  The impact of interparticle correlations on the behavior of Bose-Einstein
Condensates (BECs) is discussed using two approaches. In the first approach,
the wavefunction of a BEC is encoded in the $N$-particle sector of an extended
"catalytic state". Going to a time-dependent interaction picture, we can
organize the effective Hamiltonian by powers of ${N}^{-1/2}$. Requiring the
terms of order ${N}^{1/2}$ to vanish, we get the Gross-Pitaevskii Equation.
Going to the next order, $N^0$, we obtain the number-conserving Bogoliubov
approximation. Our approach allows one to stay in the Schr\"{o}dinger picture
and to apply many techniques from quantum optics. Moreover, it is easier to
track different orders in the Hamiltonian and to generalize to the
multi-component case. In the second approach, I consider a state of $N =l\times
n$ bosons that is derived by symmetrizing the $n$-fold tensor product of an
arbitrary $l$-boson state. Particularly, we are interested in the pure state
case for $l=2$, which we call the Pair-Correlated State (PCS). I show that PCS
reproduces the number-conserving Bogoliubov approximation; moreover, it also
works in the strong interaction regime where the Bogoliubov approximation
fails. For the two-site Bose-Hubbard model, I find numerically that the error
(measured by the trace distance of the two-particle reduced density matrices)
of PCS is less than two percent over the entire parameter space, thus making
PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the
error of PCS does not increase, in the time-dependent case, as the system
evolves for longer times. I derive both time-dependent and -independent
equations for the ground state and the time evolution of the PCS ansatz. The
time complexity of simulating PCS does not depend on $N$ and is linear in the
number of orbitals in use.
