If ${\bf A} - {\bf B}$ is positive semidefinite, then is ${\bf A}^{1/2} - {\bf B}^{1/2}$ still positive semidefinite?

Question

For two real symmetric and positive semidefinite matrices, ${\bf A}$ and ${\bf B}$, of same size, can we prove

${\bf A} \succeq {\bf B} \quad \iff \quad {\bf A}^{1/2} \succeq {\bf B}^{1/2}$,

where $\cdot^{1/2}$ denotes the square root of a matrix.