Gravity waves on the surface of topological superfluid 3He-B
aa r X i v : . [ c ond - m a t . o t h e r] F e b Gravity waves on the surface of topological superfluid He-B
V.B. Eltsov, P.J. Heikkinen, and V.V. Zavjalov
O.V. Lounasmaa Laboratory, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland (Dated: July 17, 2018)We have observed waves on the free surface of He-B sample at temperatures below 0.2 mK. Thewaves are excited by vibrations of the cryostat and detected by coupling the surface to the Bose-Einstein condensate of magnon quasiparticles in the superfluid. The two lowest gravity-wave modesin our cylindrical container are identified. Damping of the waves increases with temperature linearlywith the density of thermal quasiparticles, as expected. Additionally finite damping of the wavesin the zero-temperature limit and enhancement of magnetic relaxation of magnon condensates bythe surface waves are observed. We discuss whether the latter effects may be related to Majoranafermions bound to the surface of the topological superfluid.
PACS numbers: 67.30.H-, 47.35.Bb, 03.75.Kk
Waves on the surface of a fluid in a gravitational field[1] present a universal phenomenon in a wide range ofsystems, from a glass of drink to hot astrophysical ob-jects [2] and cold superfluids [3]. Properties of the wavesprovide an important information about the fluid itselfwhich in turn can result in useful practical applications.For example using the well-known phenomenon that oilfilm stills the water waves [4] one can find oil pollutions inthe ocean by observing its calm regions from a satellite.Recently the surface properties of the fermionic Hein its superfluid B phase have attracted a lot of atten-tion owing to non-trivial topology of this superfluid: Itis expected that fermionic bound states with Majoranacharacter emerge at the surface [5–8]. While in solid-state systems complex engineering efforts are required toobtain Majorana fermions [9], the free surface of He-Bshould be naturally covered by a thin layer of such states.Could this ’film’ in a sense ’still’ the surface waves of He-B and is it possible to observe this damping in the exper-iment? While this question awaits a proper theoreticalconsideration we report here the first, to our knowledge,observation of gravity waves on the surface of He-B.For such observation it is not enough to cool He be-low its critical temperature T c ≈ − K. At tempera-tures close to T c viscosity of the normal component of He is high, oil-like, and the surface waves are over-damped [1]. Only the third sound waves in a thin film,where the normal component is clamped, have been pre-viously observed in He-B at (0 . ÷ . T c [10]. We haveperformed measurements at temperatures below 0 . T c where the normal component becomes a rarefied gas ofballistic quasiparticles and its contribution to the damp-ing of the surface waves rapidly decreases. Experiment.
The He-B sample is contained in a verti-cal quartz cylinder with internal diameter of 2 R = 6 mmand length of 150 mm, Fig. 1. The free surface of thesuperfluid is placed about 8 mm below the upper wall ofthe cylinder within the pick-up coil of the nuclear mag-netic resonance (NMR) spectrometer. To detect surfaceoscillations we use their influence on the frequency of the NMR pick-upHe-Bcoil free surfacecoil forstatic fieldminimum quartz cellquartztuning forks zH H M ββ l M FIG. 1: Experimental setup and principle of measurements. (Left)
Sample tube with the NMR pick-up coil around thefree surface and tuning-fork thermometers at the bottom,where the cylinder opens to the heat-exchanger volume ofthe nuclear cooling stage. (Right)
Bose-Einstein condensatesof magnon quasiparticles can be created in a trap, formed inthe radial direction by texture of the the orbital anisotropyaxis (sketched with the short segmented lines) and in the axialdirection by the applied minimum of the static magnetic field H . In the condensate magnetization M precesses around themagnetic field with coherent phase and at the common fre-quency, which is measured in the experiment. The frequencyis determined by the trapping potential and is modulated bythe oscillations of the surface. coherently-precessing NMR mode known as the trappedBose-Einstein condensate of magnon quasiparticles [11].The order parameter of He-B in the magnetic fieldis anisotropic. The orientation of the orbital anisotropyaxis ˆ l slowly varies in the sample (forms a texture) whichcreates a trapping potential for magnon quasiparticles inthe radial direction owing to the spin-orbit interactionenergy: F so = 45 ~ Ω B ω L sin β l | Ψ | . (1)Here ω L = γH is the Larmor frequency, Ω B is the Leggettfrequency in the B phase, β l is the deflection angle ofˆ l from the vertical direction (growing from β l = 0 atthe cylinder axis to β l = π/ | Ψ | is related to the tipping angleof magnetization β M as | Ψ | = χH (1 − cos β M ) /γ ~ .Trapping in the axial direction is provided by an addi-tional pinch coil which creates a minimum in the staticNMR field H and thus minimum in the Zeeman energy F Z = ~ ω L | Ψ | . The lowest magnon levels in this trappingpotential typically closely follow harmonic-trap relation[12] and can be enumerated by the radial and axial quan-tum numbers m and n , respectively: f mn = f L + ν r ( m + 1) + ν z ( n + 1 / . (2)Here f L = ω L / π ≈ .
