Quantitative uncertainty principles for time-frequency Gaussian decay

For real symmetric positive definite matrices $A$ and $B$, we characterize when a function $f\in L^2({\mathbb{R}}^d)$ satisfies $$\mathrm{\mid }f(x)\mathrm{\mid }\lesssim e^{-(\frac{1}{2}-\lambda )⟨Ax,x⟩}\text{and}\mathrm{\mid }\widehat{f}(\xi )\mathrm{\mid }\lesssim e^{-(\frac{1}{2}-\lambda )⟨B\xi ,\xi ⟩},\mathrm{\forall }\lambda >0,$$ or even more specified time-frequency decay estimates, in terms of the skewed Hermite series expansion of $f$. We also consider coordinate-wise time-frequency decay and determine when it becomes equivalent to the same bounds on the skewed Hermite coefficients.