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 $$|f(x)| \lesssim e^{-(\frac12 - \lambda) \langle Ax, x\rangle} \quad \text{and} \quad |\widehat{f}(\xi)| \lesssim e^{-(\frac12 - \lambda) \langle B\xi, \xi\rangle} , \qquad \forall \lambda > 0 ,$$ 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.