Bogoliubov transformation 3-mode

[Naibaf]

Hi everyone,

I'm working on spin-wave theory and I have a problem with a bogoliubov transformation.
I must do the transformation with 3 bosons and i have no idea how to do it.
I've only found the transformation for 1 and 2-mode bosons, but not for three...

It exist?

Thanks...



And sorry for my english...

 

[DrDu]

Maybe you could provide us with your hamiltonian?

 

[Naibaf]

The hamiltonian is very long, but the idea is that I have, products of bosons operator, for example: a*a + a*b + a*c + b*b + bb + b*c and so on...

Where a, b an c are boson operators.
As there are "non-diagonal" terms, the idea is use Bogoliubov Transformation for make the hamiltonian a diagonal hamiltonian... But for 2 differents bosons, the tranformation is:

alpha_{k} = cosh(theta_{k}) a_{k} - sinh(theta_{k}) b*_{k}, where b* is b-dagger

ans this transformation is like "universal" because everyone use it.
So my question is if there is a known bogoliubov transformation for 3 differents bosons in a hamiltonian...
( I hope you understand my english :) )

 

[bcrowell]

I can certainly tell you that a solution exists, at least in the fermionic case, because the Bogoliubov transformation is used routinely in nuclear physics, where you can have ~100 protons and neutrons. Sorry not to be able to supply the actual solution you want :-)

 

[Naibaf]

In this case, the 3 bosons operators are different, I have: a, b and c.
I guess that for ~100 differents protons the transformation must be huge!

I also guess that the bogoliubov transformation for fermions should not be so different from that of bosons......at least I have some hope! :)
thanks

 

[DrDu]

So, introducing the vector $$A=(a_k, a^*_{-k}, b_k, b^*_{-k},c_k, c^*_{-k})^T$$,
you should be able to write your hamiltonian as $$H=A^\dagger M(k) A$$. Then you have to find the unitary transformation U which diagonalizes the matrix M, i.e., you have to solve the eigenvalue problem for M. Then the vector of the alphas is UA.
Not a big deal, in principle.