Microscopic dynamics of superfluid 4 He: a comprehensive study by inelastic neutron scattering
K. Beauvois, J. Dawidowski, B. Fåk, H. Godfrin, E. Krotscheck, J. Ollivier, A. Sultan
MMicroscopic dynamics of superfluid He: a comprehensive studyby inelastic neutron scattering
K. Beauvois,
1, 2
J. Dawidowski, B. F˚ak, H. Godfrin, E. Krotscheck,
4, 5
J. Ollivier, and A. Sultan Institut Laue-Langevin, CS 20156, 38042 Grenoble Cedex 9, France Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France Comisi´on Nacional de Energ´ıa At´omica and CONICET,Centro At´omico Bariloche, (8400) San Carlos de Bariloche, R´ıo Negro, Argentina Department of Physics, University at Buffalo, SUNY Buffalo NY 14260, USA Institute for Theoretical Physics, Johannes Kepler University, A 4040 Linz, Austria (Dated: May 4, 2018)The dynamic structure factor of superfluid He has been investigated at very low temperaturesby inelastic neutron scattering. The measurements combine different incoming energies resultingin an unprecedentedly large dynamic range with excellent energy resolution, covering wave vectors Q up to 5 ˚A − and energies ω up to 15 meV. A detailed description of the dynamics of superfluid He is obtained from saturated vapor pressure up to solidification. The single-excitation spectrumis substantially modified at high pressures, as the maxon energy exceeds the roton-roton decaythreshold. A highly structured multi-excitation spectrum is observed at low energies, where clearthresholds and branches have been identified. Strong phonon emission branches are observed whenthe phonon or roton group velocities exceed the sound velocity. The spectrum is found to displaystrong multi-excitations whenever the single-excitations face disintegration following Pitaevskii’stype a or b criteria. At intermediate energies, an interesting pattern in the dynamic structure factoris observed in the vicinity of the recoil energy. All these features, which evolve significantly withpressure, are in very good agreement with the Dynamic Many-body calculations, even at the highestdensities, where the correlations are strongest.
I. INTRODUCTION
Understanding the dynamics of correlated bosons isa subject of general interest in several fields of physics.Bose-Einstein condensation and superfluidity , firstfound in He, are fundamental phenomena that imprintremarkable signatures on the dynamics of these systems.Experimentally, superfluid He is the simplest example ofstrongly correlated bosons. The interaction potential isparticularly well known, and substantial effort has beendevoted to develop a coherent theoretical framework ableto describe and explain the extraordinary properties ofthis quantum fluid . The theoretical methods can begeneralized to other many-body problems, including forinstance up-to-date approaches of the complex case ofcorrelated fermions .The prediction by Landau of the phonon-roton exci-tation spectrum of superfluid He and its direct observa-tion in the dynamic structure factor S ( Q, ω ) using neu-tron scattering techniques are cornerstones of modernphysics, at the origin of the present microscopic descrip-tions of matter . The dynamics of superfluid Heat very low temperatures, in the vicinity of the groundstate, is dominated by the “phonon-maxon-roton” exci-tation branch. The corresponding excitations, extremelysharp, correspond essentially to poles of the dynamicdensity-density response function. They are referred toas “single-excitations” in the neutron scattering litera-ture, and as “quasi-particles” in theoretical works. Aneffective description of the dynamics of such systems canbe obtained in terms of these modes, allowing for in-stance a very accurate statistical evaluation of the low temperature thermodynamic properties .Sharp excitations are absent above twice the rotonenergy , and the dynamics at intermediate ener-gies is described in terms of broad excitations, named“multi-excitations” for reasons described below. Multi-excitations still have a significant statistical weight inthe dynamic structure factor . Their spectrum isknown to display some structure since the early measure-ments of Svensson, Martel, Sears and Woods . More re-cent investigations showed that some features couldbe ascribed to multi-excitations. These were relatedto pairs of high density-of-states roton (R) and maxon(M) modes (noted hereafter 2R, 2M, and MR). Thebroad ridges observed in S ( Q, ω ) at SVP (see Figure 1of Ref. 24), and at 20 bars (see Figure 1 of Ref. 29) wereconsistent with the calculated energies of the main com-binations (2R, 2M and MR).A much finer structure in the dynamic response wasobserved in our recent work at zero pressure , includ-ing sharp thresholds, narrow branches, and a new two-phonon decay process, the “ghost phonon”. Explainingthis rich dynamic response, observed from the continuumlimit to subatomic distances, constitutes a challenge andan opportunity for microscopic theories.Finally, at high energies, the dynamic structure fac-tor gradually approaches a quasi-free-particle behavior described by the impulse approximation .Even though helium is one of the most intensively in-vestigated physical substances, measurements covering alarge kinetic range are scarce. The canonical results byCowley and Woods , Dietrich et al. or Svensson etal. have a low resolution by modern standards, while a r X i v : . [ c ond - m a t . o t h e r] M a y later measurements specialize in specific ranges .Our extensive high-resolution measurements, presentedin Fig. 1, provide a detailed and complete map of thedynamics of superfluid He. In addition to its aestheticmerits, the picture shows new features which are the ob-ject of this manuscript.Helium is highly compressible. Since the atomic corre-lations depend on the density, it is interesting to investi-gate the pressure dependence of the density excitations.Much of the earlier work has been focused on the effectof pressure on the single-excitation response, in order todetermine, for example, the Landau parameters charac-terizing the dispersion curve. The multi-excitation spec-trum has also been found experimentally andtheoretically to be strongly modified by the pres-sure. It was therefore desirable to extend our recent highsensitivity measurements to finite pressures, and moreparticularly near solidification, where theory predictedradical changes in the dynamics.In this manuscript, we present a detailed investigationof the effect of pressure on the dynamics of superfluid He. We cover a large energy and wave vector range whilepreserving the resolution needed to observe the fine struc-ture of the spectra. High resolution maps of the dynamicstructure factor S ( Q, ω ) have been obtained at saturatedvapor pressure (SVP) and at P = 24 bars, close to so-lidification, as shown in Fig. 1. Additional measurementshave been made in a smaller dynamic range at the inter-mediate pressures 5 and 10 bars. We finally compare ourdata to microscopic calculations of S ( Q, ω ) within theDynamic Many-Body theory performed at the densitiescorresponding to the experimental pressure conditions. II. EXPERIMENTAL DETAILS
