Coherent and squeezed phonons in single wall carbon nanotubes

Coherent states of phonon manifest in macroscopic beatings of reflectance or transmittance following the phonons frequencies when a material is excited with an ultrashort pulse. Such beatings predominantly arise from a single phonon excitation. The two-phonon (overtone or combination modes) excitations, such as the G' (or 2D) band phonons in single wall carbon nanotubes (SWNTs), on the other hand, are not compatible with the coherent phonon states picture. Nevertheless, their macroscopic beatings are observed in the experiment. Here we formulate a more general framework involving the so-called squeezed states of phonon to explain the origin of the two-phonon signals in the pump-probe experiment of SWNTs. For a given SWNT chirality, the G' band phonon intensity in terms of the squeezed states of phonons is compared with that in terms of coherent states of phonons. Our calculation reveals that the G' band intensity from the squeezed states of phonons is some orders of magnitude larger than that from the coherent states of phonons. We also compare the G' band phonon intensity with the intensities of other coherent phonon modes generated in ultrafast spectroscopy such as the G band. Furthermore, the coherent G band and squeezed G' band phonon intensities are found to be sensitive to the laser pulse width used in ultrafast spectroscopy.