1 Giant Grüneisen parameter in a strain-tuned superconducting quantum paraelectric: A consequence of the vanishing ferroelectric phonon energy J. Franklin 1 , B. Xu 1 , D. Davino 1 , A. V. Balatsky 1,2 , U. Aschauer 3 , I. Sochnikov 1,4 1 Physics Department, University of Connecticut, Storrs, CT USA, 06269 2 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden 3 Department of Chemistry and Biochemistry, University of Bern, Bern Switzerland 4 Institue of Material Science, University of Connecticut, Storrs, CT USA, 06269 Superconductivity and ferroelectricity are typically incompatible because the former needs free carriers, but the latter is usually suppressed by free carriers, unless their concentration is low. In the case of strontium titanate with low carrier concentration, unconventional superconductivity and ferroelectricity were shown to be correlated. Here, we report theoretically and experimentally evaluated Grüneisen parameters whose divergence under tensile stress indicates that the dominant phonon mode that enhances the superconducting order is the ferroelectric transverse soft-mode. This finding rules out all other phonon modes as the main contributors to the enhanced superconductivity in strained strontium titanate. This methodology shown here can be applied to many other quantum materials. Lightly doped strontium titanate (SrTiO3) has one of the lowest carrier densities among low carrier-densities superconductors. Its Fermi energy [1,2] is lower than its Debye energy [3,4]. It shows paraelectric-like or ferroelectric-like properties even in the metallic state [5]. Superconducting SrTiO3 cannot be described by a conventional theory of superconductivity and was suggested to be quantum critical [6]. Despite increased research activity, anti-adiabatic quantum-critical low-carrier-concentration superconducting SrTiO3 is still an unresolved fundamental mystery in the field of quantum materials and unconventional superconductors [7–23,1,24,25]. Furthermore, due to its multifaceted nature, SrTiO3 can be useful in designing remarkable new functionalities [26]. On the theory side, the debate focuses on the superconducting pairing mechanism. Specifically, there is much interest in what phonons provide the superconducting pairing and how [27–35]. Some recent experiments on approaching the quantum paraelectric to