The study of the non-equilibrium dynamics of many-body open quantum systems has attracted increasing attention in recent years, due to the progress in several experimental areas. The time evolution of these systems is dictated by the Liouville-von-Neumann master equation which considers the interplay between the Hamiltonian dynamics of the system and the driven-dissipative processes. Typically, this dynamics leads to a non-equilibrium steady state for which a multitude of novel phenomena are expected, including the emergence of dissipative phase transitions. However, the theoretical modeling of out of equilibrium systems presents a major challenge since the computational eort scales exponentially with the system size. The study of these systems calls for the development of new eective methods.
In this talk I will discuss novel numerical techniques to the modeling of open many-body quantum systems. Firstly, I will present the Driven Dissipative Quantum Monte Carlo approach, a real-time pro- jector Monte Carlo method to stochastically sample the master equation. I demonstrate the eciency of our approach by applying it to the driven-dissipative two-dimensional spin lattice governed by the Heisenberg XYZ Hamiltonian.
Finally, I will also discuss some preliminary results on a variational approach based on the minimization of a suitable norm of the quantum master equation. I apply this method to the driven-dissipative one dimensional Bose-Hubbard model showing how the choice of the variational ansatz for the density matrix effects the accuracy of the steady state expectation values of certain observables.