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arXiv:1411.3983v4 [cond-mat.other] 17 Nov 2015 hr stegpi h emoi pcrm sthe is Φ spectrum, fermionic the in gap and the is ∆ where etdi em ftertto axis rotation the of terms in sented oeseicwv velocity mode-specific a n ria ere ffedm h matrix The freedom. of degrees orbital and aee ntezr antcfil is field magnetic zero the in rameter ig oe hc a xeietlyobserved. amplitude experimentally one was and bro- which mode is phase mode, NG phase Higgs one wave-function to leads the This of with ken. change symmetry the the only to number, respect complex single a of eter state. of corre- normal symmetry continuous mode the broken NG spontaneously Each to sponds modes. 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Physics, Theoretical for Institute Landau and nsprud h re aaee,wihdsrbsspontan describes which parameter, order the superfluids In sartto arx hc oncsspin connects which matrix, rotation a is .V Zavjalov, V. V. A αi θ L eemn -iesoa subspace 4-dimensional a determine ∆ = Introduction ω n spin and 1 = 0 R r ig oe n tesare others and modes Higgs are e αi i Φ 1 (or at nvriy OBx110 I006ALO Finland AALTO, FI-00076 15100, Box PO University, Aalto o eprtr aoaoy eateto ple Physics Applied of Department Laboratory, Temperature Low R c 1, αi 3 n a o mass) (or gap a and ω e emoi isotope fermionic a He, ˆn ∗ × ( 2 ˆn and ( .Autti, S. k arxo complex of matrix 3 θ , 5 = ) ˆn S ) n h re pa- order the and θ n angle and , .Tecorre- The 1. = r pnwaves. spin are ) ω 0 2 3 +( R etesym- the He 1 αi k c .B Eltsov, B. V. 3 eBoesapsiiiyta uhseai a eraie i realized be can scenario such that possibility a opens He-B Dtd pi ,2021) 3, April (Dated: srepre- is ) 2–4 2 θ with , Pa- . ω (1) 0 . 1 enivsiae o ogtm ohtheoretically have both modes time Higgs long heavy experimentally. a These and for ∆. investigated of order been the of gaps Gbsn eoemsiedet h Anderson-Higgs the to due massive mechanism. become bosons NG ilto ftesmer fteBpaelast ap- to leads phase the B for the gap the of of symmetry pearance the of violation stb h rtcltemperature critical the by (set nadtoa,lgt ig oo.TegpvleΩ value gap The boson. Higgs light, additional, an r als,we iwdfo h cl f∆ of scale the from viewed when gapless, are to aldLgetangle Leggett called nrysaeof scale energy antcfil h qiiru tt orsod to corresponds state equilibrium the field magnetic neato.A o eprtrsΩ spin-orbit the temperatures of low measure At a interaction. is it frequency, Leggett called w ekeet eoesgicn:si-ri interac- field spin-orbit magnetic applied significant: and become experiments, tion (NMR) effects resonance weak two magnetic nuclear our of curstegpeult h amrfrequency Larmor the to magnon, as equal optical way the gap same modes, the the the in acquires of modes One two ferromagnets. these in splits field the and omldfiiino uhfil in field. field Higgs Little such the of called definition is Formal Higgs, NG light includes a which and modes, modes low-energy of set such physics ytecoe ytmo egt equations. Leggett of system closed the by 18). modes Ref. NG in the see of spectrum by discussion quadratic symmetry (general with reversal field time spectrum the magnetic the of the violation of a form spectrum from unusual comes its Such but quadratic. gapless, becomes remains magnon, acoustic the γH .J Heikkinen, J. P. nsprodcosadi h tnadMdl the Model, Standard the in and superconductors In w te pnwv oe r siltosof oscillations are modes spin-wave other Two pnobtitrcinlfstedgnrc ihrespect with degeneracy the lifts interaction Spin-orbit l he o rqec pnwv oe r described are modes spin-wave frequency low three All 3 θ eB hr n ftreNambu-Goldstone three of one where He-B, inaemsls ab-odtn bosons. Nambu-Goldstone massless are tion n h iiu nrycrepnst h so- the to corresponds energy minimum the and , (where eato.Ohrmdsbcm optical become modes Other teraction. . a 442 hroook,Russia Chernogolovka, 142432, 1a, v., c.Oclain ftefil amplitude field the of Oscillations ics. nti odnaeo pia magnons optical of condensate instein sbe nldn h tnadModel. Standard the including ssible, xeso fteSadr oe,in Model, Standard the of extension ossmer raig sa ana- an is breaking, symmetry eous omto falgtHgsfo a from Higgs light a of Formation . 15–17 γ slgtcmae oelectroweak to compared light is , ilto fahde symmetry. hidden a of violation o stegrmgei ai) nte one, Another ratio). gyromagnetic the is ∼ neetial neutral electrically In 1 H,crepnigt h frequency the to corresponding MHz, 1 n .E Volovik E. G. and 10–14 θ L arccos( = ei superfluid in te , θ oe hsmd becomes mode This mode. H T 3 i.1b. Fig. , eBi ie nSup- in given is He-B − c B 1 ∼ ∼ / ,2 1, ) hsexplicit This 4). 3 10 0 kHz 100 eteemodes these He − n 3 1 ) tlow At K). ∼ nparticle In 3 ˆn 0 MHz 100 ≪ He-B nthe In . ˆn ω ∆. L B k 6–9 H 19 = is 2

