Deep Inelastic Scattering on Ultracold Gases

We discuss Bragg scattering on both Bose and Fermi gases with strong short-range interactions in the deep inelastic regime of large wave vector transfer $q$, where the dynamic structure factor is dominated by a resonance near the free-particle energy $\hslash \omega ={ϵ}_{\mathbf{q}}={\hslash }^2{q}^2/2m$. Using a systematic short-distance expansion, the structure factor at high momentum is shown to exhibit a nontrivial dependence on frequency characterized by two separate scaling regimes. First, for frequencies that differ from the single-particle energy by terms of order $\mathcal{O}(q)$ (i.e., small deviations compared to the single-particle energy), the dynamic structure factor is described by the impulse approximation of Hohenberg and Platzman. Second, deviations of order $\mathcal{O}(q^2)$ (i.e., of the same order or larger than the single-particle energy) are described by the operator product expansion, with a universal crossover connecting both regimes. The scaling is consistent with the leading asymptotics for a number of sum rules in the large momentum limit. Furthermore, we derive an exact expression for the shift and width of the single-particle peak at large momentum due to interactions, thus extending a result by Beliaev [J. Exp. Theor. Phys. 7, 299 (1958)] for the low-density Bose gas to arbitrary values of the scattering length $a$. The shift exhibits a maximum around $qa\simeq 1$, which is connected with a maximum in the static structure factor due to strong short-range correlations. For Bose gases with moderate interaction strengths, the theoretically predicted shift is consistent with the value observed by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008)]. Finally, we develop a diagrammatic theory for the dynamic structure factor which accounts for the correlations beyond Bogoliubov theory. It covers the full range of momenta and frequencies and provides an explicit example for the emergence of asymptotic scaling at large momentum.

Popular Summary

Understanding the structure of matter on the smallest possible scales—what comprises subatomic particles—relies heavily on scattering experiments, where a beam of high-energy photons or electrons is fired at a sample of material. Analysis of how the beam scatters can reveal valuable information about the makeup of the target. Such experiments led to confirmation that quarks and gluons are the building blocks of hadrons such as protons and neutrons. Interpretation of these experiments, which are now common in high-energy physics, relies on simplified assumptions that might not apply at all energies. We produce a theoretical framework that provides a quantitative understanding of experiments that probe the excitation spectrum of ultracold quantum gases, an exotic state of matter that appears at temperatures below $10^{-7}$ K.

Physics at short distances is usually believed to depend sensitively on the details of interparticle interactions. In the case of ultracold gases, this turns out not to be the case. We find that the complicated interactions between atoms are fully characterized by just a few parameters, such as the probability that two atoms are found at the same point in space. Our work shows that methods developed in a high-energy context can be extended to ensembles of ultracold atoms. We can also qualitatively explain a frequency shift observed in experiments nearly 10 years ago that has remained unexplained so far. The physical origin of the shift turns out to be connected to rotons, quasiparticles of the superfluid helium-4.

Our framework is an important step for the development of theoretical tools used to understand other properties of strongly interacting gases and might be extended to gases with strong dipolar interactions such as magnetic quantum liquids.

Figure 1

Sketch of the asymptotic structure of the dynamic structure factor at large momentum $q\tilde{\xi }\gg 1$ and $q|a|\gg 1$, where $\tilde{\xi }$ is the characteristic length scale of the gas [such as $k_n^{-1}=(6{\pi }^{2}n{)}^{-1/3}$ or ${\lambda }_T=\hslash /\sqrt{2\pi mT}$]. Note that this scaling does not necessarily require $k_n|a|\gg 1$. Small deviations in energy from the single-particle peak of order $\mathcal{O}(q)$ are described by the impulse approximation (IA), whose range of applicability shrinks with increasing momentum. Large-energy deviations of order $\mathcal{O}(q^2)$ are in turn described by the operator product expansion (OPE), which predicts asymmetric tails on the left- and right-hand side of the single-particle peak.

Figure 2

High-momentum excitations that determine the deep inelastic form of the dynamic structure factor (in a 2D plane) at large momentum transfer $q\gg {\tilde{\xi }}^{-1}$, where $\tilde{\xi }$ is the characteristic length scale which sets the typical wave vector of the atoms. We consider two cases. First, a single atom with wave number $\mathbf{k}$ (empty red square) can be transferred to a state with large wave vector $\mathbf{k}+\mathbf{q}$ (red filled square). Second, two initial atoms $({\mathbf{k}}^{\prime },-{\mathbf{k}}^{\prime })$ with large and opposite momenta (blue circles) can be excited to high momenta $({\mathbf{k}}^{\prime }+{\mathbf{q}}_1,-{\mathbf{k}}^{\prime }+\mathbf{q}-{\mathbf{q}}_1)$ (blue dots). Energy conservation implies for the probe energy in the first case $\hslash \omega -{ϵ}_{\mathbf{q}}=\mathcal{O}(q^1)$—the quasifree regime—and in the second case $\hslash \omega -{ϵ}_{\mathbf{q}}=\mathcal{O}(q^2)$—the multiparticle regime.

