- #1

matteo137

- 43

- 9

The Hamiltonian of the system is the following:

[tex] H= \Delta \sum_j a^\dagger_j a_j + U \sum_j a^\dagger_j a^\dagger_j a_j a_j + J \sum_{\langle j,j^\prime\rangle} a^\dagger_j a_{j^\prime} + F_p \sum_j \left(a^\dagger_j + a_j \right)

[/tex]

where ##\Delta, U, J, F_p ## are real constants, and the index ##j## goes from 1 to M.

##F_p ## is the amplitude of the driving field, and the dissipations are introduced through a master equation

[tex] \dfrac{d\rho}{\text{dt}} = - i \left[ H,\rho\right] + \dfrac{\kappa}{2} \sum_j \left(2 a_j \rho a^\dagger_j -a^\dagger_j a_j \rho - \rho a^\dagger_j a_j \right)

[/tex]

where ##\kappa## is the real constant.

**Case 1:**for the case of ##M=3## sites (trimer), I described each site in the Fock basis. i

*.e.*with the vectors ##(1,0,0,...)=\vert 0\rangle##, ##(0,1,0,...)=\vert 1\rangle##, ... up to a cutoff ##\vert n_{\text{max}}\rangle##. Starting from this basis, I expressed the operators ##a##, ##a^\dagger## as matrices, and I built the Hamiltonian matrix, of size ##(n_{\text{max}})^M##, using Kronecker products of the kind ##a_1 = a\otimes 1\otimes 1##.

The matrix ##\rho(t)## is then obtained propagating a starting ##\rho(0)## with the Runge-Kutta algorithm.

This method does not seems very efficient and stable...

would it be better to work in the "occupation" basis? i

*.e.*writing the operators in the basis built from the vectors ##(0,0,0)=\vert 0,0,0\rangle##, ##(1,0,0)=\vert 1,0,0\rangle##, ...

Or, for example, in the momentum basis, mean-field, ...?

**Case 2:**what about in two dimensions? ...is it better to work with a mean-field description?