Bose-Einstein Coherence in Two Dimensional Superfluid $^4$He

We present high-resolution measurements of the momentum distribution of atoms in liquid $^4$He films adsorbed in nanoporous MCM-41, with 45 {\AA} mean pore diameter. The measurements were performed at temperatures $T=0.3$ K and $T=2.3$ K and saturated vapor pressure (SVP) in the wavevectors range $24\le Q\le 29$ {\AA$^{-1}$} using the MARI time-of-flight (TOF) chopper spectrometer at the ISIS spallation neutron source. The main goal is to determine whether there is a Bose-Einstein condensate (or coherence) in a finite-size two dimensional (2D) Bose fluid at low temperatures. It is also to investigate the 2D-3D dimensional crossover in the condensate proprieties. We find clear evidence of a condensate parameter, $n_0$, at $T=0.3$ K in the films investigated. In the thinnest film ($\sim$ approximately one atomic layer thick), the observed condensate fraction is greater than but consistent with the bulk superfluid 4he value of 7.25\% within precision; i.e. $n_0=(9.34\pm3.84)$\%. As more $^4$He is adsorbed in the substrate pores, $n_0$ appears to decrease below the bulk value, possibly due to the disorder introduced by the confining media; i.e. $n_0=(2.45\pm2.54)$ \% near full pore filling.