How to Calculate Second Order Correlation from Master Equation?

[asei]

Hi,
I have a problem in calculating the second order correlation (coherence) from the master equation for the operators $$\sigma$$ and a a[+][/SUP] , because I don't know if
<aa^{+}a^{+}a> can be factorized to<a><n><a>.
I want to do this calculation directly from the density matrix solution.
thanks

 

[Cthugha]

This problem is not immediately clear to me. Do you want to calculate the second-order coherence for some light field?

If that is the case, I assume the operators are bosonic, but what exactly does $$\sigma$$ denote?
And how exactly do you get the $$\langle \hat{a} \hat{a}^\dagger \hat{a}^\dagger \hat{a} \rangle$$ term? Usually you consider normal-ordering of the operators to account for the effect of the measurement on the light field and get terms like $$\langle \hat{a}^\dagger \hat{a}^\dagger \hat{a} \hat{a} \rangle$$.

 

[asei]

yes you right about the normal ordering, the sigma are the atomic transition operators.
the problem is that when I calculate the mean values of these operators in time, how can I know the second order coherence. can I factorize the expression <A+A+AA> (the plus is dagger) to(<n><n(t)>)? if you know on some reference I will really appreciate it too.

 

[Cthugha]

Unfortunately, the $$\langle \hat{a}^\dagger \hat{a}^\dagger \hat{a} \hat{a} \rangle$$- term does not factorize to $$\langle \hat{n} \hat{n} \rangle$$.
Starting with the equal time correlation, you have $$\hat{a}^\dagger \hat{a} =\hat{a}\hat{a}^\dagger -1$$, so that the above term factorizes to $$\langle \hat{n} (\hat{n}-1) \rangle$$.

This makes sense as the detection of a photon changes the light field by destroying that photon. However, if you are interested in the time dependence of the correlation function, the term $$\hat{a}^\dagger (t+\tau) \hat{a}(t)$$ can be anything from $$\hat{a} (t) \hat{a}^\dagger(t+\tau) -1$$ to $$\hat{a} (t) \hat{a}^\dagger(t+\tau)$$ depending on the magnitude of $$\tau$$ compared to the coherence time of the light.

Finding a solution for this problem is rather demanding and depends on the kind of light field you are interested in. I suppose you are interested in lasers. "Classical" atom lasers are for example discussed within a birth-death model in "Photon statistics of a cavity-QED laser: A comment on the laser–phase-transition analogy" by P.R. Rice and H.J. Carmichael, Phys. Rev. A 50, 4318–4329 (1994). Semiconductor lasers are treated using the cluster expansion method in "Semiconductor model for quantum-dot-based microcavity lasers" by C. Gies et al., Phys. Rev. A 75, 013803 (2007).

If you could tell me what kind of system or light field you have in mind, I might be able to come up with more suitable references for your case.

 

[asei]

Hi
thanks for the help, actually I read the papers as you suggested and I track for more and I find the thing that I want. apparently, in order to calculate the the $$\left\langle[/a^{+}/a^{+}aa]\right\rangle we have to use the Heisenberg equation for it which depend on the other operators, where in some place I can use factorization in order to get a solution. let's say I have to generated quantum fields a_{1},a_{2} so in order to find their cross correlation I have to calculate the \left\langle a^{+}_{1}(t)a^{+}_{2}(t+\tau)a_{2}(t+tau)a_{1}(t)\right\rangle in this case what is the difference between t and t+\tau. and maybe because of that we can factorize earlier in the derivations.<br /> thanks for your help$$