Introduction

# Stochastics

In addition to the standard time evolution, QuantumOptics.jl also features various possibilities to treat stochastic problems. In general, the usage of the Stochastics module is very similar to the standard time evolution. The main difference is, that additional terms that are proportional to the noise in the equation have to be passed to the corresponding functions.

tout, ψt = stochastic.schroedinger(T, ψ0, H, Hs; dt=1e-2)
tout, ρt = stochastic.master(T, ψ0, H, J, Js; dt=1e-1)

Note, that we need to set the keyword dt here, since the default algorithm is a fixed time step method (see below). Like the Time-evolution module, the stochastic solvers are built around DifferentialEquations.jl using its stochastic module StochasticDiffEq. Many of the options available for stochastic problems treated with DifferentialEquations.jl like, for example, the choice of algorithm can be used seamlessly within QuantumOptics.jl.

### Default algorithm and noise

The default algorithm is a basic Euler-Maruyama method with fixed step size. This choice has been made, since this algorithm is versatile yet easy to understand. Note, that this means that by default, stochastic problems are solved in the Ito sense.

To override the default algorithm, simply set the alg keyword argument with one of the solvers you found here, e.g.

tout, ψt = stochastic.schroedinger(T, ψ0, H, Hs; alg=StochasticDiffEq.EulerHeun(), dt=1e-2)

Note, that the switch to the EulerHeun method solves the problem in the Stratonovich sense.

The default noise is uncorrelated (white noise). Furthermore, since most equations involving quantum noise feature Hermitian noise operators, the noise is chosen to be real. For example,

tout, ψt = stochastic.schroedinger(T, ψ0, H, Hs; noise=StochasticDiffEq.RealWienerProcess!(0.0, [0.0]), dt=1e-1)

corresponds to the default for a single noise term in the Schrödinger equation. Note, that the default is complex noise for semiclassical stochastic equations, where only classical noise is included (for details see stochastic semiclassical systems)

For details on the available algorithms and further control over the solvers, we refer to the documentation of DifferentialEquations.jl.