Hybrid discrete-continuous truncatedWigner approximation for driven, dissipative spin systems

We present a systematic approach for the semiclassical treatment of many-body dynamics of interacting, open spin systems. Our approach overcomes some of the shortcomings of the recently developed discrete truncated Wigner approximation (DTWA) based on Monte Carlo sampling in a discrete phase space that improves the classical treatment by accounting for lowest-order quantum fluctuations. We provide a rigorous derivation of the DTWA by embedding it in a continuous phase space, thereby introducing a hybrid discrete-continuous truncated Wigner approximation. We derive a set of operator-differential mappings that yield an exact equation of motion (EOM) for the continuous SU(2) Wigner function of spins. The standard DTWA is then recovered by a systematic neglection of specific terms in this exact EOM. The hybrid approach permits us to determine the validity conditions and to gain a detailed understanding of the quality of the approximation, paving the way for systematic improvements. Furthermore, we show that the continuous embedding allows for a straightforward extension of the method to open spin systems subject to dephasing, losses, and incoherent drive, while preserving the key advantages of the discrete approach, such as a positive definite Wigner distribution of typical initial states. We derive exact stochastic differential equations for processes which cannot be described by the standard DTWA due to the presence of nonclassical noise. We illustrate our approach by applying it to the dissipative dynamics of Rydberg excitation of one-dimensional arrays of laser-driven atoms and compare it to exact results for small systems.

Automated summary

In this study, we propose a systematic approach for the semiclassical treatment of many-body dynamics of interacting, open spin systems. This approach improves the classical treatment by accounting for lowest-order quantum fluctuations and overcomes some of the limitations of the existing discrete truncated Wigner approximation. By embedding the discrete truncated Wigner approximation in a continuous phase space, we derive an exact equation of motion for the continuous SU(2) Wigner function of spins. By neglecting specific terms in this exact equation of motion, we recover the standard discrete truncated Wigner approximation. This hybrid approach allows us to determine validity conditions and gain a detailed understanding of the approximation quality, paving the way for systematic improvements. We also demonstrate that the continuous embedding allows for an extension of the method to open spin systems subject to dephasing, losses, and incoherent drive.