826 MHz and ν r ≈
220 Hz and ν z ≈
40 Hz are the radial and axial trapping frequencies,respectively. When magnons are pumped to the trapusing NMR they relax in sub-second time to the groundlevel [12], where spontaneous coherence appears and aBose-Einstein condensate is formed. The magnetizationof the condensate precesses around the magnetic field at f = f L + ν r + ν z / ν z by limiting the trap in the axial direction andalso modifies ν r owing to orientation of ˆ l perpendicularto the surface. When the waves modify the geometry ofthe surface, the frequency of the precession f changesas a result.In the measurements we keep a small cw pumping usu-ally at m = 2 level to compensate for the loss of themagnons from the ground level. Such pumping is ap-plied at the frequency f > f and thus it does notinterfere with the measurements of the precession of theground-level condensate. To the signal recorded from theNMR pick-up coil we apply the band-pass filter to keeponly the contribution from the ground-state condensateincluding all the side bands, resulting from the frequencymodulation. The frequency f of the precession of thecondensate is found from the time intervals between zerocrossings in the filtered signal. Surface resonances.
An example of the measured f ( t ) record and its Fourier transform are shown inFig. 2. Peaks in the frequency modulation spectrum canbe attributed to the lowest-frequency gravity-wave modesin a vertical cylinder. The height profile h ( r, θ ) of suchsurface oscillations is h ( r, θ ) = J i ( k ij r ) e iθ , i = 0 , , . . . , j = 1 , , . . . (3)Here ( r, θ ) are the polar coordinates of the point on thesurface, J i are the Bessel functions, and wave numbers k ij satisfy the equation J ′ i ( k ij R ) = 0. The spectrum ofthese modes follows simple relation for the gravity waveson deep water ω ij = gk ij , where g is the free-fall accelera-tion. This applies since the length of the sample cylindersignificantly exceeds its diameter and the surface tensionof He is small and can be neglected here [13].
FIG. 2: Frequency modulation of the NMR precession owingto the surface waves. (Top)
Frequency of the precession ofthe magnon BEC in a trap bordering the free surface of the He-B sample as a function of time. The signal is recordedat T = 0 . T c with only residual vibrations of the cryostat.Modulation of the precession frequency is caused by peri-odic distortion of the trapping potential by the surface waves. (Bottom) Spectrum of the frequency modulation of the NMRprecession. The plot shows Fourier transform of the signallike in the upper panel, but measured for 20 s. Peaks cor-responding to the two surface wave modes, cartooned in theinserts, can be clearly identified and are marked with arrows.Harmonics are shown by the × n signs. The primary mode is the non-axisymmetric (1,1) modewith k = 1 . /R and its frequency in our cylinderis ω / π = 12 . k = 3 . /R and ω / π = 17 . h can beconnected to the change of the frequency of themagnon precession ∆ f with Eq. (1) as ~ · π ∆ f ∼ ~ (Ω B /ω L )( h /R ) . With ∆ f = 50 Hz and Ω B / π ≈ Hz we get for the amplitude of the wave h ∼ . v ∼ ( kR ) h ω/π . Surface damping.