ω(k) ω(k) 4 pair breaking modes optical 5 imaginary squashing modes magnon 5 real squashing modes direct ωL Higgs modes 2∆ parametric

p12/5 ∆ ωL/2 Nambu-Goldstone modes p8/5 ∆ ΩB light Higgs mode (θ) acoustic a b magnon sound mode (Φ) 2 spin wave modes (ˆn)

0 k 0 k

3 FIG. 1: Illustration of collective-mode spectra in He-B. (a) Modes at high energy scale (ω ∆ 100 MHz). From top to bottom there are six separate branches. Three red lines show heavy Higgs modes: four degenerate∼ pair∼ breaking modes with the gap 2 ∆, five imaginary squashing modes with the gap 12/5 ∆, five real squashing modes with the gap 8/5∆. Nambu- Goldstone modes include a sound mode (propagating oscillations of Φ), a spin-wave mode which corresponds to propagating oscillations of θ, two spin-wave modes which corresponds top propagating oscillations of ˆn. These modes are gaplessp at the high 3 energy scale and have different propagation velocities c. (b) Spin- modes at low energy scale (ω 10− ∆ 100 kHz). ∼ ∼ Due to spin-orbit interaction the θ mode acquires a small gap ΩB ∆ and becomes a light . Two nˆ modes are split by the magnetic field into optical and acoustic magnons. Arrows≪ indicate decay channels observed in our experiments. plementary Note 1. Results

In an NMR experiment one follows the dynamics of Suhl instability. As a tool to study dynamics of the 3 magnetization M, or of the spin S of the sample. The mo- little Higgs field in superfluid He-B we use trapped Bose- tion of θ, corresponds to longitudinal spin waves, δS k H, Einstein condensates of optical magnons, Fig. 2. The while oscillations of ˆn correspond to transverse spin mo- tion, δS ⊥ H. Optical magnons can be directly cre- ated with traditional transverse NMR. With a suitable z H coil system one can also directly excite longitudinal spin container oscillations, or light Higgs mode.20 Coupling to short- NMR coils , arb.un. wavelength acoustic magnons is hard to achieve in a tra- x ditional NMR experiment with large excitation coils. a M x b In this work we use a technique, based on Bose- 21 Einstein condensate (BEC) of optical magnons, to coil for y 0 10 20 30 40 50 60 probe interaction and conversion between all components time, s field minimum magnon condensate of the little Higgs field in 3He-B. As a result, we ob- 6 mm ˆn serve parametric decay of optical magnons to light Higgs distribution of bosons, and both parametric and direct conversion be- 3 tween optical and acoustic magnons. The measured FIG. 2: Experimental setup. (a) Superfluid He-B is con- mass of light Higgs and propagation velocity of acoustic fined in a cylindrical quartz container, and a constant mag- netic field H is applied along the container axis. A special magnons are close to the expected values. Thus we exper- coil creates a minimum of the field magnitude H in the axial imentally confirm the little Higgs scenario in 3He-B. The 22 direction, while transverse NMR coils are used to pump opti- little Higgs field appears in quantum chromodynamics, cal magnons and to detect magnetization precession. Green where NG modes () acquire light mass due to the ex- arrows show equilibrium distribution of the ˆn vector, which plicit violation of the chiral symmetry, which is negligible together with the H profile creates a trap for optical magnons at high energy, but becomes significant at low energy.23 near the axis of the sample. In this trap magnon BEC is The relatively small mass of the 125 GeV Higgs boson formed. (b) An example of the signal from the NMR coil observed at the Large Hadron Collider suggests that it during the condensate decay measured at ω/2π = 833 kHz might be also the pseudo-Goldstone (light Higgs) boson and P = 0 bar. Its amplitude is proportional to the coher- (see e.g. Ref. 24 and references therein). ently precessing transverse magnetization of the condensate. 3

