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The loss of ultracold trapped atoms due to deeply inelastic reactions haspreviously been taken into account in effective field theories for low-energyatoms by adding local anti-Hermitian terms to the effective Hamiltonian. Herewe show that when multi-atom systems are considered, an additional modificationis required in the equation governing the density matrix. We define aneffective density matrix by tracing over the states containing high-momentumatoms produced by deeply inelastic reactions. We show that it satisfies aLindblad equation, with local Lindblad operators determined by the localanti-Hermitian terms in the effective Hamiltonian. We use the Lindblad equationto derive the universal relation for the two-atom inelastic loss rate forfermions with two spin states and the universal relation for the three-atominelastic loss rate for identical bosons.

Lindblad Equation for the Inelastic Loss of Ultracold Atoms