Figure 3

Scaling function $J_{\mathrm{OPE}}(Z)$ as given in Eq. (16) as a function of the scaling variable $Z=(\hslash \omega -{ϵ}_{\mathbf{q}})/{ϵ}_{\mathbf{q}}$. The scaling function is symmetric near $Z=0$ but is strongly asymmetric for $|Z|=\mathcal{O}(1)$.

Figure 4

Line shift of a repulsive Bose gas as a function of scattering length $a$. The line shift [Eq. (50)] predicted by the operator product expansion is indicated by a red continuous line. For comparison, we include the line shift [Eq. (53)] as predicted from the single-mode ansatz (blue dashed line). Inset: OPE prediction [Eq. (51)] for the width of the Bragg peak. The black points are the experimental results by Papp et al. [19].

Figure 5

Many-body $T$-matrix approximation to the current response function. Continuous lines denote the Bogoliubov propagator, the square box the many-body $T$ matrix defined in Appendix pp2 and Fig. 11, and filled circles the density operator that inserts a frequency $\omega$ and a wave vector $\mathbf{q}$. The first diagram denotes the one-loop Bogoliubov approximation of Eqs. (59) and (60). The second diagram denotes the $T$-matrix insertion and is given by Eq. (65).

Figure 6

(a) Elementary process where two particles with large and opposite momentum are expelled from the condensate and couple to the density probe. The continuous lines are free-particle propagators. (b) Diagrams corresponding to the free-particle high-momentum limit of the one-loop Bogoliubov diagram in Fig. 5. (c) Diagram that describes the high-momentum limit of the diagram with $T$-matrix insertion in Fig. 5.

Figure 7

Density plot of the dynamic structure factor as predicted by the $T$-matrix approximation of Sec. 5. The response starts above a two-phonon threshold $\hslash \omega =2E_{\mathbf{q}/2}$ indicated by the white continuous line. The position of the coherent Bogoliubov peak at $\hslash \omega =E_{\mathbf{q}}$ is shown by a white dashed line.

Figure 8

Dynamic structure factor as a function of frequency for increasing momentum $q\xi =5$, 10, 15, 20, 30, and 50 [(a)–(f)] (continuous blue lines). The plots clearly show the crossover to the universal scaling form predicted by the OPE (dashed green lines) at high momentum, shown in full in Fig. 3. Near the one-particle peak, the dynamic structure factor is described accurately by the impulse approximation (red dot-dashed line). The vertical arrows (not to scale) indicate the position of the condensate delta peak on top of the incoherent background, which here is centered at the Bogoliubov energy $E_{\mathbf{q}}=\sqrt{{ϵ}_{\mathbf{q}}({ϵ}_{\mathbf{q}}+2gn)}$.

Figure 9

(a) Impulse approximation $Y$-scaling function $J_{\mathrm{IA}}$ for the nondegenerate Bose gas at unitarity and $T/T_n=2$. The continuous red, orange, and green lines show the scaling function for three different values of the three-body parameter, ${\kappa }_{*}/k_n=3$, 5, and 10. For comparison, we include the impulse approximation for the noninteracting gas at the same temperature. (b) The impulse approximation shows heavy tails for large $Y$, which are in good agreement with the leading-order two-body Tan relation and a subleading three-body correction. (c) The three-body tail is indicated by black dotted lines.

Figure 10

(a) Impulse approximation $Y$-scaling function $J_{\mathrm{IA}}$ for the nondegenerate Fermi gas at unitarity. The red line denotes the scaling function of the interacting gas with $T/T_F=1.5$. For comparison, we include the scaling function of a noninteracting Fermi gas with the same density. (b) The power-law $1/Y^2$ tail Eq. (11) of the interacting gas. The dashed red line marks the asymptotic behavior with ${\lambda }_T^4{\mathcal{C}}_2=14.045$.

Figure 11

Bethe-Salpeter equation for the two-phonon $T$ matrix, indicated by a box. The wavy line denotes the contact interaction with strength $g$ and continuous lines indicate Bogoliubov propagators given in Eq. (58). Indices are Bogoliubov indices.