4 a 1.2 c 300 d 1.0 2 Ath , arb.un. 200

th 0.8 A 0 , kHz 0 20 40 60 570 575 580 585 590 595 B Ω 100 amplitude, arb.un. time, s frequency, kHz b 0 1 29.0 bar 0 10 20 30 P , bar 0 2.5 1 e 26.4 bar , arb.un.

th 2.0 0 A 1 23.1 bar , kHz

δf 1.5 1 δf /2 0 2Ω B 1.0 0 10 20 30 550 600 650 700 750 800 850 P , bar frequency, kHz

FIG. 3: Measurements of the Suhl instability. (a) An example measurement of the NMR signal amplitude during the magnon condensate decay obtained at ωL/2π = 741 kHz and P = 26.4 bar. A Suhl instability threshold at the amplitude Ath is clearly seen. (b) Dependence of the threshold amplitude Ath on the frequency of optical magnons ωopt ωL (colored symbols) ≈ for three pressures P . Sharp decrease is seen at ωopt = 2ΩB(P ), where the decay of an optical magnon to two light Higgs bosons becomes possible. Black lines are smoothed data using running average. Curves for different pressures are shifted in the vertical direction and respective zero levels are marked by horizontal lines. (c) Zoom to one measurement in the panel b shows periodic modulation, which corresponds to resonances of acoustic magnons in the experimental container. (d) Pressure dependence of the light Higgs boson mass ΩB (symbols) from measurements in the panel b. (e) Frequency separation of acoustic magnon resonances (symbols) from measurements like in the panel c. lines in the panels d and e are theoretical values based on known 3He parameters1 without fitting. condensate is well separated from the container walls, lows us to identify the decay channels. It is clear from where the strongest magnetic relaxation in 3He usually Fig. 1b that the decay of the optical magnon to a pair 25 occurs. Thus tiny relaxation effects, connected to cou- of light Higgs bosons with the frequency ωHiggs, ωopt = pling of optical magnons to other components of the little ωHiggs(k)+ ωHiggs(−k), is possible only when the preces- Higgs field, can be observed. sion frequency is larger then 2ΩB. We see a pronounced drop of the threshold amplitude at this frequency: The When number of pumped magnons is low, slow expo- threshold decreases by about an order of magnitude. nential relaxation of the precession signal is determined by spin diffusion and energy losses in the NMR pick- In Fig. 3d the measured mass of light Higgs ΩB is up circuit,26 Fig. 2b. We have found that above some plotted as a function of pressure. Measurements are in threshold amplitude the relaxation becomes much faster, a good agreement with values of the Leggett frequency Fig. 3a. The explanation is the Suhl instability,27 a from Ref. 1. well-known nonlinear effect in when a uniform Resonances of acoustic magnons. In addition to precession of magnetization (here at the optical magnon the sharp drop, connected with light Higgs mode, we frequency ω ) parametrically excites a pair of acoustic opt find periodic modulation of the threshold amplitude as a magnons with twice smaller frequency ω and opposite ac function of the frequency of the precession, Fig. 3c. These k-vectors: ω = ω (k)+ω (−k). In the case of 3He-B opt ac ac periodic peaks originate from the parametric decay of the both acoustic magnons and the light Higgs modes can be optical magnons in the BEC to acoustic magnons. The parametrically excited, Fig. 1b. The process occurs with frequency dependence is explained by quantization of conservation of energy and momentum. The threshold the magnon spectrum in the cylindrical container, which amplitude is inversely proportional to the coupling be- serves as a resonator for acoustic magnons. Consider a tween decaying and excited waves and proportional to decay of the optical magnon with frequency ω into the relaxation in the excited wave. opt acoustic magnons with frequency ωac = Nωopt, where Mass of light Higgs. The measured threshold am- for the parametric excitation N = 1/2. In the following plitude as a function of NMR frequency and pressure discussion we will use ωopt = ωL since the difference is is plotted in Fig. 3b. The frequency dependence al- negligible for trapped optical magnons. By sweeping the 4 magnetic field we can change both magnon spectrum and Discussion magnon trap and observe resonances in the cell. The sim- ple resonance condition for acoustic magnons in a cylin- To summarize, we have observed