Phonon-Roton Modes and Localized Bose-Einstein Condensation in Liquid Helium under Pressure in Nanoporous Media

We show, using inelastic neutron scattering, that liquid helium in porous media, two gelsils and MCM-41, supports a phonon-roton mode up to a pressure of 36–37 bars only. Modes having the highest energy (“maxons”) broaden and become unobservable at the lowest pressures ($p\simeq 26\mathrm{bars}$) while rotons survive to the highest pressure. By comparing with the superfluid density observed by Yamamoto and co-workers in gelsil, we propose that there is a Bose glass phase containing islands of BEC surrounding the superfluid phase.

Figure 1

Schematic phase diagram of ${}^4\mathrm{He}$ in 25 Å mean pore diameter (MPD) gelsil. The superfluid phase observed by Yamamoto et al. [23] is depicted by the solid region bounded by a solid line ($T_C$). The Bose glass phase containing islands of BEC surrounding the superfluid phase is indicated by dark patches and is bounded by a long dashed line. A solid line marks the onset of freezing [24]. Short dashed lines show the phase boundaries in bulk ${}^4\mathrm{He}$.

Figure 2

The solid lines show the phonon-roton (P-R) energy ${\omega }_Q$ dispersion curve and twice the roton energy ($2\Delta$) at saturated vapor pressure (SVP) ($p\simeq 0$). The solid dots show the present observed P-R dispersion curve at $p=31.2\mathrm{bars}$ in 34 Å MPD gelsil and the dashed line shows the corresponding $2\Delta$. At $p=31.2\mathrm{bars}$, P-R modes are observed at wave vectors in the range $1.6\le Q\le 2.4{\AA }^{-1}$ only where ${\omega }_Q\le 2\Delta$.

Figure 3

Temperature dependence of $S(Q,\omega )$ of ${}^4\mathrm{He}$ in 25 and 34 Å MPD gelsil at $Q=2.1{\AA }^{-1}$ and $p=31.2\mathrm{bars}$. There is substantial elastic scattering from the gelsil centered at $\omega =0$. The height of the roton mode peak at $\omega \simeq 0.6\mathrm{meV}$ decreases with increasing temperature until a well-defined mode is no longer observed at $T\gtrsim 1.4\mathrm{K}$. The peak at $\omega \simeq 1.1\mathrm{meV}$ arises from phonons in the bulk solid ${}^4\mathrm{He}$ that lies between the gelsil sample and the cell wall.

Figure 4

Pressure dependence of $S(Q,\omega )$ at $Q=2.1{\AA }^{-1}$ of ${}^4\mathrm{He}$ at $T=0.07\mathrm{K}$ in 25 Å MPD gelsil. The P-R mode at $\omega \simeq 0.65\mathrm{meV}$ is no longer observed at pressures $p=37.8\mathrm{bars}$ and above. The peak at $\omega \simeq 1.1\mathrm{meV}$ arises from phonons in the bulk solid ${}^4\mathrm{He}$ around the gelsil sample. The pressure 37.8 bars is the pressure at which the cell was filled at $T=2.1\mathrm{K}$ and blocked. The corresponding pressure at $T=0.07\mathrm{K}$ is estimated to be $p=36.3–36.8\mathrm{bars}$.

Figure 5

Well-defined phonon-roton (P-R) modes are observed at pressures and temperatures up to, but not above, the points with error bars joined by a long dashed line in the present porous media. Otherwise the lines are as described in Fig. 1 for ${}^4\mathrm{He}$ in 25 Å MPD gelsil.