  The dark states of a group of two-level atoms in the Tavis-Cummings resonator
with zero detuning are considered. In these states, atoms can not emit photons,
although they have non-zero energy. They are stable and can serve as a
controlled energy reservoir from which photons can be extracted by
differentiated effects on atoms, for example, their spatial separation. Dark
states are the simplest example of a subspace free of decoherence in the form
of a photon flight, and therefore are of interest for quantum computing. It is
proved that a) the dimension of the subspace of dark states of atoms is the
Catalan numbers, b) in the RWA approximation, any dark state is a linear
combination of tensor products of singlet-type states and the ground states of
individual atoms. For the exact model, in the case of the same force of
interaction of atoms with the field, the same decomposition is true, and only
singlets participate in the products and the dark states can neither emit a
photon nor absorb it. The proof is based on the method of quantization of the
amplitude of states of atomic ensembles, in which the roles of individual atoms
are interchangeable. In such an ensemble there is a possibility of
micro-causality: the trajectory of each quantum of amplitude can be uniquely
assigned.
