Dissipative Bose-Einstein condensation in contact with a thermal   reservoir
  We investigate the real-time dynamics of open quantum spin-$1/2$ or hardcore boson systems on a spatial lattice, which are governed by a Markovian quantum master equation. We derive general conditions under which the hierarchy of correlation functions closes such that their time evolution can be computed semi-analytically. Expanding our previous work [Phys. Rev. A 93, 021602 (2016)] we demonstrate the universality of a purely dissipative quantum Markov process that drives the system of spin-$1/2$ particles into a totally symmetric superposition state, corresponding to a Bose-Einstein condensate of hardcore bosons. In particular, we show that the finite-size scaling behavior of the dissipative gap is independent of the chosen boundary conditions and the underlying lattice structure. In addition, we consider the effect of a uniform magnetic field as well as a coupling to a thermal bath to investigate the susceptibility of the engineered dissipative process to unitary and nonunitary perturbations. We establish the nonequilibrium steady-state phase diagram as a function of temperature and dissipative coupling strength. For a small number of particles $N$, we identify a parameter region in which the engineered symmetrizing dissipative process performs robustly, while in the thermodynamic limit $N\rightarrow \infty$, the coupling to the thermal bath destroys any long-range order. 
Stephan Caspar, Florian Hebenstreit, David Mesterh\'azy, Uwe-Jens   Wiese
quant-ph cond-mat.quant-gas