  We investigate weakly coupled spin-1/2 ladders in a magnetic field. The work
is motivated by recent experiments on the compound
$\mathrm{(C_5H_{12}N)_2CuBr_4}$ (BPCB). We use a combination of numerical and
analytical methods, in particular the density matrix renormalization group
(DMRG) technique, to explore the phase diagram and the excitation spectra of
such a system. We give detailed results on the temperature dependence of the
magnetization and the specific heat, and the magnetic field dependence of the
nuclear magnetic resonance (NMR) relaxation rate of single ladders. For coupled
ladders, treating the weak interladder coupling within a mean field approach,
we compute the transition temperature of triplet condensation and its
corresponding antiferromagnetic order parameter. Existing experimental
measurements are discussed and compared to our theoretical results. Furthermore
we compute, using time dependent DMRG, the dynamical correlations of a single
spin ladder. Our results allow to directly describe the inelastic neutron
scattering cross section up to high energies. We focus on the evolution of the
spectra with the magnetic field and compare their behavior for different
couplings. The characteristic features of the spectra are interpreted using
different analytical approaches such as the mapping onto a spin chain, a
Luttinger liquid (LL) or onto a t-J model. For values of parameters for which
such measurements exist, we compare our results to inelastic neutron scattering
experiments on the compound BPCB and find excellent agreement. We make
additional predictions for the high energy part of the spectrum that are
potentially testable in future experiments.