The measurements were performed on the IN5 time-of-flight spectrometer at the high-flux reactor of InstitutLaue Langevin. Our previous work at low tempera-tures and saturated vapor pressure used cold neutrons ofenergy E i =3.55 meV. In the present work, we combinedata taken using four different incident neutron energies, E i =3.55, 5.11, 8.00, and 20.45 meV, for which the en-ergy resolution (FWHM) at elastic energy transfer was0.070, 0.12, 0.23 and 0.92 meV, respectively. This allowedus to obtain a complete map of the dynamic structurefactor at the most relevant pressures, i.e. , saturated va-por pressure (SVP) and near solidification ( P = 24 bars).We also investigated a few intermediate pressures using E i =3.55 meV.The cylindrical sample cell was made out of alu-minum 5083, with 1 mm wall thickness and 15 mm innerdiameter . Cadmium disks of 0.5 mm thickness wereplaced inside the cell every centimeter to reduce multi-ple scattering. The cell was thermally connected to themixing chamber of a very low temperature dilution re-frigerator using massive OFHC copper pieces. Heat ex-changers made out of sintered silver powder were used to provide a good thermal contact with the helium sample.Care was taken to thermally anchor the filling capillaryat several places along the dilution unit, in order to re-duce heat leaks to the cell. The measurements were allperformed at very low temperatures, well below 100 mK, i.e. essentially at zero temperature for the propertiesunder investigation.High purity (99.999 %) helium gas was condensed inthe cell at low temperatures, using a gas handling sys-tem including a “dipstick” cold trap operated in a heliumstorage dewar. The dipstick was used to condense the gasand to pressurize the helium sample. The pressure in thesystem was measured with a precision of 6 mbars witha 0-60 bars Digiquartz gauge located at the top of thecryostat. The corresponding precision for the pressuresinside the cell is 20 mbars, after applying helium hydro-static head corrections. The actual pressures in the cellfor the nominal 0, 5, 10 and 24 bars are essentially 0 (SVPat 100 mK), 5.01(2), 10.01(2) and 24.08(2) bars. III. DATA REDUCTION
Standard time-of-flight data-reduction was used toobtain the dynamic structure factor S ( Q, ω ) from theraw data. The contribution of the cell scattering wassubtracted, as well as that of double scattering events oftype “inelastic helium scattering plus elastic scatteringfrom the cell”. This type of double scattering is essen-tially independent of wave vector.The contribution of the multiple scattering within thehelium was corrected using Monte Carlo simulations .Due to the small diameter of our sample cell and thepresence of several cadmium plates, multiple scatteringcorrections are small (the ratio of double-scattered tosingle-scattered neutrons is on the order one percent ),but may be comparable to the multi-excitation signal. Itis therefore essential to verify that multiple scattering isnot contaminating the spectra in the energy and wavevector regions of interest, and perform the correctionswhen necessary, in particular at low Q .Since multiple scattering depends on the incident neu-tron energy, as shown in figure 2, while multi-excitationsdo not, Monte Carlo calculations can be used to selectthe most appropriate incident neutron energy for the ex-periments, and also to experimentally distinguish multi-excitations from multiple scattering.The only input needed by the Monte Carlosimulations in the present case is the initially measuredscattering function S ( Q, ω ) after corrections for multiplescattering processes involving the cell, and the coherentscattering cross section of He, σ c = 1 .
34 barns. We firstcalculate the total scattering cross section σ s ( E i ): σ s ( E i ) = N σ c k i (cid:90) QdQ (cid:90) S ( Q, ω ) dω, (1)where N is the number of scatterers and k i the incident b) P=24 barWave vector Q (Å -1 ) E n e r g y ( m e V ) a) P=0 bar FIG. 1. S ( Q, ω ) of superfluid He as a function of wave vector and energy transfer, measured at T ≤
100 mK at (a) saturatedvapor pressure ( P ≈
0) and (b) near solidification ( P = 24 bars). The plots combine data measured at different incident neutronenergies ( E i =3.55, 5.1, 8.00 and 20.45 meV) for an optimum energy resolution; the dashed black lines represent the limits ofthe corresponding kinetic ranges. The dotted red line is the free He atom recoil energy E r = (cid:126) Q M . The color-coded intensityscale is in units of meV − . neutron wave vector.We find σ s ( E i = 3 .
55 meV)=0.64 barns, about one halfof the coherent scattering cross section σ coh . The multi-ple scattering fraction is 0.8 % for E i =3.55 meV, increas-ing slightly with pressure from 0.79 % at SVP to 1.06 %at 24 bars. This agrees well with calculations using thesemi-analytical method developed by Sears , which give FIG. 2. Monte Carlo calculation of the contribution of double-scattering within the helium to S ( Q, ω ). Results are shownfor two incident neutron energies, E i =3.55 and 5.11 meV. Thecolor-coded intensity scale is in units of meV − . values increasing from 0.93 % to 1.09 % for the same pres-sures. Multiple scattering can be seen in the experimen-tal spectra at low wave vectors, thus providing a way tocheck the Monte Carlo calculations used to eliminate thiseffect. This is a crucial step in the data analysis, neededto ensure that all the features we report in S ( Q, ω ) doindeed correspond to multi-excitations.The calculated contribution due to multiple scatteringwithin the helium has been subtracted from the spectrameasured using incident neutron energies E i =3.55 and5.11 meV. This was found to be unnecessary for E i =8.00and 20.45 meV, because multiple scattering processes arenegligible in the corresponding regions of the“combined”spectra of Fig. 1.An overall scale factor was applied to S ( Q, ω ) at SVP,so that the weight of the single excitation Z ( Q ) agreeswith that of Cowley and Woods near the roton, i.e. , Z ( Q = 2 ˚A − )=0.93 at SVP. At higher pressures, thesame scaling factor was used, but corrected for the den-sity ratio ρ ( P ) /ρ ( P = 0). IV. EXPERIMENTAL RESULTSA. Spectra at SVP and P=24 bars in a largedynamic range
