Scalable quantum field simulations of conditioned systems

We demonstrate a technique for performing stochastic simulations of conditional master equations. The method is scalable for many quantum-field problems and therefore allows first-principles simulations of multimode bosonic fields undergoing continuous measurement, such as those controlled by measurement-based feedback. As examples, we demonstrate a 53-fold speed increase for the simulation of the feedback cooling of a single trapped particle, and the feedback cooling of a quantum field with 32 modes, which would be impractical using previous brute force methods.

Figure 1

(Color online) Energy vs time for a single particle undergoing measurement-based feedback averaged over $10000$ “fictitious” noises and 100 “real” noises. An initial coherent state displaced in position with an initial energy of $3\hslash \omega$ is effectively cooled. We compare simulations using the Wigner phase space method (solid, red), and our WSDEs with (dashed, green) and without (dot-dashed, blue) breeding. Dotted lines correspond to the standard errors. The estimation of the momentum variable becomes rapidly inaccurate without breeding (see inset for momentum evolution over a single “real” noise path). This inaccuracy is fed back into the equations of motion resulting in failure of the integration method. Breeding corrects this sampling problem ensuring convergence to the exact solution for longer times.

Figure 2

Energy for a multimode quantum field calculation using 20 realizations of the WSDEs method with breeding.