First observation of bright solitons in bulk superfluid He-4
Francesco Ancilotto, David Levy, Jessica Pimentel, Jussi Eloranta
FFirst observation of bright solitons in bulk superfluid He Francesco Ancilotto , David Levy , Jessica Pimentel , and Jussi Eloranta Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM,Universit`a di Padova, via Marzolo 8, 35122 Padova,Italy and CNR-IOM Democritos, via Bonomea, 265 - 34136 Trieste, Italy Department of Physics and Astronomy, California State University at Northridge, California 91330, USA Department of Chemistry and Biochemistry, California State University at Northridge, California 91330, USA (Dated: September 25, 2018)The existence of bright solitons in bulk superfluid He is demonstrated by time-resolved shad-owgraph imaging experiments and density functional theory (DFT) calculations. The initial liquidcompression that leads to the creation of non-linear waves is produced by rapidly expanding plasmafrom laser ablation. After the leading dissipative period, these waves transform into bright solitons,which exhibit three characteristic features: dispersionless propagation, negligible interaction in two-wave collision, and direct dependence between soliton amplitude and the propagation velocity. Theexperimental observations are supported by DFT calculations, which show rapid evolution of theinitially compressed liquid into bright solitons. At high amplitudes, solitons become unstable andbreak down into dispersive shock waves.
PACS numbers: 67.25.D-,67.25.bf, 67.85.dt
Solitons are localized non-linear waves in a medium,which do not disperse as a function of time and exhibitno interaction during a two-wave collision. After theirdiscovery in the early eighteen hundreds, solitons havebeen observed in many different media, which exhibitpronounced non-linear response. In recent years, solitonshave become an intense field of research due to their im-portant applications in areas such as plasma physics, elec-tronics, biology, and optics [1]. Mathematical descriptionof solitons can be formulated in terms of model depen-dent non-linear partial differential equations (e.g., thenon-linear Schr¨odinger equation). In general, it has beenestablished that non-linear excitations (i.e., shock wavesand solitons) exhibit distinct dependency between theiramplitude and propagation velocity [1].Solitons in thin He films adsorbed on solid substrateshave been studied extensively by both experiments [2–5] and theory [6–9]. The film thickness is typically onlya few atomic layers, which supports the propagation ofthird sound [10]. When the film is driven by a sufficientlylarge amplitude excitation, the response of the systembecomes non-linear and typically follows the Korteweg-de Vries (KdV) equation [4, 7]. The KdV equation isknown to support solitonic solutions, which has been con-firmed experimentally for helium films in the previouslymentioned references. Solitons have also been observedexperimentally in related systems such as Bose-Einsteincondensates (BEC) and He (magnetic solitons) [11–20].In the former case, experimental observations have beensuccessfully modeled by the Gross-Pitaevskii (GP) equa-tion [21–23]. However, bright solitons have not been ob-served in bulk superfluid He up to date. Such obser-vation would not only provide important details of theunderlying non-linear response of this quantum liquid,but it would also allow for the study of soliton dynam-ics (including dissipation) over much longer propagation distances and times than currently possible in BECs.Studies of non-linear excitations in bulk superfluid he-lium are scarce. Most experiments have concentratedon the propagation of second sound shock waves [24–26] whereas non-linear first sound has received very lit-tle attention. In the latter case, the efforts have mainlyconcentrated on the construction of cryogenic compres-sion shock tubes [27–29], which can be used to generateshock waves in the liquid and study their properties (e.g.,velocity, amplitude). Shock waves, unlike solitons, areknown to exhibit strong dissipation and dispersion [30].Semi-empirical analysis of shock waves can be carried outby the Rankine-Hugoniot theory or its extension that isapplicable in the superfluid phase [31]. As shown in arecent study, shock waves in superfluid helium evolve ona nanosecond time scale and hence time-resolved experi-ments are required for their characterization [32].Due to the lack of sufficiently accurate theoretical mod-els for bulk superfluid helium, the possible existence ofsolitons and their properties in this medium have notbeen studied previously. Note that neither GP or KdVequations