Our comprehensive results on the dynamic structurefactor S ( Q, ω ) at SVP and P = 24 bars are shown in FIG. 3. Dynamic structure factor S ( Q, ω ) combining dataat four incident neutron energies: spectra for different wavevectors Q at SVP (green diamonds) and P = 24 bars (purplesquares). The dashed lines are Gaussian fits of the resolution-limited phonon-roton peaks (off scale). The black lines repre-sent the helium recoil energy. At Q = 3 ˚A − , the purple andgreen lines represent the two-roton energy 2∆ R at SVP and P = 24 bars, respectively. At Q = 1 ˚A − , MR and MM arethe energy positions at SVP of the maxon-roton and maxon-maxon multi-excitations, respectively. Fig. 1. These maps were obtained by combining the fourdifferent neutron energies. Higher energies make a largerdynamic range accessible, but the instrumental energyresolution deteriorates rapidly (see section II). Since thecorresponding dynamic ranges have a substantial over-lap, we can select the most appropriate data set in termsof resolution, neutron counts or cleanest background foreach region of the Q − ω plane. The S ( Q, ω ) maps arebuilt in the following way: first, the spectrum measuredat E i =3.55 meV is represented; outside its useful kineticrange, the data at E i =5.11 meV are added, then the dataat E i =8.00 meV and finally, the data at E i = 20.45 meV.The constant wave vector scans presented in Fig. 3,obtained as particular “cuts” of Fig. 1, provide a com-plementary perspective on the data. The phonon-rotonsingle-excitation mode is very narrow at the scale ofFigs. 1 and 3, and the observed width is essentially a mea- sure of the experimental energy resolution (with the re-markable exception of the maxon at high pressures, whichis discussed in the next section). The influence of a finiteenergy resolution is clearly seen in Fig. 1 as a width dis-continuity in the Pitaevskii plateau , between rangescorresponding to different incident neutron energies. It isimportant to note, however, that the experimental broad-ening effects are negligible in all the multi-excitation re-gion investigated in the present work (except at the endof the Pitaevskii plateau).Merging data measured with different resolutions hasbeen successfully achieved, judging from the remarkablecontinuity in intensity between the different regions rep-resented in Fig. 1. This is essentially due to the factthat the sharpest multi-excitations are found in the lowenergy and low wave vector sector, adequately coveredby our high resolution data at E i =3.55 and 5.11 meV.Conversely, the spectra in the quasi-free particle re-gion, at high energies and wave-vectors, are intrinsicallybroad, and adequately covered by our data at 8.00 and20.45 meV, in spite of their lower resolution. Using op-timized incident neutron energies reveals the completeevolution of the system, characterized by several multi-excitation branches merging progressively at high wavevectors to form a broad but rather intense feature. In-tensity in this region was observed in early studies ,but the data where either strongly truncated , ormeasured with low resolution . This feature finally be-comes, after a strong oscillation, a less intense branchprogressively approaching the free particle parabolic dis-persion. B. High resolution spectra as a function of pressure
We present in this section the spectra obtained usingan incident neutron energy of E i =3.55 meV, for wave vec-tors up to Q = 2 . − and energies up to ω =2.22 meV.The results are shown in Fig. 4(a), where we representour earlier data at SVP, the present data at 5 and10 bars, and the data at P = 24 bars discussed in theprevious section. One can readily note that both thesingle-excitation and the multi-excitation components ofthe dynamic structure factor are modified by the pres-sure.Our results for the single-excitations dispersion mea-sured at several pressures, shown in Fig. 5a, are in excel-lent agreement with previous works .The roton parameters at each pressure have been ob-tained from fits of the single-excitations dispersion rela-tion (cid:15) Q ( Q ) to the expression (cid:15) Q = ∆ R + (cid:126) mµ R ( Q − Q R ) + b ( Q − Q R ) + c ( Q − Q R ) , (2)where ∆ R is the roton energy gap, Q R the wave vectorat the roton minimum, and µ R the roton effective mass; b and c are additional adjustable parameters. Fits were E n e r g y ω ( m e V ) Wave vector Q ( Å - ) FIG. 4. (a) S ( Q, ω ) of superfluid He measured as a func-tion of wave vector and energy transfer, at P = 0, 5, 10 and24 bars and temperature T ≤
100 mK. The incident neu-tron energy is E i =3.55 meV. (b) Dynamic many-body theorycalculation of S ( Q, ω ) at corresponding densities ( n =0.0215,0.0230, 0.0240 and 0.0255 ˚A − , see text). Note that the mainfeatures of the experimental data are well reproduced. Thecolor-coded intensity scale is in units of meV − . The inten-sity is cut off at 1 meV − in order to emphasize the multi-excitations region. The apparent width of the phonon-rotonexcitations in the experimental plot is due to an energy reso-lution of 0.07 meV, while the calculated phonon-roton disper-sion curve has been highlighted by a thick red line. made over a total wave vector range ∆Q up to 0.47 ˚A − .Due to the large number of individual detectors and thehigh neutron rate of IN5, the statistical uncertainty of thefits is very good (see Table I). The roton mass determinedin our work is lower than the one obtained by Andersenet al. using a parabolic fit of the roton minimum, 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(cid:2) P (cid:2) (cid:2) P (cid:2) (cid:2) P (cid:2)
10 bar (cid:2) P (cid:2)
24 bar Q (cid:2) (cid:3) (cid:3) (cid:3) Ε Q (cid:2) m e V (cid:3) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Q ! " ! " W Q ! m e V " FIG. 5. a) Dispersion relation (cid:15) Q ( Q ) of the single-excitationsmeasured at 0, 5, 10 and 24 bars. Note the flattening of thecurve at the maxon at high pressures. b) The wave vectordependence of the measured width (FWHM) of the single-excitation peaks. The measured width reflects the shape ofthe experimental resolution ellipsoid cut by the dispersion re-lation curve at different angles. At 24 bars, however, a physi-cal broadening of the maxon is clearly observed. but it agrees well with earlier measurements where theparabolic fit was limited to a very narrow wave vectorrange. P (bars) ∆ R (meV) Q R (˚A − ) µ R R , wave vector of the rotonminimum Q R and roton effective mass µ R ; values in paren-thesis are one standard deviation errors from least-squares fitsdescribed in the text. A similar analysis can be performed in the maxon re-gion. The corresponding maxon parameters ∆ M , Q M and µ M ( d and e are additional adjustable parameters)have been calculated by fits of (cid:15) Q in the maxon region,over a wave vector range ∆Q on the order of 0.8 ˚A − , tothe formula: (cid:15) Q = ∆ M − (cid:126) mµ M ( Q − Q M ) + d ( Q − Q M ) + e ( Q − Q M ) . (3)The results are given in Table II. P (bars) ∆ M (meV) Q M (˚A − ) µ M R M , wave vector Q M and effectivemass µ M ; values in parenthesis are one standard deviationerrors from least-squares fits described in the text. The lastcolumn gives twice the energy of the roton gap 2∆ R , for com-parison with the value of ∆ M . As expected for a system approaching localization ,the phonon and the maxon energies increase steadilywith pressure, while the energy of the roton minimumdecreases. The single-excitation data of Fig. 5a clearlyshow