Crevecoeur, de Schepper, and Montfrooij Reply:

. Crevecoeur, de Schepper, and Montfrooij Reply:The distinction made in the Comment [1] to our Letter [2 between the frequency regions "v ø kBT and "v ¿ kBT has indeed been proven to be useful in the past understand the neutron spectra Ssq, vd in purely classical fluids s"v ø kBT d and superfluid Hes"v ¿ kBT d. However, as a consequence, this has led to two quite ferent physical descriptions of Ssq, vd: the two-Lorentzian DHO (phonon) model for superfluid He and the thre Lorentzian model for classical fluids (i.e., the hydrody namic Rayleigh-Brillouin triplet extended to larger q). In our Letter we consider He at 4 K (intermediate) purpose to determine the physical connection between both regim using a unified theoretical description. To do so, we use thexact Mori-Zwanzig projection operator formalism since it is valid for all q and v, for both classical as well as quantum fluids. In particula it takes the asymmetry of Ssq, vd, which develops when one enters the quantum regime, exactly into account. this formalism, dynamic correlation functions like Ssq, vd [or xnnsq, vd] are expressed in terms of ( v-independent) coupling parameters between a number of , relevant microscopic variables (microscopic density, longitudin momentum, temperature, and so on), with , $ 2, since the continuity equation implies that a fluctuation in th number density is always (and exclusively) coupled to momentum fluctuation which can then be coupled to oth microscopic variables. These coupling parameters given by the, 3 , matrix Hsqd. The effective number , of contributing parameters can be determined in princip from theory and experiment, by including increasing more microscopic variables until an accurate descripti of Ssq, vd has been obtained. This formalism is derive in Ref. [13] of Ref. [2] forall q andv, and not only for the low frequency domain "v ø kBT as Griffin stated in his Comment. From the good fits to our experimental da we have shown that , 2 for He at 4.0 K in the region covered by our neutron experiment, implying that th only two relevant microscopic variables are the dens and longitudinal momentum. Therefore the observ fluctuations in density must necessarily be propagat (phonons) or diffusive (overdamped phonons). Thus, n only have we shown that the spectra can be well describ by a damped harmonic oscillator form, we have al shown this form to be theoretically sound outside i hitherto assumed region of validity. By no means ha we simply taken an expression for the low energy regi s"v ø kBT d and assumed its validity throughout th entire spectrum as suggested in the Comment. We n here that there is no theoretical derivation yet for the fa that, 2 for He at 4 K. An illuminating discussion on the physical nature o overdamped phonons has been given by Kirkpatri [3]. He considers for classical hard spheres a 2 3 2