Creation/Annihilation Operators to Numerics?
Basically as the title says, how would I go about doing numerical calculations from second quantisation? I'm mainly interested in circuit QED, and on the analytical side it's done mainly with second quantisation and creation/annihilation operators. However, if I try to proceed naively, where I go to the Fock space basis and write out the creation operator, for Bosons I'm limited by size of my matrix. The highest n number element is cut out, and when I try to do the commutator relation I can't get the identity back. The closest thing I've managed to find to representing creation/annihilation operators in a finite basis is the Jordan-Wigner transformation, but that's mapping Fermionic operators onto Bosons, and there doesn't seem to be a simple way to invert it. I could alternatively just put in more states than I need, but this would become very expensive very quickly (I would like to even model the environment and dissipation eventually).
What are some key terms that I should try to look for?