in addition a substantial flattening at the level ofthe maxon in the dispersion curve corresponding to apressure of 24 bars. Earlier results at this pressure didnot detect this effect , while more recent systematicresults by Gibbs et al. were limited to pressures below20 bars. The data at 24 bars are qualitatively differentfrom those at low pressures because the maxon energyexceeds twice the roton energy. At high pressures, themaxon excitation can therefore decay by phonon emis-sion, exactly as in the case of higher wave vectors, at thePitaevskii’s plateau .We also observe the corresponding broadening of themaxon single-excitation (Fig. 5b): the measured maxontotal width of 0.012 meV, obtained after subtraction ofthe instrumental resolution, is substantial compared totypical phonon and roton widths (see Ref. 42 and ref-erences therein). The excitations in the maxon re-gion broaden until they become unobservable in confinedhelium , where very high pressures can be reachedbefore solidification.We now concentrate on the multi-excitation region,shown in Fig. 4(a), which displays highly structured spec-tra for all pressures. The data for the pressures 5 and 10bars are qualitatively similar to our previous results atsaturated vapor pressure . The high resolution spec-tra display very clearly a threshold in energy at about1.5 meV. This feature, which corresponds to the decay ofan excitation into a pair of rotons, depends on pressure,since the roton energy depends on pressure. In addition,we observe several well-defined multi-excitation branchesdisplaying substantial dispersion. Their gradual evolu-tion reflects, as will be shown in section VI, the changewith pressure of the single-excitation dispersion.We also observe important qualitative changes at highpressures. We examine first the multi-excitation region of the “ghost-phonon”. This multi-phonon excitation,observed in our previous work, appears as a linear exten-sion of the phonon branch . We observe in the presentwork that the ghost-phonon intensity strongly decreaseswith pressure until it disappears at some pressure below24 bars.We also see very clearly in Fig. 1(a) a multi-phononregion just above the roton branch for wave vectors ofthe order of 2.2 to 2.4 ˚A − . The high resolution spectraat E i =3.55 meV only show part of this multi-excitationregion, but the results have been completed by spectrataken at E i = 5.11 meV at SVP and 24 bars, shown inFig. 1. The intensity of these multi-excitations, describedin detail in Section VI C, decreases strongly with pres-sure, behaving similarly as the ghost-phonon.The multi-excitation spectra are strongly modified athigh pressures, as the maxon enters the multi-excitationscontinuum. Fig. 4 shows that substantial intensity devel-ops at this pressure for energies just above the maxon.Similar effects were also observed by Graf et al. , Tal-bot et al. , and by Gibbs et al. at a lower pressure(20 bars). The present data benefit from a sharper reso-lution, as can be seen by directly comparing spectra at Q ≈ − around the maxon peak.All these effects will discussed in detail in section VI inthe context of a comparison with theoretical calculations. V. CALCULATIONS WITH THE DYNAMICMANY-BODY THEORY
We present in this section our calculations of the dy-namic structure factor of superfluid He at zero tempera-ture obtained within the Dynamic Many-Body theory . A. State of the art of Theory
Theoretical studies of the dynamic structure functionin He began with the work of Feynman , and Feynmanand Cohen . The Feynman theory of elementary exci-tations was developed in a systematic Brillouin-Wignerperturbation theory by Jackson and Feenberg . Animportant contribution was the identification of classes oftheories for the dynamic structure function that satisfythe ω and ω sum rules exactly.The most complete evaluation of the phonon-roton dis-persion relation in terms of Brillouin-Wigner perturba-tion theory was carried out by Lee and Lee who ob-tained an impressive agreement with the experimentalphonon-roton spectrum up the wave vector of 2.5 ˚A − .The major drawback with these early calculations wasthat the required input, pair and three-body distributionfunctions, were poorly known.Manousakis and Pandharipande used input statesof the Brillouin-Wigner perturbation theory including“backflow” correlations in the spirit of Feynman and Co-hen. Through the gradient operator acting on the wavefunction, specific dynamic correlations are introduced toall orders. The “backflow-function” is, however, chosenper physical intuition rather than by fundamental prin-ciples, and the evaluation of the perturbative series be-comes very complicated. Topologically, diagrams similarto those of Lee and Lee were included. While the accu-racy of the theoretical roton energy is comparable to thatof Lee and Lee, one can clearly see an inconsistency sincethe energy of the Pitaevskii plateau lies below twicethe energy of the roton gap.The first theoretical descriptions of the multi-excitations were qualitatively in agreement withthe early multi-excitations data . The simplestversion of Correlated Basis Functions theory producesphonon, maxon and roton modes, as well as multi-phonons. In this approximation, the calculated multi-excitations decay into Feynman modes instead of thecorrect single-excitations; large gaps are found in thespectrum, and many predicted features are not seen inthe experiments. Other features calculated in the multi-excitation region do indeed survive in recent theories, likethe presence of intensity above the phonon branch andthat of a well-defined 2-roton threshold (these effects aredescribed below). These calculations, as well as manyothers addressing specific aspects of the multi-excitationdynamics, could not be quantitatively compared to theexperimental results, but they motivated further inves-tigations on multi-particle dynamics. Reviews can befound in Ref. 4 and 5.More recent calculations used a hybrid approachof Brillouin-Wigner perturbation theory and equationsof motion for time-dependent multi-particle correlationfunctions to derive a self-consistent theory of the dynamicdensity-density response of He. The self-consistencyof this semi-analytic method allows the identification ofmode-mode coupling processes that lead to observablefeatures in the dynamic structure function. The underly-ing physical mechanisms, their relationship to the groundstate structure, and the consequences on the analyticproperties of the dynamic structure function, emerge di-rectly from the theory.A very different approach involves novel numericalmethods that give access to dynamic propertiesof quantum fluids. These important algorithmic devel-opments will reproduce, extend and complete the exper-imental data with the future development of computingpower; their present accuracy and consistency, however,are still limited in the multi-excitations region investi-gated here.