are applicable for superfluid helium. Onlynon-local phenomenological models, such as density func-tional theory (DFT) [33, 34], can describe the atomic-scale static and dynamic response of superfluid helium ac-curately. While previous time-dependent DFT (TDDFT)calculations have noted the existence of supersonic non-linear waves [35, 36], their nature and properties werenot studied further. Consequently, no experimental ef-forts have been put forward to prove (or disprove) theexistence of solitons in this medium. The obvious differ-ences between helium films and the bulk liquid are thedimensionality (2-D vs. 3-D) and the presence of the filmsupporting substrate. The latter influences the sound ve-locity as a function of depth, which makes the applicationof KdV-type equation attractive [37]. a r X i v : . [ c ond - m a t . o t h e r] J a n
355 nm
Target
PrimarySecondary TargetWindows mm FIG. 1: Metal target geometry (scale accuracy ± µ m). Theprimary waves (red) originate from the expanding plasma cre-ated by laser ablation (red circle) whereas the planar waves(black) are created by rapid boiling on the target surface. Theoptical axis for imaging is perpendicular to the plane shown.A photograph of the target is shown on the right. In this work, we report on the first experimental ob-servation of bright solitons in bulk superfluid He, whichare created by rapidly expanding plasma and boiling ona metal target surface. In addition to the experimentalevidence, their existence and dynamic properties are alsostudied by TDDFT.The experiments employed a focused (spot diam. 50 µ m) laser pulse (3rd harmonic 355 nm; 9 ns pulse length;0.5 GW/cm ; Continuum Minilite-II Nd-YAG laser) togenerate plasma on the surface of a solid copper targetimmersed in bulk superfluid helium between 1.7 and 2.1K at saturated vapor pressure (Oxford Variox or Janis8DT cryostat) [32]. A schematic target configuration isdepicted in Fig. 1. The initial radial plasma expansion[38–40] leads to non-linear excitation of the surround-ing liquid, which was visualized by time-resolved shadow-graph photography using a monochrome charge-coupleddevice (CCD; Imaging Source DMK23U445) equippedwith 180X zoom lens (working dist. 95 mm; max. resol.1.7 µ m/pixel and focal depth ± µ m) and a delayedlaser pulse (2nd harmonic 532 nm; 9 ns pulse length;Continuum Surelite-II Nd-YAG laser) as the backgroundlight. The contrast in the images is given by the Lapla-cian of the liquid density, which identifies the propagat-ing wave edges. Due to scattering of the backlight, theimages also show some contrast inside the wave.A closeup of the system at early times is shown in Fig.2. The primary wave emission is produced directly by theexpanding plasma (half-spherical geometry) whereas thesecondary planar wave originates from boiling of liquidhelium on the target surface and the subsequent rapidgas expansion. The latter process is a consequence of thefast heat transfer on the metal surface (propagation ve-locity up to 10 m/s) following the ablation event. In thelong-time regime, both the primary half-spherical (width ca. µ m) and secondary planar waves propagate insuperfluid helium without dispersion (rate < t = 0.2 μs t = 1.0 μs t = 1.5 μs
500 μm
FIG. 2: Snapshots of the waves emitted from the target atgiven times t ( T = 1 . A v e r a g e v e l o c it y ( m / s ) Time ( µ s) I n t e n s it y d i ff e r e n ce Spherical solitonPlanar soliton
10 15 20 25 30 35 40220240260280300
Sound 9.1 mm
FIG. 3: The top panel shows the time evolution of the averagesoliton velocity (accuracy better than ± The time evolution of normalized shadowgraph inten-sity difference in front of the soliton vs. immediately be-hind is shown in the bottom panel of Fig. 3. Assumingthat the nature of the left-over liquid excitations (i.e.,spatial variations in liquid density) does not evolve intime, this difference reflects the wave dissipation rate [41].During the first ca. µ s, rapid dissipation of both spher-ical and planar solitons takes place along with the associ-ated decrease in their propagation velocity (top panel ofFig. 3). The second regime ( t > µ s) exhibits lower wavedissipation rate and the decrease in wave velocity beginsto level off. As shown in the inset, the long-time ( t > µ s) soliton propagation appears nearly dissipationless asthe velocity remains constant at slightly above the speed t = 3.8 m s(before collision) t = 4.4 m s(collision) t = 5.5 m s(after collision) FIG. 4: Time evolution of the primary (spherical) and sec-ondary (planar from top) solitons before, during, and aftercollision ( T = 1 . t . of first sound (instantaneous velocity ca.
