Here, θ ( t ) is the step function and we have set the volume equal to unity. The Fourier transform Im χ$_{nn}$ ( q , ω ) of the imaginary part of the density response function is related to the dynamic structure factor S ( q , ω ) by Im χ$_{nn}$ ( q , ω + i 0 $^{+}$) = − nπ [ S ( q , ω ) − S ( − q , − ω )] = − nπ S ( q , ω )(1 − e − β ℏ $^{ω}$) , (2) where n is the density, β = 1 /T is the inverse temperature, and in the last step we have made use of the “detailed balance” relation, S ( q , − ω ) = e − $^{βω}$S ( q , ω ) and also the inversion symmetry of the system which gives S ( − q , ω ) = S ( q , ω ). Here and throughout this paper we set ℏ = k$_{B}$ = 1. Equation (2) can also be re-written as S ( q , ω ) = − 1 πn [ N $^{0}$( ω ) + 1]Im χ$_{nn}$ ( q , ω + i 0 $^{+}$) , (3) where N $^{0}$( ω ) = ( e βω − 1) − 1 is the Bose distribution function. The latter arises in both Bose and Fermi fluids because density fluctuations always obey Bose statistics. We see from (2) that Im χ$_{nn}$ ( q , ω ) is the antisymmetric (frequency) part of the dynamic structure factor S ( q , ω ). The dynamic structure factor S ( q , ω ) is measured by using inelastic scattering (e.g., neu- tron and Brillouin light scattering) of probe particles, which transfer momentum q and energy ω to the system. S ( q , ω ) is discussed and calculated in all standard texts on many body physics (convenient reviews are given in [18, 19]). In contrast, the density response function χ$_{nn}$ ( q , ω ) describes the density fluctuation induced by an external perturbing po- tential. As an example, in atomic gases, the imaginary part of the density response function is probed in two-photon Bragg scattering and also describes the density pulse produced by a blue-detuned laser. Brillouin and neutron scattering have been used extensively to study the collective modes in $^{4}$He. These techniques cannot be applied to dilute atomic gases, however, since the gases are far too dilute to generate an appreciable signal. Instead, spec- troscopic probes of dilute gases have successfully used Bragg scattering since this technique makes use of the stimulated light scattering processes induced by two counter-propagating laser beams, resulting in a strong enhancement of the signal. In Bragg scattering experiments (see, e.g., [20]), by measuring the total momentum trans- ferred to the sample, one measures the imaginary part of the density response function χ$_{nn}$ ( q , ω ), where q and ω are the momentum and energy transferred by the stimulated ab- sorption and emission of the photons [18]. If the Bragg pulse is short compared to 2 π/ω$_{z}$ , 5