  Quantum phase transitions are often embodied by the critical behavior of
purely quantum quantities such as entanglement or quantum fluctuations. In
critical regions, we underline a general scaling relation between the
entanglement entropy and one of the most fundamental and simplest measure of
the quantum fluctuations, the Heisenberg uncertainty principle. Then, we show
that the latter represents a sensitive probe of superradiant quantum phase
transitions in standard models of photons such as the Dicke Hamiltonian, which
embodies an ensemble of two-level systems interacting with one quadrature of a
single and uniform bosonic field. We derive exact results in the thermodynamic
limit and for a finite number N of two-level systems: as a reminiscence of the
entanglement properties between light and the two-level systems, the product
$\Delta x\Delta p$ diverges at the quantum critical point as $N^{1/6}$. We
generalize our results to the double quadrature Dicke model where the two
quadratures of the bosonic field are now coupled to two independent sets of two
level systems. Our findings, which show that the entanglement properties
between light and matter can be accessed through the Heisenberg uncertainty
principle, can be tested using Bose-Einstein condensates in optical cavities
and circuit quantum electrodynamics
