Fröhlich discovered the remarkable condensation of polar vibrations into the lowest frequency mode when the system is pumped externally. For a full understanding of the Fröhlich condensate one needs to go beyond the mean field level to describe critical behavior as well as quantum fluctuations. The energy redistribution among vibrational modes with nonlinearity included is shown to be essential for realizing the condensate and the phonon-number distribution, revealing the transition from quasi-thermal to super-Poissonian statistics with the pump. We further study the spectroscopic properties of the Fröhlich condensate, which are especially revealed by the narrow linewidth. This gives the long-lived coherence and the collective motion of the condensate. Finally we show that the proteins such as Bovine Serum Albumin (BSA) and lysozyme are most likely the candidates for observing such collective modes in THz regime by means of Raman or infrared (IR) spectroscopy.This supplementary material will provide the derivations for some equations in main text and also some detail analysis of Fröhlich condensate in support of the main text.
DESCRIPTION OF MOLECULES IMMERSED IN SOLVENTWhen the molecules are placed in dense medium, i.e., solvent, the full description usually includes the energy of electrons, nucleis of molecules and nucleis of the medium, as well as their interactions. In practice, however, modeling to include so many degrees of freedoms is overwhelmly complicated, which makes the theory elusive and unaccessible. Fortunately, thank to their distinct energy and time scales, we do not have to deal with such a large amount of DOF in practice. This enables us to simplify the model which contains certain DOF of interests. Since we are focusing on the low-frequency (THz) vibrations of molecules where the electron motions will not be altered much in experiments, the effective description of the system contains only the nucleis of molecules and surrounding medium (solvent), dropping the electron energy. In this sense, the Hamiltonian of our interest reads