  In the framework of noisy quantum homodyne tomography with efficiency
parameter $1/2 < \eta \leq 1$, we propose a novel estimator of a quantum state
whose density matrix elements $\rho_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$,
for fixed $C\geq 1$, $B>0$ and $0<r\leq 2$. On the contrary to previous works,
we focus on the case where $r$, $C$ and $B$ are unknown. The procedure
estimates the matrix coefficients by a projection method on the pattern
functions, and then by soft-thresholding the estimated coefficients.
  We prove that under the $\mathbb{L}_2$ -loss our procedure is adaptive
rate-optimal, in the sense that it achieves the same rate of conversgence as
the best possible procedure relying on the knowledge of $(r,B,C)$. Finite
sample behaviour of our adaptive procedure are explored through numerical
experiments.
