Dynamics of Integrable Systems

We demonstrate that local integrals of motion, exact and approximate, play important roles in the dissipative operator dynamics of many-body quantum systems. In real-time dynamics, they give rise to "slow modes", characterized by much slower decay rates compared to a typical local operator. Therefore, any local operator which has non-zero overlap with integrals of motion evolves into a superposition of these modes after a fast transient. Equivalently, these slow modes appear as low-lying eigenvalues in the Liouvillian spectrum. We develop a perturbative procedure to calculate the decay rate, or Liouvillian eigenvalue, of these modes. These results indicate that introducing weak dissipation to unitary many-body evolution enables the identification of local integrals of motion, either numerically through the diagonalization of a Lindbladian operator, or experimentally by probing dissipative dynamics on a quantum simulator.