  For the one-dimensional spin-1/2 XX model with either periodic or open
boundary conditions, it is shown by using a fermionic approach that the matrix
element of the spin operator $S^-_j$ ($S^-_{j}S^+_{j'}$) between two
eigenstates with numbers of excitations $n$ and $n+1$ ($n$ and $n$) can be
expressed as the determinant of an appropriate $(n+1)\times (n+1)$ matrix whose
entries involve the coefficients of the canonical transformations diagonalizing
the model. In the special case of a homogeneous periodic XX chain, the matrix
element of $S^-_j$ reduces to a variant of the Cauchy determinant that can be
evaluated analytically to yield a factorized expression. The obtained compact
representations of these matrix elements are then applied to two physical
scenarios: (i) Nonlinear optical response of molecular aggregates, for which
the determinant representation of the transition dipole matrix elements between
eigenstates provides a convenient way to calculate the third-order nonlinear
responses for aggregates from small to large sizes compared with the optical
wavelength, and (ii) real-time dynamics of an interacting Dicke model
consisting of a single bosonic mode coupled to a one-dimensional XX spin bath.
In this setup, full quantum calculation up to $N\leq 16$ spins for vanishing
intrabath coupling shows that the decay of the reduced bosonic occupation
number approaches a finite plateau value (in the long-time limit) that depends
on the ratio between the number of excitations and the total number of spins.
Our results can find useful applications in various "system-bath" systems, with
the system part inhomogeneously coupled to an interacting XX chain.
