  The loss of ultracold trapped atoms due to deeply inelastic reactions has
previously been taken into account in effective field theories for low-energy
atoms by adding local anti-Hermitian terms to the effective Hamiltonian. Here
we show that when multi-atom systems are considered, an additional modification
is required in the equation governing the density matrix. We define an
effective density matrix by tracing over the states containing high-momentum
atoms produced by deeply inelastic reactions. We show that it satisfies a
Lindblad equation, with local Lindblad operators determined by the local
anti-Hermitian terms in the effective Hamiltonian. We use the Lindblad equation
to derive the universal relation for the two-atom inelastic loss rate for
fermions with two spin states and the universal relation for the three-atom
inelastic loss rate for identical bosons.