B. Dynamic Many-Body Theory calculation
In order to calculate quantitatively both the single-excitation and the multi-excitation response, our calcu-lations include up to three-body dynamic fluctuations inthe correlation functions of the equations of motion . Wederive the self-consistent density-density response of He χ (Q, ω ), expressed as χ ( Q, ω ) = S ( Q ) ω − Σ( Q, ω ) + S ( Q ) − ω − Σ( Q, − ω ) (4)where S ( Q ) is the static structure factor, and the self-energy Σ( Q, ω ) is determined by the integral equationΣ(
Q, ω ) = (cid:15) ( Q ) + 12 (cid:90) d pd k (2 π ) n δ ( (cid:126)Q − (cid:126)p − (cid:126)k ) ×| V (3) ( (cid:126)Q ; (cid:126)p, (cid:126)k ) | ω − Σ( p, ω − (cid:15) ( k )) − Σ( k, ω − (cid:15) ( p )) . (5)In this expression, (cid:15) ( Q ) is the Feynman dispersion rela-tion, and V (3) the three-body coupling matrix element.The simplest approximation for V (3) , the so-called con-volution approximation , including static ground statetriplet correlations , improves the density–dependenceof the roton minimum visibly. The most advancedcalculation , which is taken here and in Ref. 9, sums aninfinite series of diagrams, the so-called “fan-diagrams”which is the minimum set of diagrams that must be in-cluded to reproduce exact features of V (3) for both, longwavelength and short distances.Linear response theory provides the relation betweenthe experimental dynamic structure factor and the dy-namic susceptibility calculated by the theory describedabove: the dynamic structure factor S ( Q, ω ) is propor-tional to the imaginary part of the dynamic susceptibility χ ( Q, ω ), the linear response of the system to a densityfluctuation.Full maps of S ( Q, ω ) have been calculated for differentatomic densities, see Fig. 10 in Ref. 9. The data shownin Figs. 1 and 4 correspond to n = 0 . − , values which provide the best overallagreement with the experiment. They turn out to bevery close to the experimental results for P = 0, 5, 10and 24 bars, n exp =0.0218, 0.0230, 0.0239 and 0.0258 ˚A − .The small shift in density is within the expected accuracyof the theoretical calculations.The calculations presented here have been performedusing only the most relevant diagrams . This approx-imation is sufficient to provide an excellent descriptionof the dynamics, but minor discrepancies can still beseen. The most salient effect is that the roton energyis overestimated; at zero pressure, for instance, the cal-culated value is 0.83 meV while the measured value is0.7416(10) meV. This discrepancy could be resolved byincluding additional diagrams, but it does not seem nec-essary to perform such a tedious calculation given thequality of the agreement already achieved at this stage.The calculation provides absolute values for the struc-ture factor. In our previous work , the calculated val-ues were multiplied by an overall normalization factor of1.28 in order to have Z ( Q = 2 ˚A − )=0.93 near the roton.Here, this normalization has not been applied. Given thefinite number of diagrams involved in the calculations, afactor of this order is within their estimated absolute ac-curacy. C. Mode-mode coupling
Multi-excitations arise from the enhanced response ofthe system at particular energies and wave vectors cor-responding to two or more single-excitations into whichthey can decay. The theory considers (see equation 5)the most relevant processes where a density fluctuation( (cid:126)Q, ω ) of wave vector (cid:126)Q and energy ω decays into a pairof single-excitations with corresponding values ( (cid:126)p, ω p )and ( (cid:126)k, ω k ). The calculations have been shown to bein excellent agreement with experiment at saturated va-por pressure . Here we investigate the general pressuredependence of the dynamics, and several particularlyintense mode-mode couplings. The latter were exam-ined theoretically in Ref. 9, and additional calculationsspecialized to the main mode-mode couplings (phonon-phonon, phonon-roton, roton-roton, maxon-roton) canbe found in Ref. 58. The next section provides a de-tailed comparison between the theory and the experi-mental data. VI. IDENTIFICATION OF THEMULTI-EXCITATIONS
Above the sharp and intense phonon-maxon-roton dis-persion curve, we observe a highly-structured multi-excitation region. Multi-excitations are relatively strongif they can decay into a pair of high intensity single-excitation modes. The energy and momentum of thesepair combinations is directly related, by the conservationof energy and momentum, to those of the underlyingelementary excitations. It is possible to determine theposition of the main multi-excitation resonances in thedynamic structure factor map (2-Phonons, 2-Rotons, 2-Maxons and Maxon-Roton) from pure kinematic consid-erations, i.e. energy and momentum conservation. Thechallenge for microscopic theories is to predict the inten-sity of the multi-excitation spectrum, if possible in a largedynamic range. Obtaining the fine structure we observerequires a quantitative calculation of mode couplings.We first present in this Section a brief description ofthe kinematic constraints for different pair-excitations,setting the framework for their identification. The fol-lowing two subsections concentrate on new features ob-served in the multi-excitation spectrum, that we named“ghost-phonon” and “ghost-roton”. We then describe adifferent type of multi-excitations, associated to roton-roton coupling, which we observed in particular “abovethe maxon” and “beyond the roton”. We conclude thisSection by a discussion on higher order multi-excitations,and the progressive evolution to the high energy regime.
A. Kinematic constraints for pair-excitations
The kinematic constraints calculated for the main lowenergy multi-excitations are shown in Fig. 6. We use be-low the notation R − and R + to distinguish rotons oneach side of the roton minimum. ω ( m e V ) Q (Å −1 )phonon−phonon processesphonon−R− processesphonon−R+ processes FIG. 6. Kinematically allowed regions for different multi-phonon processes: P-P (including P-M − ), P-R − , and P-R + .See text for details. The allowed regions are necessarily located above thesingle-excitations dispersion curve. The P-P region isfound at low wave vectors. Beyond the maxon, P-R − excitations are allowed in a large region delimited bythe dispersion curve and two lines starting at the maxonmaximum and at the roton minimum, with slopes equalto − c and + c , respectively, where c is the speed of sound.P-R + excitations occupy a region delimited by the dis-persion curve and a line starting from the roton minimumwith slope − c . There is a large overlap with the P-R − region.The case of 2R, not shown, is particularly simple, witha threshold at twice the roton energy, 2∆ R . The situa-tion for 2M processes is similar, with an upper limit equalto 2∆ M . M-R combinations of excitations may lead tobranches with substantial dispersion. The kinematic con-straints are sufficient to determine unambiguously whichare the dominant processes in some multi-excitations re-gions, in particular at low Q above the phonon dispersion,and inside the roton parabolic dispersion curve.The evolution of the observed multi-excitations in alarge energy range, for different pressures, is illustrated inFigs. 1 and 4. We can distinguish different types of multi-excitations. Several narrow branches are easily identified,as indicated in Fig. 7, as corresponding to 2P, 2R, 2M andM-R processes. The 2R feature is observed in Fig. 4 as aclear threshold, both in the theoretical and experimentaldata. P+R
Wave vector Q (Å -1 ) E n e r g y ( m e V ) S ( Q , )( m e V - ) FIG. 7. Map of S ( Q, ω ) at SVP identifying remarkablemode-mode coupling regions: phonon-phonon (2P, with anellipse around the “ghost-phonon”), phonon-roton (P+R, aregion marked by a triangle, which includes an ellipse indicat-ing more specifically a high-intensity “ghost-roton” region),roton-roton (2R, marked by a rectangle around 1.5 meV), andat higher energies the roton-maxon (R+M) and maxon-maxon(2M) regions. The description of the different lines is givenin Fig. 1.