the interplay of all der with the radius R gives the distance between the three spin wave modes, which form a little Higgs field in resonances (Supplementary Note 2): superfluid 3He-B. In particular, we have found two chan- nels of parametric decay of optical magnons: to a pair 1 c of light Higgs bosons and to a pair of acoustic magnons. δfN = . (2) While the search for similar resonant production of pairs 4R pN(1 + N) of Higgs bosons reported by the ATLAS collaboration32 has not succeeded yet, our results sup- where c is the relevant spin-wave velocity. port the basic physical idea behind this effort. Another system where the light Higgs mode can be observed is the In Fig. 3e the measured acoustic magnon resonance multicomponent condensate in cold gases33, where inter- 1 period δf /2 is plotted as a function of pressure. The action between components can be set up to produce the results are in a good agreement with Eq. (2), where val- hidden symmetry. ues of the spin-wave velocity are taken from our recent We find that the low-energy physics in superfluid 3He 4 measurements. has many common features of the Higgs scenario in Stan- Effect of quantized vortices. An additional relax- dard Model: both are described by the SU(2) and U(1) ation mechanism for the magnon condensate is found symmetry groups; the acoustic and optical magnons cor- when quantized vortices are formed in the sample. In the presence of these localized topological objects the mo- a 1 mentum k of the spin-wave modes is not conserved, and − 1.4 one expects direct excitation of acoustic magnons by the 1.2 optical mode. We can rotate the sample with angular ve- 1.0 locities up to Ω = 2 radian per second to create a cluster of rectilinear quantized vortices, which cross the whole 0.8 b experimental region including the magnon BEC, Fig. 4a. 0.6 Relaxation rate, s 825 830 835 840 845 850 In this state the relaxation rate, plotted as a function of frequency, kHz the frequency in Fig. 4b, reveals several periodic sets of magnon BEC peaks. We attribute these peaks to resonances of acoustic magnons with frequencies ωL, 2ωL, etc. 2.5 c In 3He-B the rotational symmetry of a vortex is spon- taneously broken and the vortex core can be treated as 2 δf1 a of two half-quantum vortices which can 2.0 4 δf4 rotate around the vortex axis. Dynamics of the vortex is 30,31 affected by the precessing magnetization . Precession 2 δf2 of S and ˆn in the magnon BEC produces torsional oscil- , kHz 1.5 lations of the vortex core. The fact that the equilibrium δf δf1 position of ˆn deviates from the vertical direction within the magnon BEC makes this oscillations unharmonic. As 1.0 a result, acoustic magnons with frequencies NωL can be emitted. The amplitudes of the various resonances depend on a 0 5 10 15 20 25 30 distribution of vortex cores and the wave nodes of acous- P , bar tic magnons. For example, an axially symmetric dis- FIG. 4: Excitation of acoustic magnons by vortices. tribution of vortices can excite only symmetric waves, 3 which means doubling of the observed resonance pe- (a) Schematic plot of vortices in rotating He-B. Precession of magnetization of the magnon condensate cause oscillations riod. In our experiment acoustic magnons (with wave of non-axisymmetric vortex cores. The oscillations produce length 5 − 10 µm) are emitted by vortices, which are acoustic magnons and increase relaxation of the condensate. within the magnon condensate. The distance between (b) Relaxation rate of the magnon condensate as a func- vortices 0.1 − 0.2 mm is comparable with the size of the tion of frequency, measured at Ω = 1 radian per second and trapped condensate 0.2 − 0.4 mm. Thus the amplitudes P = 23.4 bar. Two sets of acoustic magnon resonances with of resonances are sensitive to details, such as order pa- periods about 1 kHz produce a clearly seen beat with a pe- rameter texture, rotation and pressure and we do not see riod of 5 kHz. (c) Measured periods of magnon condensate all the harmonics at all pressures. Nevertheless the res- relaxation peaks (symbols) as a function of pressure. Lines onance periods plotted in Fig. 4c follow the theoretical are plotted using equation (2) for acoustic magnon resonances without fitting parameters. values (2) or their multiples (denoted as 2 δf1, etc.). 5

