- #1
Baggio
- 211
- 1
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
<br /> \[<br /> \begin{array}{l}<br /> \zeta _1 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t - \delta \tau /2) - i(\omega - \Delta /2)){\rm{ }}\forall {\rm{ }}t - \delta \tau /2 > 0 \\ <br /> \zeta _1 (t) = 0{\rm{ }}\forall {\rm{ }}t - \delta \tau /2 < 0 \\ <br /> \zeta _2 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t + \delta \tau /2) - i(\omega + \Delta /2)){\rm{ }}\forall {\rm{ }}t + \delta \tau /2 > 0 \\ <br /> \zeta _2 (t) = 0{\rm{ }}\forall {\rm{ }}t + \delta \tau /2 < 0 \\ <br /> \end{array}<br /> \]<br />
Here \tau, \delta\tau, \omega, \Delta and t are real
The function that I expect to derive is:
<br /> \[<br /> P(\tau ,\delta \tau ,\Delta ) = \frac{1}{2}\exp \left( { - 2\left| {\delta \tau - \tau } \right|} \right) + \frac{1}{2}\exp \left( { - 2\left| {\delta \tau + \tau } \right|} \right) - \cos \left( {\left| \tau \right|\Delta } \right)\frac{1}{2}\exp \left( { - 2\left| {\delta \tau } \right| + \left| \tau \right|} \right)<br /> \]<br />
|...| 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
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
<br /> \[<br /> \begin{array}{l}<br /> \zeta _1 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t - \delta \tau /2) - i(\omega - \Delta /2)){\rm{ }}\forall {\rm{ }}t - \delta \tau /2 > 0 \\ <br /> \zeta _1 (t) = 0{\rm{ }}\forall {\rm{ }}t - \delta \tau /2 < 0 \\ <br /> \zeta _2 (t) = \sqrt[4]{{\frac{2}{\pi }}}\exp ( - (t + \delta \tau /2) - i(\omega + \Delta /2)){\rm{ }}\forall {\rm{ }}t + \delta \tau /2 > 0 \\ <br /> \zeta _2 (t) = 0{\rm{ }}\forall {\rm{ }}t + \delta \tau /2 < 0 \\ <br /> \end{array}<br /> \]<br />
Here \tau, \delta\tau, \omega, \Delta and t are real
The function that I expect to derive is:
<br /> \[<br /> P(\tau ,\delta \tau ,\Delta ) = \frac{1}{2}\exp \left( { - 2\left| {\delta \tau - \tau } \right|} \right) + \frac{1}{2}\exp \left( { - 2\left| {\delta \tau + \tau } \right|} \right) - \cos \left( {\left| \tau \right|\Delta } \right)\frac{1}{2}\exp \left( { - 2\left| {\delta \tau } \right| + \left| \tau \right|} \right)<br /> \]<br />
|...| 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
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