B. Phonon-phonon coupling: the ghost-phonon
The ghost-phonon (see Section IV B and Fig. 7)corresponds to a process where a high energy multi-excitation decays into a pair of phonons of lower en-ergy. In the case of phonon single-excitations , anoma-lous dispersion opens the phase space needed for suchprocesses. The anomalous character of the phonon dis-persion strongly decreases with increasing pressure, andnormal dispersion is recovered at high pressures .The ghost-phonon intensity follows this trend: the pres-sure dependence is strong, and the ghost-phonon isclearly suppressed at P = 24 bars, as shown in the ex-perimental and theoretical results in Fig. 4, and in moredetail in Fig. 8.Cuts of S ( Q, ω ) at several wave vectors at the ghost-phonon level are presented for P =0, 5 and 10 bars inFig. 9. The ghost-phonon peaks for the different wavevectors are clearly located on the extension of the linearpart of the phonon branch. According to the calcula-tions [see Eq. (6.4) of Ref. 9], the ghost-phonon remainsvisible until twice the wave vector up to which the dis-persion relation is to a good approximation linear. In-deed, Fig. 9 shows that the energy, strength and shape ofthe calculated ghost-phonon are in excellent quantitativeagreement with the experiment at all pressures. E n e r g y ω ( m e V ) Wave vector Q ( Å - ) FIG. 8. Left: measured dynamic structure factor S ( Q, ω )in the ghost-phonon region at P = 0, 5, 10 and 24 bars.The dashed straight lines correspond to the sound disper-sion curve at each pressure, taken from direct measurementsof the sound velocity . Right: calculated dynamic structurefactor at corresponding densities, n = 0 . − , respectively (see text). The dashed straightlines correspond to the calculated sound velocities. The color-coded intensity scale is in units of meV − . C. Phonon-roton coupling and the emergence ofthe ghost-roton
One notes in Fig. 4, for all pressures, the presenceof substantial intensity in the region within the rotonparabola. Near the roton minimum, where P-R processesare expected to dominate, we observe that the intensity isnot symmetric with respect to the roton minimum wavevector Q R : a faint branch, clearly related to the kine-matic limitation for P-R + processes (see Fig. 6) is seen0 FIG. 9. Dynamic structure factor S ( Q, ω ) in the ghost-phonon region: spectra for different wave vectors Q at (a) P = 0 bar, (b) P = 5 bars and (c) P = 10 bars. Filled cir-cles: Experimental S ( Q, ω ). Theoretical dynamic structurefactor spectra shown as solid lines at densities n = 0 . − . Dashed lines: Intensity of thephonon-roton mode (cut off) calculated directly from the selfenergy and convolved with the instrumental resolution of0.07 meV. The blue lines represent the linear phonon disper-sion (cid:15) Q ( P ) / (cid:126) Q = C ( P ), where C ( P ) is the sound velocityat a given pressure . for Q < Q R , while a strong branch is formed just abovethe dispersion curve for Q > Q R . These new features,and in particular the one for Q > Q R , provide a signifi-cant contribution to the multi-excitations weight at lowpressures (Fig. 10). They appear as an extension of theroton parabolic dispersion towards higher energies, andby analogy with the ghost-phonon, we call these multi-excitations “ghost-rotons”.It is remarkable that the intensity in this region ofthe P-R multi-excitations, as was the case for the ghost-phonon, is high at P = 0, but is suppressed in the 24 barsdata, as shown in Figs. 10, 11 and 12. The origin of theseeffects is discussed below.Spectra for several wave vectors in the region of theghost-roton are shown in Fig. 12 at P = 0 and 24 bars (ex-periment), and in Fig. 13 for the corresponding densities n = 0 . − (theory). We observe a goodagreement between theory and experiment, even at thehighest densities, near solidification. Studies of mode-mode couplings can therefore be most convenientlyperformed in the ghost phonon and the ghost-roton re-gions, rather than looking for a very small broadening ofsingle excitations.Pitaevskii described the decay of single-excitations when their group velocity reaches the velocity of sound. S ( Q , ω )( m e V - ) S ( Q , ω )( m e V - ) FIG. 10. Theoretical and experimental results for S ( Q, ω )at saturated vapor pressure displaying enhanced multi-excitations (“ghost-rotons”) above the R + roton branch, inthe supersonic rotons region. The dashed lines represent thelimits of different neutron kinetic ranges, see Fig. 1. The smalloscillations observed along some contours should be disre-garded, they result from numerical discretization. He named this mechanism of single-excitation broaden-ing ”‘type a”’. The process considered here, however,is the emission of phonons by multi-excitations in thevicinity of nearly supersonic single-excitations. The gen-eration of multi-excitations by neutron scattering in theR + rotons region by this mechanism was qualitativelypredicted by Burkova . Here we show that the ghost-roton corresponds to this effect, that the ghost-phonon isa similar effect, involving supersonic phonons, and thatboth are correctly predicted by the Dynamic Many-BodyTheory .It has been observed by Dietrich et al. and confirmedby several groups (see and references therein) that theR + rotons group velocity reaches the sound velocity for Q ≈ − at low pressures, but remains below thespeed of sound near the melting pressure. We show inFigs. 14 and 15 our measured and calculated curves forthe group velocity of the single-excitations, for differentpressures. Two regions of interest are clearly seen: thefirst one, at low wave vectors, corresponds to the anoma-lous dispersion region and gives rise to the ghost-phonon,while the second occurs for wave vectors somewhat abovethat of the roton minimum (and slightly below the roton1 S ( Q , ω )( m e V - ) S ( Q , ω )( m e V - ) FIG. 11. Theoretical and experimental results for S ( Q, ω )at P = 24 bars. A comparison with Fig. 10 shows that athigh pressures, ghost-roton multi-excitations are strongly sup-pressed. They are masked by the finite energy resolution inthe experimental graph, but still visible in the calculation. minimum, but with a much smaller intensity), producingthe ghost-roton.According to the analytic calculations by Burkova ,the neutron-scattering spectrum which corresponds tothe production of one roton should have a linear wingon the high-energy side, with a slope which depends onthe wave vector. This is not really observed, neither inthe experimental data, nor in the Dynamic Many-Bodycalculation: the linear part, if any, is probably not visibleat the scale of the graphs (see Figs. 10, 11, 12 and 13), oris buried inside a broadened single-excitations branch.Several effects are thus observed when the roton single-excitations approach the speed of sound: a broadening ofthe roton branch, a downward bending of the dispersioncurve, and the appearance of a multi-phonon region justabove the distorted dispersion curve. These effects arelarge at low pressures; the rapid increase of the soundvelocity with pressure is responsible for the suppressionof the ghost-roton multi-excitations at 24 bars. D. Roton-roton coupling
