  We study double quantum dots coupled to a quasistatic cavity mode with high
mode-volume compression allowing for strong light-matter coupling. Besides the
cavity-mediated interaction, electrons in different double quantum dots
interact with each other via dipole-dipole (Coulomb) interaction. For
attractive dipolar interaction, a cavity-induced ferroelectric quantum phase
transition emerges leading to ordered dipole moments. Surprisingly, we find
that the phase transition can be either continuous or discontinuous, depending
on the ratio between the strengths of cavity-mediated and Coulomb interactions.
We show that, in the strong coupling regime, both the ground and the first
excited states of an array of double quantum dots are squeezed Schr\"{o}dinger
cat states. Such states are actively discussed as high-fidelity qubits for
quantum computing, and thus our proposal provides a platform for semiconductor
implementation of such qubits. We also calculate gauge-invariant observables
such as the net dipole moment, the optical conductivity, and the absorption
spectrum beyond the semiclassical approximation.
