Quantum filaments in dipolar Bose-Einstein condensates F. W¨achtler and L. Santos Institut fu¨r Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, DE-30167 Hannover, Germany Collapse in dipolar Bose-Einstein condensates may be arrested by quantum fluctuations. Due to theanisotropyofthedipole-dipoleinteractions,thedipole-drivencollapseinducedbysoftexcitations iscompensatedbytherepulsiveLee-Huang-Yangcontributionresultingfromquantumfluctuations of hard excitations, in a similar mechanism as that recently proposed for Bose-Bose mixtures. The arrestedcollapseresultsinself-boundfilament-likedroplets,providinganexplanationtorecentdys- prosiumexperiments. Arrestedinstabilityanddropletformationarenovelgeneralfeaturesdirectly 6 linked to the nature of the dipole-dipole interactions, and should hence play an important role in 1 all future experiments with strongly dipolar gases. 0 2 PACSnumbers: n a J Dipole-dipole interactions (DDI) lead to qualitatively (a) (b) 8 new physics for dipolar gases compared to non-dipolar 1 ones [1, 2]. As a result, this physics constitute the fo- cus of a large interest, including experiments on mag- ] netic atoms [3–6], polar molecules [7–10], and Rydberg- s a dressed atoms [11]. A characteristic feature of dipo- g lar Bose-Einstein condensates (BECs) is their geometry- - t dependent stability [12]. If the condensate is elongated n along the dipole orientation, the DDI are attractive in a u average, and the BEC may become unstable, in a simi- FIG. 1: (Color online) Crystal-like droplet arrangements of q lar,butnotidentical,wayasaBECwithnegatives-wave n (x,y)/N, with n (x,y) = (cid:82) dzn(r), for a BEC of XY XY t. scatteringlength, a<0. Chromiumexperimentsshowed N=7500 atoms (top), and 15000 atoms (bottom), initially a that, as for a < 0, the unstable BEC collapses, albeit formed with a=120aB, 20ms after a quench to a=70aB. m with a peculiar d-wave post-collapse dynamics [13]. - d Thispicturehasbeenchallengedbyrecentdysprosium orientation. As a result, the LHY correction resulting n experiments [14], in which destabilization, induced by a fromthehardmodesprovidesarepulsivetermthatdomi- o quenchtoasufficientlylowa,isnotfollowedbycollapse, natesatlargedensitiesarrestinglocalcollapses,resulting c but rather by the formation of stable droplets that are [ in the nucleation of droplets (see Figs. 1). We show by only destroyed in a large time scale by weak three-body means of a generalized nonlocal non-linear Schr¨odinger 1 losses (3BL). This surprising result, which resembles the equation (NLNLSE) that this mechanism accounts for v Rosensweiginstabilityinferrofluids[15,16],pointstoan 1 the Dy experiments. We stress that this effect results 0 up to now unknown stabilization mechanism that plays fromthepeculiarnatureoftheDDI,beinghenceachar- 5 a similar role as that of surface tension in classical fer- acteristic novel feature of strongly dipolar gases, which 4 rofluids. It has been recently suggested that large con- shouldplayanimportantroleinfutureexperimentswith 0 servative three-body forces, with a strength several or- highly magnetic atoms and polar molecules. . 1 ders of magnitude larger than the 3BL, may account for Generalized NLNLSE– We consider a BEC of mag- 0 the observation [17, 18]. There is however no justifica- netic dipoles of mass m and dipole moment µ oriented 6 tion of why large three-body forces should be present, or 1 alongthezdirectionbyanexternalmagneticfield(equiv- whether there is a link between them and the DDI. : alent results can be found for electric dipoles). In mean- v This Letter explores an alternative mechanism, based field (MF), the physics is given by the NLNLSE [1]: i X on quantum fluctuations, which is suggested by very (cid:20)−(cid:126)2∇2 (cid:21) r recent experiments [19]. As recently shown [20], Lee- i(cid:126)ψ˙(r)= +V(r)+g|ψ(r)|2+Φ(r) ψ(r), (1) a Huang-Yang (LHY) corrections may stabilize droplets 2m in unstable Bose-Bose mixtures. This interesting effect with