Thermodynamic length and work optimisation for Gaussian quantum states

Constructing optimal thermodynamic processes in quantum systems relies on managing the balance between the average excess work and its stochastic fluctuations. Recently it has been shown that two different quantum generalisations of thermodynamic length can be utilised to determine protocols with either minimal excess work or minimal work variance. These lengths measure the distance between points on a manifold of control parameters, and optimal protocols are achieved by following the relevant geodesic paths given some fixed boundary conditions. Here we explore this problem in the context of Gaussian quantum states that are weakly coupled to an environment and derive general expressions for these two forms of thermodynamic length. We then use this to compute optimal thermodynamic protocols for various examples of externally driven Gaussian systems with multiple control parameters.