Field theory of open quantum systems

Abstract

We apply and further develop field theory methods to study the Lindbladian evolution of many-body open quantum systems. The Keldysh functional integral technique is used to obtain dynamicalinformation, the stationary state, and response features for generic non interacting bosonic and fermionic Lindbladians. Special consideration is given to the relationship between the spectrum and stationary state, which are shown to be determined respectively by a certain non-Hermitian operator and as the solution to an operator-valued Lyapunov equation. This formalism is subsequently applied and further developed to study several specific topics. Non-Linear parametrically driven quantum oscillators can display bistability, in which spontaneous breaking of parity symmetry produces multiple long-lived states. We show that symmetry is restored at long times due to rare switching events between these states. We obtain the switching rates for parametric oscillators with both weak and strong parity symmetry from real-time instanton contributions to the Keldysh path integral. We also study disorder effects in non-interacting Lindbladian fermions, in which we observe non-concurrent phase transitions in the spectrum and stationary state. In random matrix models, this occurs by the non-concurrent opening of spectral gaps in either the spectrum or stationary distribution. We also consider a Lindbladian generalization of Anderson localization in finite spatial dimensions, which demonstrates localization transitions in the Lindbladian spectrum and stationary state occur at different critical disorder strengths. This demonstrates the existence of unconventional phases with a mixture of localized and delocalized features. Finally, we demonstrate an exact equivalence between classical population dynamics and Lindbladian evolution admitting a dark state and obeying a set of certain weak local symmetries. We then develop a simplified field theory method for studying “quantum population dynamics” models, in which this local symmetry condition is relaxed. In the population language, such theories allow non-classical processes in which animals behave like Schrödinger’s cat and enter superpositions of alive and dead states. A prototypical “Schrödinger cat” population model on a d-dimensional lattice is studied, which exhibits a phase transition between a dark extinct phase and an active phase that supports a stable quantum population. Critical behavior is obtained using perturbative renormalization group methods and is found to be different from both classical population dynamics and standard quantum phase transitions.