Initial states and apodisation for quantum field simulations in phase-space

Abstract

Bosonic quantum fields can be simulated with "quantum software" in phase-space. The positive-P, Wigner and Q-function phase-space methods are reviewed. Initial quantum states and boundaries for infinite domains are considered in detail. The quantum initial conditions treated include both multi-mode Gaussian states and number states, which are sampled using phase-shifted weighting techniques. This is more efficient than direct probabilistic sampling. Algorithms for treating periodic boundaries within infinite domains are developed via apodisation, or absorbers near the boundary. Similar truncation or anti-aliasing issues arise in the frequency domain with Fourier transform methods. Complex apodisation, in which there are phase-shifts as well as absorption, improves accuracy by reducing unwanted errors, and this is illustrated numerically. The complex power law apodisation techniques described here can be applied to numerical calculations or even directly in optics. Phase-space simulations, used for both quantum optical and Bose-Einstein condensate quantum dynamical calculations, require quantum noise terms to be included. A method is described to analyze number conservation in apodised calculations. Some techniques developed here are potentially applicable to all digital simulations, whether using classical or quantum logic.