250 m/s; seeRef. [32]). This limiting velocity follows the same tem-perature dependence as the first sound. We attribute thefast initial dissipation and reduction in the propagationvelocity to wave crest breaking process where the highdensity liquid is left behind as shocks. Note that a smalldecay in the velocity is also expected due to the finiteviscosity present in the experiments [6] and the changein volume of the spherical soliton with increasing radius.Another inherent property of solitons is that theyemerge from a two-wave collision without any apparentchange to their shape (apart from a possible change inphase). Collision between two solitons is shown in Fig. 4,where the primary half-spherical wave collides with theplanar wave originated from the top section of the target.Due to the geometry of the emitted waves (half sphere vs.plane), the two waves must intersect at the focal plane ofthe imagining system. The shadowgraph images clearlyshow that the solitons do not interact and continue topropagate unchanged after the collision. Furthermore,collision of the solitons with a metal surface (not shown)leads to effective reflection, but this is accompanied byenergy loss as evidenced by an audible mechanical shockemitted into the metal. In the long-time regime, the re-flected solitons from the cryostat walls can be observedto reach the target region again (total travel dist. 10 cm).In addition to the experimental observations discussedabove, we have also carried out TDDFT calculations in3-D [42] to identify solitonic solutions in bulk superfluid He and study their dynamic properties. Within thismodel, helium is described by a complex valued orderparameter Ψ( r , t ), which is related to the atomic densityas ρ ( r , t ) = | Ψ( r , t ) | . The TDDFT equation is ı (cid:126) ∂∂t Ψ( r , t ) = (cid:26) − (cid:126) m ∇ + δ E c δρ (cid:27) Ψ( r , t ) (1)where m is the mass of He and the functional E c [ ρ ] was FIG. 5: Snapshots of superfluid He density along the direc-tion of soliton propagation ( x -axis) from TDDFT. The initialcompression was created on the simulation box boundary with n = 30 (see text). Due to periodic boundary condition, thesolitons propagate as shown in (a) - (d), collide in (e), andcontinue propagating almost unchanged in (f) - (h). Whencompared with experiments, the x -axis is oriented perpendic-ular to the ablation target. taken from Ref. 34. This functional includes both finite-range and non-local corrections that are required to de-scribe the T = 0 response of liquid He accurately on the˚Angstr¨om-scale. Note that this model does not includeviscous dissipation and cannot be propagated over longtimes (microseconds) due to limitations in current com-putational resources. For this reason, TDDFT cannot beused to study the related dissipative effects observed inthe experiments. Furthermore, the accessible length scaleis also very different from the experiments (i.e., nm vs. µ m). However, as discussed below, the TDDFT resultscan be scaled up to match the experiments.To mimic the initial condition in the experiments (i.e.,sudden compression by expanding plasma), the initial or-der parameter, Ψ( r ,
0) = (cid:112) ρ ( r , ρ ( r ,
0) = ρ [1 + (∆ ρ/ρ ) sin ( πx/λ c )Θ w ( x )] (2)where Θ w represents a “box function” centered at x withwidth w = nλ c (with n integer), i.e., Θ w ( x ) = 1 when x − w/ < x < x + w/ w = 0 otherwise. Eq. (2)represents a square profile with average value ∆ ρ/ ρ (0 . − at T = 0) and modulated along the x -axis with wave-length λ c = 3 .