To determine the damping of thesurface waves we have measured the resonance width∆ f surf of the forced oscillations in the primary mode,Fig. 3. Oscillations are excited by periodically tilting the W i d t h o f s u rf ace w a v e r e s on a n ce ( H z ) T / T c T = 0.158 T c f exc (Hz) A ( H z ) FIG. 3: Damping of the surface waves as a function of temper-ature. The measured width of the primary surface wave reso-nance ∆ f surf is plotted with circles against vertical axis. Bot-tom horizontal axis shows the resonance width of the quartztuning fork used as a thermometer [26]. Corresponding tem-perature values are shown on the upper axis. Solid line is alinear fit to data points. (Insert) An example of a measure-ment of the surface wave resonance. Surface oscillations aremechanically excited at frequency f exc and the amplitude ofthe frequency modulation A in the precession of the magnoncondensate is measured (circles). Solid line shows a fit tothe square of the standard Lorentzian response, from which∆ f surf is determined. cryostat at a frequency f exc using the active air-springdampers on which the cryostat is suspended. With thevertical distance from the suspension point to the sam-ple surface of about 1.5 m this transforms essentially tohorizontal oscillations of the sample container. We mea-sure f ( t ) dependence like in Fig. 2(top) and extract theamplitude of the response A at 2 f exc frequency using alock-in-like detection.As seen in Fig. 3 the damping increases with increas-ing temperature as the density of the normal compo-nent grows. We are not aware of the rigorous calcula-tion of the damping applicable at these temperatures,where the normal component of He-B presents a gasof ballistic quasiparticles, which would include also theeffects of Andreev reflection from the surface [14] andfrom the flow along it. A simple model of thermaldamping of quartz tuning forks in the ballistic regime[15] predicts that the width of the resonance is propor-tional to ( d/M ) exp( − ∆ /T ), where d is the size of theobject perpendicular to the direction of the oscillations, M is the effective mass and the exponential factor re-flects temperature-dependent density of thermal quasi-particles. Using this expression we can roughly scalethe thermal effect from the thermometer fork, which has d/M ≈
270 cm/g [16], to the oscillations of the sur-face. For the primary mode of the oscillating surfacewe estimate d ∼ R and M ∼ ρ He R /k which gives d/M ∼
200 cm/g. Thus we may expect that the ther-mal contribution to the width of the resonant responsewill be of the same order for our quartz tuning forks andfor the oscillating surface. This is indeed demonstratedby the data in Fig. 3, where the slope of the fit line isclose to 1. We note, however, that this slope has notbeen exactly reproducible in the runs which differ by theazimuthal orientation of the cryostat (and thus differentdirections of the forcing with respect to the residual mis-alignment of the sample axis and the vertical direction),which is probably related to the sensitivity of the sur-face wave pattern in a vibrating cylinder to the exactconditions of the forcing [17].Another feature seen by Fig. 3 is the finite value of thesurface resonance width (0.3 Hz) when extrapolated tozero temperature. This zero-temperature damping hasbeen reproducible in all measurement runs. Among pos-sible explanations for this damping is the surface frictionat the cylindrical wall of the sample or non-linear in-teractions with other surface wave modes and possiblecreation of wave turbulence [18, 19]. An intriguing pos-sibility is contribution to the damping from the surface-bound Majorana fermions. No calculations of such con-tribution exist up to date, though, and even the physi-cal mechanism of possible damping is not entirely clear.It can be similar to the recently proposed temperature-independent but frequency-dependent dissipation mech-anism in the motion of quantized vortices originatingfrom the vortex-core-bound fermions [20]. Alternativelysurface-bound fermions can directly mediate energy andmomentum transfer to the container walls without trans-ferring them to the bulk quasiparticles first. Such con-tribution to damping should have power-law dependenceon temperature which in our experimental temperaturerange would mimic a finite zero-temperature damping.
Relaxation of the magnon BEC.
