  We introduce a new density for the representation of quantum states on phase
space. It is constructed as a weighted difference of two smooth probability
densities using the Husimi function and first-order Hermite spectrograms. In
contrast to the Wigner function, it is accessible by sampling strategies for
positive densities. In the semiclassical regime, the new density allows to
approximate expectation values to second order with respect to the high
frequency parameter and is thus more accurate than the uncorrected Husimi
function. As an application, we combine the new phase space density with
Egorov's theorem for the numerical simulation of time-evolved quantum
expectations by an ensemble of classical trajectories. We present supporting
numerical experiments in different settings and dimensions.
