Understanding 2nd Order Correlation in Fock States and Density Functions

[babylonia]

Hi all,

I read on some paper that for a system of Fock state |...n_k...>, and with the field operator expanded as
$$\Psi$$(r)=$$\sum$$a_k $$\phi$$_k(r), the second order density correlation function can be expressed as
G⁽²⁾=<$$\Psi$$⁺(r)$$\Psi$$⁺(r')$$\Psi$$(r')$$\Psi$$(r)>=<n(r)><n(r')>+|<$$\Psi$$(r)⁺$$\Psi$$(r')>|²-$$\sum$$$$^{N}_{k}$$ n_k ( n_k +1) |$$\phi$$^*(r)|²|$$\phi$$(r')|².
I have no idea how the last term, ie. the term after the minus sign, come out? If I use the Wick's theorem for
<a⁺_ka⁺_la_ma_n>=<a⁺_ka_m><a⁺_la_n>$$\delta$$_k,m$$\delta$$_l,n+<a⁺_ka_n><a⁺_la_m>$$\delta$$_k,n$$\delta$$_l,m,
so why in the 2nd correlation there are additional terms after '-'?

This seems really strange, can anybody help me? Thank you.

 

[cgk]

I didn't check your formula for the correlation function, but your Wick theorem looks wrong to me. You have a two-body density matrix on the left and on the right you have only one-body density matrices. Did you apply a relation like
$$\langle a^+_k a_m a^+_l a_n\rangle = \langle a^+_k a_m\rangle\langle a^+_l a_n\rangle?$$
Because such a relation does generally *not* hold.

 

[babylonia]

I didn't check your formula for the correlation function, but your Wick theorem looks wrong to me. You have a two-body density matrix on the left and on the right you have only one-body density matrices. Did you apply a relation like
$$\langle a^+_k a_m a^+_l a_n\rangle = \langle a^+_k a_m\rangle\langle a^+_l a_n\rangle?$$
Because such a relation does generally *not* hold.

Hi,

Thanks a lot for your reply. I think for Fock state the Wick theorem leads to $$\langle a^+_k a^+_l a_m a_n\rangle = \langle a^+_k a_m\rangle\langle a^+_l a_n\rangle \delta_{k,m}\delta_{l,n}+\langle a^+_k a_n\rangle\langle a^+_l a_m\rangle\delta_{k,n}\delta_{l,m}$$, because particle number conservation requires the other terms in the full expression given by wick theorem to vanish. And sorry I forgot to mention that my problem is for bosons. If this is wrong, why, and what is the correct form?

Thanks

 

[cgk]

I think for Fock state the Wick theorem leads to $$\langle a^+_k a^+_l a_m a_n\rangle = \langle a^+_k a_m\rangle\langle a^+_l a_n\rangle \delta_{k,m}\delta_{l,n}+\langle a^+_k a_n\rangle\langle a^+_l a_m\rangle\delta_{k,n}\delta_{l,m}$$, because particle number conservation requires the other terms in the full expression given by wick theorem to vanish. And sorry I forgot to mention that my problem is for bosons. If this is wrong, why, and what is the correct form?

Sorry, I might have misunderstood your post: By "In Fock Space", do you mean for a single permanent[1] (or for some mean field approximation?)? Because for a general superposition of permanents no such relation holds, and the two-body reduced density matrix $$\langle a^+_k a^+_l a_m a_n\rangle$$ in general cannot be reduced to anything which is itself less than a two-body (mixed) density matrix. So by applying (only) the Wick theorem, you could have, for example, something like [2]
$$\langle a^+_k a^+_l a_m a_n\rangle = \langle a^+_k a_m a^+_l a_n\rangle + \delta_{ml} \langle a^+_k a_n\rangle$$
but that still has a two-body density matrix in it and is still far from your expression for G. You might need to apply some other relations, too.

[1] that's the positive-symmetry version of a determinant
[2] how exactly the Wick theorem looks depends on whether there is a normal order imposed on the operators, and if it is, which reference it applies to.