Quantum dynamics in the partial Wigner picture
Geoff Beck, Alessandro Sergi · Journal of Physics A Mathematical and Theoretical · 2013
Recently we have shown how the partial Wigner representation of quantum mechanics can be used to study hybrid quantum models where a system with a finitenumber ofenergy levels iscoupled tolinearornonlinear oscillators(Beck and Sergi 2013 Phys. Lett. A 377 1047). The purpose of this work is to provide a detailed derivation of the partially Wigner-transformed quantum equations of motion for nonlinear oscillator subsystems under the action of general polynomial potentials. Such equations can be written in terms of a propagator, which can then be expanded in a power series. The linear terms of the series describe quantum-classical dynamics while the nonlinear terms provide the corrections needed to restore the fully quantum character of the evolution. In thecaseofpolynomialpotentialsandpositiondependentcouplings,thenumber of nonlinear terms is finite and the corrections can be calculated explicitly. In this work we show how to implement numerically the above scheme where, in principle, no assumption about the strength of the coupling must be taken. We illustrate the formalism by studying a two-level system interacting with an asymmetric quartic oscillator. We integrate the quantum dynamics of the total system and provide a comparison with the case of the quantum-classical dynamics of the quartic oscillator. The approach presented here is expected to be effective for studying hybrid quantum circuits in quantum information theoryandforwitnessingthequantum-to-classicaltransitioninnano-oscillators coupled to pseudo-spins.