ψ(r) the BEC wavefunction, V(r) the trapping po- results from the presence of soft and hard elementary tential, g = 4π(cid:126)2a, and Φ(r) = (cid:82) d3r(cid:48)V (r−r(cid:48))|ψ(r(cid:48))|2, excitations. Whereas soft modes may become unstable, m dd quantumfluctuationsofthehardmodesmaybalancethe withVdd(r)= µ40π|µr3|2(1−3cos2θ),whereµ0isthevacuum instability,resultinginanequilibriumdroplet. Asshown permittivity, and θ is the angle between r and µ. below, due to the anisotropy of the DDI, a dipolar BEC In the homogeneous case, V(r)=0 with density n, el- also presents soft and hard modes, characterized in free ementary excitations with momentum k have an energy spacebymomentaperpendicularorparalleltothedipole E(k) = (cid:112)(cid:15) ((cid:15) +2gnf((cid:15) ,θ )), where (cid:15) = (cid:126)2k2, and k k dd k k 2m 2 f((cid:15) ,θ )=1+(cid:15) (3cos2θ −1), with (cid:15) = µ0|µ|2, and effect is well recovered by Eq. (3). We postpone for a dd k dd k dd 3g θ the angle between k and µ. Due to the anisotropy of future analysis the detailed study of the effect of long k the DDI, excitations with cos2θ > 1/3 become harder wavelength excitations. In addition, the validity of the k with growing (cid:15) , whereas those with cos2θ < 1/3 be- generalized NLNLSE demands a small quantum deple- dd k come softer. For (cid:15)dd > 1, long wave-length excitations tion [21, 22], η(r) ≡ ∆nn(r(r)) = 3√8π(cid:112)n(r)a3FD((cid:15)dd), with with θk = π/2 drive the BEC unstable. Quantum fluc- F ((cid:15) )= 1(cid:82) dθ sinθ f((cid:15) ,θ )3/2. Inoursimulations, tuations of the excitations result in the LHY correction η(Dr)(cid:46)dd0.012at anky poinkt andddtimk e. of the chemical potential [21, 22]: Droplet nucleation– In the following we employ 32 √ Eq. (3) to study the formation of BEC droplets in re- ∆µ(n,(cid:15)dd)= 3√πgn na3F((cid:15)dd), (2) centDyexperiments[14]. WeconsideraBECwithN Dy atoms,with|µ|=10µ ,withµ theBohrmagneton. In B B with F((cid:15) ) = 1(cid:82) dθ sinθ f((cid:15) ,θ )5/2. In the vicinity order to compare our results with recent experiments we dd 2 k k dd k oftheinstability,(cid:15)dd ∼1,theoverwhelmingcontribution assume a trap with ωx,y,z/2π =(44,46,133)Hz [25]. We to F((cid:15) ) stems from hard modes (cos2θ > 1/3). Cru- employ imaginary time evolution of Eq. (3) to form an dd k cially, thisistrueevenwhentheBECbecomesunstable. initial BEC with a = 120aB, with aB the Bohr radius. This situation, with unstable soft modes and LHY cor- Under these conditions the BEC, with a wavefunction rection dominated by stable hard modes, resembles the ψ0(r), is stable and in the TF regime. At finite temper- recentlydiscussedcaseofBose-Bosemixtures[20]. Asfor ature, T, thermal fluctuations seed the modulational in- thatscenario,thecontributionoftheunstablesoftmodes stability after the quench of a discussed below, and may isnegligiblefor(cid:15) ∼1, andquantumfluctuationsofthe hence influence droplet nucleation. Following Ref. [18] dd hard modes result in a repulsive LHY correction ∝n3/2. we add thermal fluctuations (for T =20nK) in the form (cid:80) Let us consider at this point a harmonically trapped ψ(r,t=0)=ψ0(r)+ nαnφn,whereφn areeigenmodes BEC,V(r)= 1m(ω2x2+ω2y2+ω2z2). Thetreatmentof of the harmonic trap with eigenenergies (cid:15)n, the sum is 2 x y z beyondMFcorrectionsisingeneralmuchmoreinvolved. restricted to (cid:15)n < 2kBT, and αn is a complex Gaussian In the Thomas-Fermi (TF) regime one may evaluate the random variable with (cid:104)|αn|2(cid:105)= 21+(e(cid:15)n/kBT −1)−1 [26]. effectofquantumfluctuationsbytreatingtheexcitations At t = 0 we perform a quench in 0.5ms to a final quasi-classically and employing local density approxima- a = 70a that destabilizes the BEC [27]. The most B tion (LDA), obtaining a corrected equation of state [21, unstable Bogoliubov mode has a non-zero angular mo- 22]: µ(n(r))=V(r)+µ (n(r),(cid:15) )+∆µ(n(r),(cid:15) ), with mentum, a so-called angular roton [28], and as a result 0 dd dd µ (n(r),(cid:15) ) = gn(r)+(cid:82) d3r(cid:48)V (r−r(cid:48))n(r(cid:48)). One may at T = 0 the BEC develops an initial ring-like modula- 0 dd dd then insert this correction in a generalized NLNLSE: tional instability on the xy plane, followed by azimuthal symmetry breaking into droplets. At finite T droplets (cid:104) (cid:105) i(cid:126)ψ˙(r)= Hˆ +µ (n(r),(cid:15) )+∆µ(n(r),(cid:15) ) ψ(r), (3) may nucleate from thermal fluctuations before the ring- 0 0 dd dd like structure associated with the angular-roton insta- with Hˆ ≡ −(cid:126)2∇2 + V(r). This equation is appealing bility develops (as it is observed in experiments [29]). 0 2m Both cases are characterized by the formation of sta- since it allows for a simplified analysis of the effects of ble droplets in few ms, which eventually arrange in a quantum fluctuations in the TF regime, and because it quasi-crystalline structure as those of Figs. 1, in excel- may be simulated using the same numerical techniques lentagreementwiththeexperimentalresultsofRef.[14]. employedforEq.(1)[1,23]. HowevertheuseoftheLDA Droplet nucleation does not involve however the whole toevaluatetheeffectsofquantumfluctuationsinquench condensate. A significant amount of atoms remains in a experiments must be carefully considered. The droplets halo-like background too dilute to gather particles into a discussedbelowareintheTFregimealongthedipoledi- stable droplet (approximately 30% in Figs. 1, although rectioninallcases,whereasonlylargedropletsareinthe it is barely visible due to the contrast). TF regime also along xy. For small droplets, with less than 4000 atoms in the calculations below, the xy den- Droplet features – The droplets result from the com- sity profile approaches rather a Gaussian. We may eval- pensation of the attractive MF term µ0 ∝ n(r) by the uate the contribution of quasi-classical excitations with effective repulsion introduced by the LHY term, ∆µ ∝ momenta k, such that |k|R(θ ) (cid:29) 1 where R(θ ) is a n(r)3/2. In order to study the properties of individ- k k typicaldistancefordensityvariationinthedropletalong ual droplets, we evolve Eq. (3) in imaginary time for the direction given by the angle θk. This contribution a = 70aB and different particle numbers. In order to is for the smallest droplets presented below of the or- guarantee the controlled formation of a single droplet in der of ∼80% of the total LHY correction expected from thenumerics,weemployasinitialconditionfortheimag- LDA (for details of this estimation see [24]). The cor- inary time evolution a cigar-like Gaussian wavefunction rection due to long wave-length modes may hence mod- atthetrap centerverycompressedonthe xy plane[30]. ify the prefactor of the correction, but the bulk of the Figure2showsthedropletenergy,E ,asafunctionof D 3 4 10 (a) 0.07 9 2 0.06 8 ~-E/()h(cid:116)D ---- 86420 n(0,0,z)/N 00000.....00000 123450-3 -2 -1 z(µ 0m) 1 2 3 Number of droplets 234567 Average particle number 11111 012349000000000000 5000 10000 15000 20000 -10 1 Total number of particles 0 5000 10000 15000 20000 5000 7500 10000 12500 15000 17500 20000 ND Number of atoms (b) FIG. 2: (Color online) Droplet energy, E (in units of (cid:126)ω˜, D with ω˜ =(ω ω ω )1/3) as a function of the number of parti- x y z cles in the droplet, N , for a=70a ; for N <N (cid:39)900 D B D min nostabledropletisfound. Dropletswithpositiveinternalen- ergy occur for N (cid:46)1500. In the inset, we show the density D profile (solid line with crosses) of a droplet with N = 1000 D atthetrapcenterforthecutx=y=0. Atthecenterofthe droplet, n(0,0,z)∝(1−z2/Z2)2/3 (dotted curve). FIG.3: (Coloronline)(a)Numberofdroplets(seetext)asa function of the initial number of atoms 20ms after a quench froma=120a to70a ;theinsetshowsthenumberofparti- B B the number of particles in the droplet, ND. Two impor- clesperdropletasafunctionofN underthesameconditions. tant features are worth mentioning. There is a minimal Inbothcasestheaveragevalueisdenotedbyabluecross,and particle number, N (cid:39)900, such that for N <N the variance by the error bar; (b) Histogram of the number min D min of particles in a droplet again for the same conditions. The no stable droplet may form. If the local density does histogram was evaluated from a sample of 260 droplets. not allow for the gathering of that critical number, then nodropletisformed,accountingforthebackgroundhalo. Second,E (N )presentsanon-monotonousdependence D D with N, showing a minimum (at N (cid:39)13000 in Fig. 2), that overwhelms the repulsive contact MF term. In ad- D being only positive at N values close to N (E =0 dition, as mentioned above, the droplets are in any case D min D atN (cid:39)1500inFig.2). Thisisparticularlyrelevantfor well within the TF regime along the z direction (see the D the droplet nucleation after a quench. After the quench, inset of Fig. 2 for ND = 1000). Large droplets, with theBECenergy, whichisinitiallypositive[31], isalmost ND >8000,arealsowellwithintheTFregimealongthe conserved, just decreasing slowly due to 3BL. The final xy direction. On the contrary as already noted, small droplet gas is characterized by the internal energy of the droplets with ND < 4000 have approximately a Gaus- droplets, the center of mass (CM), kinetic and poten- sian profile along xy. This remains true for the droplets tial, energy of the droplets, and the inter-droplet dipole- found in the simulations of quench experiments. Quan- dipole repulsion (the halo, being much more dilute has tum pressure is hence non-negligible for small droplets, a comparatively small contribution to the BEC energy). butitisnotcrucialforthedropletstability,whichispro- Although the CM energy and the repulsive inter-droplet vided by the compensation of the attractive MF interac- interaction are obviously positive, they cannot balance a tion and the LHY correction. This is in stark contrast negative internal energy of the droplets, as required by with self-bound solitons that require necessarily quan- the quasi-conservation of the energy in the absence of tum pressure to compensate attractive interactions, and strong dissipation. This explains why, as discussed be- hencecannotoccurinanycaseintheTFregime. Dueto low, inthequenchexperimentsdropletsformwithparti- the LHY term, the droplets do not present an inverted- cle numbers between 900 and 1500, despite the fact that paraboloid profile even in the TF regime. At the center, bigger droplets could be in principle stable (Fig. 2). n(x=0,y =0,z)(cid:39)(1−z2/Z2)2/3 (inset of Fig. 2). The shape of the droplets is also significant for the Droplet statistics in quench experiments– We have overall discussion. The droplets are markedly elongated performedfordifferentN simulationsoftheBECdynam- alongthedipoledirection. ForN (cid:39)1000,thezhalf-size ics after the quench of a, starting from different initial D is(cid:39)2µm,whereasalongxyis(cid:46)0.3µm. Thisisexpected, conditions given by random thermal fluctuations. As in since a cigar-like shape is required for an attractive DDI the experiments, we observe that the droplets arrange in 4 crystal-likepatterns(seeFig.1),althoughtheypresenta 2.5 residual dynamics. Our simulations show that the num- 10000 berofparticlesinadropletvariesfromdroplettodroplet 2 inasingleshotandbetweenshots(asintheactualexper- 8000 W iments). This variance results from the fact that stable S ms dvnimaurolmpupoeblrseettraNsn,oDcfaeds>roosfhpNtolhemwetisnbnia.sacabkTdoghdvreoeitu,iconomndrarahelylsayplboaoe.ffndIefnocintraemgddedvbdaiytrifiotoahnrne,cddeviraffoorefpiarltebehntleest pectral weight, 1 .15 46000000 Number of ato S formedatthevergeofinstabilityN (cid:39)N ,mayeven- D min 0.5 2000 tuallybecomeunstableanddiluteinthehalo, andhence the number of apparent droplets may vary in time. In order to measure objectively the number of droplets, we 0 0 0 5 1015 50 100 150 200 250 300 350 400 (cid:82) have obtained the column density nXY(x,y)≡ dzn(r) time (ms) after 20 ms of post-instability dynamics, and defined