Continuous versus discrete truncated Wigner approximation for driven, dissipative spin systems

We present an alternative derivation for the recently proposed discrete truncated Wigner approximation (DTWA) for the description of the many-body dynamics of interacting spin-1/2 systems. The DTWA is a semi-classical approach based on Monte-Carlo sampling in a discrete phase space which improves the classical treatment by accounting for lowest-order quantum fluctuations.

We provide a rigorous derivation of the DTWA based on an embedding in a continuous phase space. We derive a set of operator-differential mappings that yield an exact equation of motion (EOM) for the continuous  Wigner function of spins. The truncation approximation is then identified as neglecting specific terms in the exact EOM, allowing for a detailed understanding of the quality of the approximation and possible systematic improvements. Furthermore, we show that the continuous TWA (CTWA) yields a straightforward extension to open spin systems.

We derive exact stochastic differential equations for dephasing, decay and incoherent pump processes, which in the standard DTWA suffer from problems such as non-positive diffusion. We illustrate the CTWA by studying the dynamics of dissipative 1D Rydberg arrays and compare it to exact results for small systems.