Additional informa-tion about the surface waves is provided by the measure-ments of the relaxation of the magnon condensates afterswitching the NMR pumping off. For the magnon con-densate in a time-independent trap which is completelyin bulk the relaxation is determined by the bulk thermalquasiparticles [21] and by the interaction with a NMRpick-up circuit, which will be described elsewhere. Forour condensates in a time-dependent potential modulatedby the surface waves we observe an additional relaxationchannel so that the life time of magnon condensates de-creases with increasing amplitude of the surface oscilla-tions, Fig. 4.In principle, for a quantum-mechanical system in atime-dependent potential one can expect transitions toother states. However, for our trapped condensates inthe ground state we expect the highest probability of ex-citation at modulation frequencies corresponding to tran- R e l a x a ti on ti m e ( s ) A = 11.6 Hz τ = 13.8 stime (s) M ⊥ ( a r b . un . ) FIG. 4: Relaxation of the magnon condensate in a periodi-cally modulated trap. Relaxation time is plotted as a functionof the amplitude A of the modulation of the ground level inthe trap. Circles show the modulation owing to the surfacewaves and squares modulation with the trapping magneticfield. Modulation frequency in both cases is 25 Hz. Insert presents an example of a measurement of the relaxation time τ . The amplitude of the magnon condensate signal in theNMR pick-up coil, which is proportional to the transversemagnetization M ⊥ , is shown as a function of time after switch-ing the pumping off (circles). The line is an exponential fit M ⊥ = M ⊥ exp( − t/τ ) from which τ is determined. sitions between different levels in the trap. Such resonantdepletion was observed in Bose-Einstein condensates ofcold atoms [22]. In our case the modulation of the trap-ping potential is at the frequency ω /π ≈
25 Hz whilethe lowest excited level is at 2 ν z ≈
80 Hz and thus theresonant conditions are not satisfied. We have also mea-sured the relaxation rate in the case when the trappingpotential is periodically modulated with the same 25 Hzfrequency using the current in the minimum field coil. Insuch a case no enhancement of the relaxation has beenobserved (squares in Fig. 4).Thus it seems likely that the modulation of the trap-ping potential in itself can not explain the effect of thesurface waves on the magnon relaxation and one maywonder whether this effect is related to the surface-boundMajorana fermions. It has been suggested [7] that Ma-jorana fermions can be probed with the relaxation of thespin of an electron, which is localized in a bubble closeto the surface of He-B. Magnon condensates have sig-nificantly larger coherently precessing spin than a singleelectron and so should be more sensitive to the surfacerelaxation effects. Oscillating surface might enhance re-laxation processes in the bound-states subsystem whichwould allow for a faster energy transfer from the conden-sate to the bound states.
To conclude, we have observed gravity waves on a freesurface of the ultra-cold superfluid He-B. In the tem-perature range below 0 . T c the waves are only weaklydamped and are easily excited by minute vibrations of the sample container. With decreasing temperature thedamping decreases linearly with the density of bulk ther-mal quasiparticles, but extrapolates to a finite value inthe T → He-B a theoretical model ofthe relevant phenomena should be built. An interest-ing development will be to study the shallow-water case,where surface waves with relativistic spectrum can beused to construct analogues of black holes [23] includingHawking radiation [24]. Ultra-cold He-B has no viscos-ity and provides coupling of the waves to the fermionicquantum vacuum of the superfluid, which should makesuch analogue systems much richer compared to waterwaves. Another possibility is to use similarities betweenthe magnon BEC confined with the ˆ l texture field andthe confined quarks in the MIT bag model [25] to study’high-energy physics’ on a moving brane, role of which isplayed by the surface of He.We thank G.E. Volovik, M. Krusius, M.A. Silaev andI.A. Todoshchenko for stimulating discussions. The workis supported by the Academy of Finland (Center of Excel-lence 2012-2017) and the EU 7th Framework Programme(grant 228464 Microkelvin). P.J.H. acknowledges finan-cial support from the V¨ais¨al¨a Foundation. [1] L. D. Landau and E. M. Lifshitz,
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