a dropletassuchifitreachesamaximaln /N >0.3(see X,Y FIG. 4: (Color online) Number of atoms (blue dotted) and e.g. Figs. 1). Figure 3(a) summarizes our results for the spectral weight (red solid) as a function of the time after a dependence with N of the number of droplets formed quench from a = 120a to 70a for a BEC with initially B B after 20 ms of post-instability dynamics. Although as N =10000 atoms. mentioned above the number of droplets presents a rel- evant statistical variance, the average number shows in agreement with experiments an approximate linear de- tual nucleation of further droplets, and especially in the pendence. The deviation at larger N is due to the fact formation of crystal-like droplet arrangements as those that at 20ms there are a number of droplets that are of Fig. 1, which minimize inter-droplet interaction. The about to be nucleated in the outer halo regions but are results of Fig. 4 are in very good agreement with exper- not fully formed (according to the previous criterion). iments, showing a growth of SW up to t ∼ 10ms, and a The deviation at low N is due to the longer time needed subsequent decrease in a much longer time scale, accom- tofullydevelopdroplets(e.g. N =5000developsupto3 panied by the corresponding particle loss due to 3BL. dropletsafter40ms). Theapproximatelineardependence Conclusions and outlook– Quantumfluctuationspre- of Fig. 3(a) stems from the local character of the nucle- vent local collapses in unstable dipolar BECs. In par- ation, which results in a number of particles per droplet ticular, our results show the appearance of filament-like that is basically independent of N (inset of Fig. 3(a)). droplets,accountingforrecentresultsinDycondensates. The histogram of Fig. 3(b) shows that, as expected from SincetheLHYcorrectiondependsonna3,weexpectthat our discussion of the droplet energy, the particle number droplets should collapse for lower a values, providing a per droplet lies overwhelmingly between 900 and 1500, clear criterion to discern LHY stabilization from stabi- with an average of approximately 1200, again in good lization based on large three-body forces [17, 18]. Our agreement with experiments. results, based on a simplified treatment using a general- Three-body losses– We observe in our Dy simula- ized NLNLSE are already in very good agreement with tions peak densities of ∼ 2 × 1021m−3 [32]. At these the experiments [14, 19], although a more precise analy- densities, albeit low, 3BL become relevant in the long sis of the effects of long-wave length excitations in small run. In order to take them into account, we add a droplets may be necessary to provide a fully quantita- term −i(cid:126)L3|ψ(r)|4ψ(r) [13] to the right hand side of tive comparison, in particular in what concerns the peak 2 Eq. (3), with L =1.2×10−41m6/s [33]. Figure 4 shows densityinthedroplets. WestressthatLHYstabilization 3 for N = 10000 the number of atoms as a function of resultsfromtheanisotropyofthedipolarinteractions. It time. We show in the same figure, the spectral weight was absent in previous Cr experiments [13] because the SW =(cid:82) d2kn˜ (k), wheren˜ (k)istheFouriertrans- BECbecameunstableforavalueofa(cid:39)10timessmaller XY XY (cid:82) form of the column density n (x,y) ≡ dzn(r) and than in Dy (m|µ| in Dy is 10 times larger than in Cr), XY themomentumintegralextendsfromk =1.5µm−1 to and LHY stabilization would demand n ∼ 1024m−3, a min k =5µm−1. ThisfunctionwasintroducedinRef.[14] density which is never reached due to 3BL. In contrast, max to characterize the appearance and disappearance of the LHYstabilizationanddropletnucleationareacharacter- droplet pattern. The losses not only decrease the atom isticgeneralfeatureinducedbytheDDIthatmayplaya number, but also lead to the eventual destruction of the roleinallfutureexperimentswithstronglydipolargases droplets, which may loose too much particles to remain of highly-magnetic atoms and polar molecules. stableagainstmeltinginthebackground. Moreover,3BL Acknowledgements.