Initial value and dissipative formulation of a quantum harmonic oscillator

The in-in formalism and its influence functional generalization are widely used to describe the out-of-equilibrium dynamics of unitary and open quantum systems. In this talk, I will build on these techniques and discuss the effective theory of a quantum harmonic oscillator that is initialized in a pure/mixed state and undergoes dissipative dynamics. I will first restrict to a Gaussian initial state and time-local linear dissipation to obtain exact expressions for the oscillator's Green's functions. Using the resulting Green's functions, I will solve for the evolution of correlators and the purity of the oscillator, find when the fluctuation-dissipation relation is satisfied, and show why parameters in the effective action are constrained. I will next discuss how to go beyond Gaussian initial states and time-local linear dynamics, and briefly discuss the generalization to quantum field theory.