Nonlinear optical signatures of the tensor order in Cd$_2$Re$_2$O$_7$

The pyrochlore oxide Cd$_2$Re$_2$O$_7$ undergoes a structural phase transition at $T=200$~K with an unusual tensor character.\footnote{J. C. Petersen \textit{et al.}, Nature Phys. \textbf{2}, 605 (2006); C. A. Kendziora \textit{et al.}, PRL \textbf{95}, 125503 (2005); I. A. Sergienko \textit{et al.}, PRL \textbf{92}, 065501 (2004).} The order parameter for this state is two-dimensional, and associated with a phonon mode with $E_u$ symmetry. In magnetically frustrated pyrochlores, magnetoelastic coupling to similar modes can induce fascinating magnetically ordered states;\footnote{Y. Yamashita and K. Ueda, PRL \textbf{85}, 4960 (2000); O. Tchernyshyov \textit{et al.}, PRL \textbf{88}, 067203 (2002) and PRB \textbf{66}, 064403 (2002).} Cd$_2$Re$_2$O$_7$ is nonmagnetic, so it is possible to study the structural instabilities of the pyrochlore lattice in isolation. We have used optical second harmonic generation with polarization sensitivity to resolve an ambiguity in the low-temperature crystal structure, and verify an auxiliary condition on the structure that is implied by the order parameter symmetry. We also show that the temperature-dependence of the order parameter is consistent with thermal occupation of a Goldstone mode that results from the $E_u$ order parameter symmetry. The methodology that we have developed may be applied more widely in characterizing ordered states in matter.