–WethankD.PetrovandY.Lifor eliminates high-energy atoms at the BEC maxima. This interesting comments, and H. Kadau, I. Ferrier-Barbut, energy dissipation plays an important role in the even- andT.Pfauforinsightfuldiscussionsandforprovidingus 5 withtheirexperimentalresults. Weacknowledgesupport q (cos2θ+λ2sin2θ)1/2, with λ the aspect ratio of the z by the cluster QUEST, and the DFG Research Training droplet, and qz the cut-off along z. Introducing this Group 1729. cut-off in the LHY calculation for the homogeneous space at a given density n, results in a modified cor- rection ∆µc = 15√2 (cid:82)0πdθsinθχ((cid:15)dd,θ) where χ((cid:15) ,θ)= ∆µ 16 (cid:82)0πdθsinθf((cid:15)dd,θ)5/2 dd 25/2(cid:16)2f((cid:15)dd,θ) − qc(θ)2(cid:17)(cid:16)qc(θ)2 +f((cid:15) ,θ)(cid:17)3/2+qc(θ)5+ 15 2 2 dd 5 [1] See e.g. T. Lahaye et al., Rep. Prog. Phys. 72, 126401 qc(θ)3f((cid:15) ,θ)− qc(θ)f((cid:15) ,θ)2, and q (θ) is in units of 3 dd 2 dd c (2009), and references therein. ξ−1, with ξ = (8πna)−1/2. A droplet with N = 1000 D [2] See e.g. M. A. Baranov, M. Dalmonte, G. Pupillo, and particles has for a = 70a a z size of (cid:39) 2µm (cid:39) 25ξ B P. Zoller, Chem. Rev. 112, 5012 (2012), and references (with ξ calculated for an averaged central density of therein. 1.5×1021m−3), and an aspect ratio λ(cid:39)6. For a z cut- [3] A.Griesmaieretal.,Phys.Rev.Lett.94,160401(2005). off q ξ (cid:39) 0.25, excitations with |q(θ)| > q (θ) may be z c [4] M.Lu,N.Q.Burdick,S.H.Youn,andB.L.Lev,Phys. considered as quasi-classical. For this cut-off, we obtain Rev. Lett. 107, 190401 (2011). ∆µc (cid:39)0.8,showingalargecontributionofquasi-classical [5] K. Aikawa et al., Phys. Rev. Lett. 108, 210401 (2012). ∆µ excitations to the LHY correction. [6] A. de Paz et al., Phys. Rev. Lett. 111, 185305 (2013). [25] Droplets may form in other trap geometries as well, but [7] K. K. Ni et al., Science 322, 231 (2008). the details of the stability threshold, as well as of the [8] B. Yan et al., Nature 501, 521 (2013). droplet nucleation vary with the precise trap. [9] T.Takekoshietal.,Phys.Rev.Lett.113,205301(2014). [26] We obtain similar results using a stochastic Gross- [10] J.W.Park,S.A.Will,andM.W.Zwierlein,Phys.Rev. Pitaevskii equation to create the thermal excitations. Lett. 114, 205302 (2015). 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A 92, tion of (metastable) droplet structures should occur not 061603(R) (2015). only in the post-quench dynamics, but also when di- [19] I. Ferrier-Barbut et al., arXiv:1601.03318 rectly forming the condensates at sufficiently low scat- [20] D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015). tering lengths, as discussed in Ref. [14]. [21] A. R. P. Lima and A. Pelster, Phys. Rev. A 84, [31] Of the order of 8(cid:126)ω˜ in Figs. 1, with ω˜ ≡(ω ω ω )1/3. 041604(R) (2011). x y z [32] These densities are larger than those in recent experi- [22] A. R. P. Lima and A. Pelster, Phys. Rev. A 86, 063609 ments[19]byafactorof3–4.NotethattheLHYdepends (2012). onna3andhencealargeramayreducethedensity.Even [23] The simulation of Eq. (3) is performed using split oper- morerelevantly,asmentionedinthemaintext,thepref- atortechniques,andtreatingtheDDIusingconvolution actor of the LHY may be modified for small droplets theorem and fast-Fourier transformation. Following [S. byshort-wavelengthexcitations.Anincreaseinthepre- Ronen, D. C. E. Bortolotti, and J. L. Bohn, Phys. Rev. factor will further decrease the peak density. A 74, 013623 (2006)] we employ a cut-off of the dipole- [33] The exact value of L is not yet known but it should be dipole potential to reduce spurious boundary effects. 3 in the lower 10−41m6/s. See M. Lu, Quantum Bose and [24] The relative importance of short- and long-wave length Fermi gases of dysprosium: production and initial study, excitations in the LHY correction may be estimated as PhD Thesis, Stanford University (2014). follows. We consider a low-momentum cut-off q (θ) = c