta /2)t){\rm{ }}\forall {\rm{ }}t + \delta \tau /2 > 0 \\ \zeta _2 (t) = 0{\rm{ }}\forall {\rm{ }}t + \delta \tau /2 < 0 \\ \end{array}$$$

it's just math :/

bump..