Open quantum systems beyond the Markovian regime via Schwinger-Keldysh field theory

We develop a Schwinger-Keldysh field theory (SKFT) for open quantum systems and apply it to the spin-boson model as an archetypical example where the environment is composed of infinitely many harmonic oscillators. Prior field-theoretic formulations have been restricted to the Markovian regime, where they offer an alternative to a conventional Lindblad quantum master equation (QME) that is a time-local (i.e., memoryless) matrix differential equation evolving reduced density matrix of the system. Here we use the two-particle irreducible (2PI) action, effectively resumming a class of Feynman diagrams to an infinite order, where the diagrams are obtained from 1/N expansion with N being the number of Schwinger bosons to which the spin operators are mapped. Our SKFT+2PI yields a nonperturbative solution that reproduces time evolution in the Markovian regime (benchmarked against the Lindblad QME) and closely mimics time evolution in the non-Markovian regime (benchmarked against both hierarchical equations of motion and tensor networks methods). The SKFT+2PI approach can also access particularly challenging cases, such as zero-temperature combined with a sub-Ohmic bosonic bath, as well as enable evolution for arbitrary long times. Although the non-Markovian regime—where taking into account memory effects and revival of genuine quantum properties becomes essential—is needed for applications of the spin-boson model in quantum computing, such as understanding of qubit decoherence, it poses a challenge for numerous previously developed methods when trying to incorporate arbitrary properties of the bath, system-bath coupling and length of time evolution. Taking into account favorable numerical cost of solving the obtained integro-differential equations with increasing number of spins, time steps or spatial dimensionality—which eventually render all previously developed methods inapplicable—the SKFT+2PI approach offers a promising route for simulation of driven-dissipative systems in quantum computing or quantum magnonics and spintronics in the presence of a single or multiple dissipative environments.