58 ˚A. This ansatz is based on the followingassumptions: (i) when the liquid is rapidly compressed,the local density is increased with respect to the bulk and(ii) liquid He flowing at a velocity greater than the Lan-dau critical velocity ( v L ) undergoes a transition from aspatially homogeneous liquid to a layered state character-ized by a periodic density modulation along the directionof propagation (wavelength λ c and amplitude determinedby v − v L ) [43, 44]. Such layered structures with densi-ties higher than the bulk have also been observed in DFTsimulations of fast moving particles in liquid He [45].During the early stages of the time evolution of ρ ( r , t ),dispersive low-amplitude supersonic waves with wave-length ∼ λ c were produced (not shown). For the sakeof clarity, we show smoothed density profiles after tak-ing a local average of the density within a space windowof ± λ c . We wish to stress that this procedure is justpost-processing and therefore it does not affect the timeevolution itself. Note that the applied theoretical modelmust be able to describe the underlying atomic scale in-ternal structure of the soliton.When the initial state given by Eq. (2) is propagatedin time using Eq. (1), it splits rapidly into two counterpropagating bright solitons as shown in Fig. 5. In con-trast, due to the presence of the expanding plasma andthe target in the experiments, only one soliton may formfollowing the initial compression. The initial position x of the square profile was placed at the simulation boxboundary and, due to the periodic boundary condition,the two solitons resulting from the initial splitting moveaway from the boundaries towards the center. Note thatthe soliton width ( ∼
20 nm for the case shown) andheight are well preserved during the time evolution. Incomparison, a gaussian wave packet with the same widthand amplitude would disperse rapidly at 90 m/s. Thesolitons were also found to be stable with respect to ran-dom distortions introduced into the order parameter.The relationship between the average soliton height ρ s ,which is controlled by the value of ∆ ρ/ρ in Eq. (2), andits propagation velocity exhibits nearly linear behavior atlow amplitudes as shown in Fig. 6. In the limit of verysmall amplitudes, the velocity approaches the speed ofsound. When the amplitude ρ s /ρ is increased above1.3, the system becomes unstable and evolves rapidlyinto a series of shock waves. This instability may berelated to the previously mentioned wave crest breakingphenomenon, which was observed before 3 µ s in Fig. 3.The calculated maximum stable soliton velocity ( ca. µ s inFig. 3). The long-time wave propagation velocity in theexperiments remained slightly above the speed of sound,which corresponds to ca.
3% density increase at the soli-ton with respect to the bulk liquid (cf. Figs. 3 and 6).Both the lack of dispersion and the distinct amplitude-velocity dependence are characteristic to solitons.When the initial width of the compression in Eq. (2) isincreased, the width of the emitted solitons increases ac-
FIG. 6: Soliton propagation velocity ( v s ) vs. amplitude( ρ s /ρ ) from TDDFT. The horizontal dotted line indicatesthe velocity of first sound in superfluid He at T = 0. cordingly. Therefore, despite of the obvious difference inthe length scale between TDDFT and the experiments,this suggests that the presented nanometer-scale mecha-nism can scale up to micrometers. We also note that thepresented solitonic waves from Eq. (2) can only be ob-served using a finite-range non-local energy density func-tional whereas local models, such as GP fitted to repro-duce the speed of sound, do not support such solutions.A collision between two solitons from TDDFT is shownin Fig. 5. Based on the simulations, the amplitude,shape, and velocity of the solitons are well preserved af-ter the collision. This observation is in agreement withthe experimental images shown in Fig. 4. At the pointof collision shown in panel (e), the solitons interfere con-structively as they both have a common phase factor (i.e.,identical origin). If a soliton is made to collide with anexponentially repulsive wall (not shown), TDDFT calcu-lations show that it loses its shape partially and dissipatessome of the energy as shock waves. This behavior is alsoconsistent with the experimental observations.In summary, we have shown for the first time that bulksuperfluid He can support bright solitonic waves. Thisis evidenced by both direct experimental observations aswell as theoretical modeling based on TDDFT. The liq-uid compression created by the expanding plasma is suf-ficiently high such that the resulting non-linear responsecan counteract the dispersive effects. This is in contrastto previously studied thin liquid helium films where thepresence of the supporting substrate played a major rolein producing the necessary non-linear response. In bulksuperfluid helium, solitons become unstable when theiramplitude exceeds a critical threshold, which correspondsto a velocity slightly above 400 m/s.This work was supported by National Science Founda-tion grant DMR-1205734. The authors thank M. Bar-ranco, M. Pi, and L. Salasnich for helpful discussions. [1] P.G. Drazin and R.S. Johnson,
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