Phonon-roton modes, superfluidity and a Bose glass phase in nanoscale liquid $^4$He

We present neutron scattering measurements of the elementary phonon-roton modes of liquid $^4$He confined in nanoporous media. The aim is to compare phonon-roton (P-R) and superfluid density measurements in helium at nanoscales and in disorder. A specific goal is to determine the region of temperature and pressure in which well defined phonon-roton modes (and therefore BEC) exist and compare this with the superfluid phase region, i.e. with the superfluid-normal phase critical temperature T$_c$ and pressure p$_c$ in porous media. We find well defined P-R modes (BEC) extend to temperatures above T$_c$ (up to T$_\lambda$ = 2.17 K at SVP) and to pressures above p$_c$ (up to a pressure p = 36.3-36.8 bars at T $\simeq$ 0 but no modes above this pressure [1]). This suggests that there is a Bose glass phase consisting of local regions of BEC separated by normal liquid so that there is no phase coherence across the sample lying between the superfluid and normal liquid phase. The Bose glass phase surrounds the superfluid phase at all p and T. We compare this phase diagram with other dirty Bose systems. [1] Bossy et al. Phys. Rev. Lett. 101, 025301 (2008), Phys. Rev. B (in press)(2008)