We discuss now a different type of multi-excitations ,related to Pitaevskii’s “type b” single-excitations decay (cid:16) (cid:13) (cid:8) (cid:5)(cid:7) (cid:6) (cid:1)(cid:28) (cid:4)(cid:7) (cid:16) (cid:13) (cid:8) (cid:5)(cid:7) (cid:12) (cid:1)(cid:28) (cid:4)(cid:7) (cid:16) (cid:13) (cid:8) (cid:5)(cid:9) (cid:8) (cid:1)(cid:28) (cid:4)(cid:7) S( w ) (meV-1) (cid:16) (cid:13) (cid:8) (cid:5)(cid:10) (cid:7) (cid:1)(cid:28) (cid:4)(cid:7) (cid:1) (cid:16) (cid:13) (cid:8) (cid:5)(cid:11) (cid:6) (cid:1)(cid:28) (cid:4)(cid:7) (cid:15) (cid:24)(cid:20) (cid:25) (cid:25) (cid:26) (cid:24)(cid:20) (cid:1)(cid:13) (cid:1)(cid:6) (cid:1)(cid:16) (cid:13) (cid:8) (cid:5)(cid:11) (cid:12) (cid:1)(cid:28) (cid:4)(cid:7) (cid:14) (cid:23) (cid:20) (cid:24)(cid:21) (cid:27) (cid:1) w (cid:1)(cid:1)(cid:2)(cid:22) (cid:20) (cid:17) (cid:3) (cid:16) (cid:13) (cid:8) (cid:5)(cid:7) (cid:6) (cid:1)(cid:28) (cid:4)(cid:7) (cid:16) (cid:13) (cid:8) (cid:5)(cid:7) (cid:12) (cid:1)(cid:28) (cid:4)(cid:7) (cid:16) (cid:13) (cid:8) (cid:5)(cid:9) (cid:8) (cid:1)(cid:28) (cid:4)(cid:7) S( w ) (meV-1) (cid:16) (cid:13) (cid:8) (cid:5)(cid:10) (cid:7) (cid:1)(cid:28) (cid:4)(cid:7) (cid:16) (cid:13) (cid:8) (cid:5)(cid:11) (cid:6) (cid:1)(cid:28) (cid:4)(cid:7) (cid:15) (cid:24)(cid:20) (cid:25) (cid:25) (cid:26) (cid:24)(cid:20) (cid:1)(cid:13) (cid:1)(cid:8) (cid:10) (cid:1)(cid:19) (cid:18) (cid:24)(cid:1) (cid:16) (cid:13) (cid:8) (cid:5)(cid:11) (cid:12) (cid:1)(cid:28) (cid:4)(cid:7) (cid:14) (cid:23) (cid:20) (cid:24)(cid:21) (cid:27) (cid:1) w (cid:1)(cid:1)(cid:2)(cid:22) (cid:20) (cid:17) (cid:3) FIG. 12. S ( Q, ω ) measured in the region of the ghost-rotonat P = 0 (upper graph). At P = 24 bars (lower graph) oneobserves the suppression of the ghost-roton. Dashed lines areGaussian fits of the single-excitation peaks. A comparisonwith Figs. 10 and 11 clarifies the origin of the observed roton-peak asymmetry for some wave vectors. processes where the disintegration of a single-excitationoccurs as its energy exceeds twice the roton gap .At high pressures, the maxon energy exceeds twice theroton gap, and a maxon can decay into two rotons. Wedescribed in Section IV B the broadening of the maxonas it enters the continuum. At 24 bars, the maxon is inthe continuum of the multi-excitations for wave vectors2 S( w ) (meV-1) (cid:15) (cid:1) (cid:2) (cid:30) (cid:4)(cid:7) (cid:3) (cid:1) (cid:1) (cid:8) (cid:5) (cid:6) (cid:7)(cid:1) (cid:8) (cid:5) (cid:7) (cid:6)(cid:1) (cid:8) (cid:5) (cid:7) (cid:12)(cid:1) (cid:8) (cid:5) (cid:9) (cid:8)(cid:1) (cid:8) (cid:5) (cid:10) (cid:7)(cid:1) (cid:8) (cid:5) (cid:11) (cid:6)(cid:1) (cid:8) (cid:5) (cid:11) (cid:12) (cid:16) (cid:22) (cid:19) (cid:25) (cid:26) (cid:29) (cid:1)(cid:1)(cid:24) (cid:13) (cid:6) (cid:5)(cid:6) (cid:8) (cid:11) (cid:11) (cid:1)(cid:30) (cid:4)(cid:9) S( w ) (meV-1) w (cid:1)(cid:2) (cid:23) (cid:19) (cid:17) (cid:3)(cid:16) (cid:22) (cid:19) (cid:25) (cid:26) (cid:29) (cid:1)(cid:1)(cid:24) (cid:13) (cid:6) (cid:5)(cid:6) (cid:8) (cid:7) (cid:11) (cid:1)(cid:30) (cid:4)(cid:9) FIG. 13. S ( Q, ω ) calculated for wave vectors in the regionof the ghost-roton, at densities n = 0 . − ,associated to P = 0 and P = 24 bars respectively. P (cid:2) (cid:2) (cid:2)
10 barP (cid:2)
24 bar0.0 0.5 1.0 1.5 2.0 2.5 (cid:3) (cid:3) Q (cid:2) (cid:2) (cid:3) (cid:3) Ω (cid:5) (cid:2) Q (cid:3) (cid:4) c FIG. 14. The experimentally determined group velocity ofsingle-excitations normalized by the sound velocity , as afunction of wave vector for several pressures. between Q = 0 . Q = 1 . − . Under these con-ditions, a strong multi-excitation intensity is observedabove the maxon (Figs. 16 and 17). The very character-istic “rainbow-like” measured spectrum is in very goodagreement with the theoretical calculation, showing inparticular that the weight of the maxon is transferred tothe two-roton excitations.The multi-excitations discussed above, observed abovethe maxon at high pressure, are a special case of roton- −1.0−0.5 0.0 0.5 1.0 1.50.0 0.5 1.0 1.5 2.0 2.5 ω ’ ( (cid:52) ) / c (cid:52) (Å −1 ) (cid:81)(cid:32)(cid:3) −3 (cid:81)(cid:32)(cid:3) −3 (cid:81)(cid:32)(cid:3) −3 (cid:81)(cid:32) −3 (cid:81)(cid:32)(cid:3) −3 FIG. 15. The theoretically calculated group velocity of single-excitations normalized by the calculated sound velocity, as afunction of the wave vector, for several densities. The experi-mental values of the densities for P = 0, 5, 10 and 24 bars are n exp =0.0218, 0.0230, 0.0239 and 0.0258 ˚A − . roton decay. In fact, a sharp roton-roton threshold isobserved at all wave vectors (Figs. 1, 4 and 17), in re-gions located far from single-excitations. The roton-roton threshold is, in particular, observed at low Q inthe present work. It is also clear, in fact, that the in-tensity of the RR threshold is enhanced in the vicinityof single-excitations, as was the case above the maxonat 24 bars, but also in the region above the Pitaevskiiplateau. Theory and experiment display a similar shapeof the spectra and intensity pattern around the roton-roton threshold, at all pressures (see Figs. 1 and 4). E. Higher order multi-excitations
The sharp “branches” described above correspond todecay mechanisms into 2-excitations. Phase-space argu-ments show that the signal of higher order processes willbe distributed in a rather featureless way in the energy-wave vector space, due to the vector addition of mo-menta. However, the data of Fig. 1 show that the multi-excitations region at wave vectors on the order of 1.5 ˚A − extends to rather high energies, on the order of 4 meV.This last value constitutes a clear experimental demon-stration that multi-excitations of higher order, related to3 and 4 single-excitations (the energy of rotons and max-ons is on the order of 1 meV), play a significant role inthe dynamics of superfluid He.One can also examine the corresponding effect on thewave vector axis, beyond the end-point of the Pitaevskiiplateau. The plateau could be expected to end at2 Q R , for a multi-excitation of energy 2∆ R decayinginto two rotons of colinear wave vectors. Previousmeasurements have found that the plateau in-tensity vanishes at Q = 3 . − , considerably below2 Q R = 3 .
