Localized Bose-Einstein Condensation in films of Liquid $^4$He in Disorder

When porous media is only partially filled, the helium is deposited as films on the porous media walls. The initial helium is tightly bound on the media walls, denoted ``dead layers". In subsequent fillings there is a transition to a superfluid phase below a critical temperature $T_c$ and $T_c$ increases with increasing filling. We present measurements of phonon-roton modes in liquid $^4$He films on 25 A diameter gelsil at fillings greater than 70\%. P-R modes are observed at low temperature and up to a maximum temperature denoted $T_{PR}$ which also increases with filling. Above $T_{PR}$ well-defined P-R modes are not observed. Since well-defined P-R modes exist where there is Bose-Einstein condensation (BEC), $T_{PR}$ is associated with $T_{BEC}$, the critical temperature for BEC in films. $T_{PR}$ lies above $T_c$ observed by Yamamoto et al. (Phys. Rev. Lett. 93, 075302 (2004)) in the same gelsil. There is a temperature range $T_c$ $ < T <$ $T_{BEC}$ where there is BEC but no superflow. This is identified as a ``localized" BEC region in which the BEC exists in islands or blobs separated by normal liquid. The P-R mode data and LBEC region will be discussed.