84 ˚A − . This is also observed in the presentwork, as seen in Figs. 1 and 3. This effect has been at-tributed to the decay into two rotons with an attractive3 S ( Q , ω )( m e V - ) S ( Q , ω )( m e V - ) FIG. 16. S ( Q, ω ) in the region of the maxon at P = 24 bars(experiment) and at the corresponding density of 0.0255 ˚A − (theory). R-R interaction , but other possible interpretations ofthe data are presently debated. We also note that theintensity does not extend to higher Q -values at higherenergies as expected for decays into 3- and 4-excitationsprocesses, an effect which is probably related to the smallphase-space available for colinear combinations of wavevectors. As discussed above, the energy, a scalar, is a bet-ter probe for detecting high order multi-excitation pro-cesses. The data at 24 bars display similar effects witha simple shift towards higher wave vectors, due to thelarger value of Q R =2.06 ˚A − at this pressure.We now concentrate on the multi-excitations regionlocated slightly below the free-particle dispersion curve,around 2.5 ˚A − (see Fig. 1). Earlier studies ob-served a rather intense broad feature extending to higherenergies. We find here a much finer structure than previ-ously believed, and also that it depends rather stronglyon the pressure. Multi-excitations in this region can onlydecay into 3 or more single-excitations, which is there-fore of interest for mode-mode coupling theories. Thefact that we observe a high intensity peak is probablyrelated, at these relatively high energies, to an enhancedsystem response in the vicinity of the free-particle disper-sion curve, which is the asymptotic behavior at higher FIG. 17. Dynamic structure factor S ( Q, ω ): spectra for dif-ferent wave vectors Q in the maxon region, at P = 24 bars.Filled circles: Experimental S ( Q, ω ). Solid lines: theory atthe density n =0.0255 ˚A − . Dashed lines: Intensity of thephonon-roton mode (cut off) calculated directly from the selfenergy and convolved with the instrumental resolution of0.07 meV. Blue line: energy of the roton-roton threshold. PRindicates phonon-roton multi-excitations. energies. The peak at 24 bars is less intense than thecorresponding one at SVP, suggesting that the maxon,strongly reduced at this pressure, is involved in the cor-responding decay processes.Finally, at the highest energies explored here, S ( Q, ω ) progressively converges towards the free-particleparabola, remaining below it (see Fig. 1). The so-called“glory” oscillations seen as a function of Q , both in thepeak position and the width, are well documented in the4literature . Directly related to the corresponding os-cillations in the static structure factor S ( Q ), they resultfrom the hard core part of the He- He interaction poten-tial and from quantum coherence effects. Earlier workscould not fit the spectra of the first oscillation with a sin-gle peak. The highly structured multi-excitations seen inthe present work show that this peak of unusual shaperesults in fact from the superposition of a few multi-excitation “branches” corresponding to decays into a fewsingle-excitations. Again, the dynamic structure factorin this region depends on pressure, and the spectra for Q ≈ . − are strongly affected by the collapse of themaxon. VII. CONCLUSION
A comprehensive understanding of the dynamics of in-teracting Bose systems, going from the Landau quasi-particles and multi-excitations regimes, up to the high-energy limit where the independent particle dynamicsis recovered, emerges from our combined experimentaland theoretical work. The up-to-now largely unexploredmulti-excitations regime has been systematically investi-gated. Ghost-phonon and ghost-roton regimes have beenobserved, associated to phonon emission in the regionof nearly supersonic multi-excitations, by a Cherenkov-like process qualitatively predicted by Burkova’s exten- sion of Pitaevskii’s theory. Several other multi-excitationbranches or thresholds have been observed and identifiedin the low energy sector, where an excellent quantitativeagreement is found with the predictions of the DynamicMany-Body theory. This agreement extends even to highpressures, near solidification, as shown for example forthe remarkable case of the maxon disintegration into tworotons. The calculations including specific multiparti-cle fluctuations to all orders provide a good descriptionof the dynamics for energies as high as 2 meV. Abovethis value, higher order processes dominate the dynam-ics. Our high energy/wave vector data call for furthertheoretical developments able to describe quantitativelythe behavior observed at higher energies, above the sim-ple multi-excitations region but still substantially belowthe quasi-free particle (impulse-approximation) sector. VIII. ACKNOWLEDGEMENTS
We are grateful to X. Tonon for his help with the ex-periments. This work was supported, in part, by theAustrian Science Fund FWF grant I602, the French grantANR-2010-INTB-403-01, the European Community Re-search Infrastructures under the FP7 Capacities SpecificProgramme, Microkelvin project number 228464, and theEuropean Microkelvin Platform. D. Pines and P. Nozi`eres,
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