− respond to the doublet of W+ and W gauge bosons at the bottom of the sample cylinder. The heat leak to the which spectrum also splits in magnetic field;3 the light sample was measured in earlier work to be about 12 pW.35 Higgs mode has parallel with the 125 GeV Higgs boson. The measurements are performed at pressures 0 29 bar and 3 in magnetic fields H = 17 26 mT with corresponding− NMR However, in addition, the He-B has the high-energy sec- − frequencies ωL/2π = 550 830 kHz. tor with 14 heavy Higgs modes. This suggests that in − the same manner the 125GeV Higgs boson belongs to Magnon trap. Minimum of the axial magnetic field forms the low energy sector of , and if so, one a trapping potential for optical magnon quasiparticles in the may expect the existence of the heavy Higgs bosons at axial direction. Trapping in the radial direction is provided TeV scale. by the spin-orbit interaction via the equilibrium distribution of the order parameter. In this geometry it forms the so- We have demonstrated that the short-wavelength called flare-out texture: ˆn is parallel to H on the cell axis and acoustic magnons can be emitted and detected with the tilted near walls because of boundary conditions.36 The com- BEC of optical magnons. Acoustic magnons can be bined magneto-textural trap is nearly harmonic with trapping lensed by non-uniform magnetic fields and the order- length about 0.3 mm in the radial direction and 1 mm in the parameter texture, and thus might serve in future as a axial direction (see Supplementary Note 3 for details). powerful local probe to study topological superfluidity of Measurements of magnon BEC. Owing to the geom- 3 He, including Majorana on the boundaries of etry, the coils couple only to optical magnons with k 0. the superfluid and in the cores of quantized vortices. With a short rf pulse in the NMR coils non-equilibrium opti-≈ cal magnons are created. At temperatures of our experiment equilibration within the magnon subsystem proceeds much Methods faster that the decay of magnon number, and the pumped magnons are condensed to the ground level of the trap within The sample geometry and NMR setup. Superfluid 0.1 s from the pulse. Manifestation of Bose-Einstein condensa- 3 tion is the spontaneously coherent precession of the conden- He-B is placed in a cylindrical container with the inner di- 37,38 ameter of 2R = 5.85 mm, made from fused quartz, Fig. 2. sate magnetization, which induces current in the NMR The container has closed top end and open bottom end, which coils. The amplified signal is recorded by a digital oscillo- provides the thermal contact to the nuclear demagnetization scope; an example record is in Fig. 2b. We then perform slid- ing Fourier transform of the signal with the window 0.3 1 s. refrigerator. Static magnetic field is applied parallel to the − container axis. A special coil creates a controlled minimum In the resulting sharp peak in the spectrum the frequency of the field magnitude along the axial direction. Transverse determines the BEC precession frequency ωopt, while the am- NMR coils, made from copper wire, are used to create and plitude (such as shown in Fig. 3a) is proportional to the square detect magnetization precession. Coils are part of a tuned root of the number of magnons in the trap. tank circuit with the Q value of around 130. Frequency tun- Rotation. The sample is installed in the rotating nuclear ing is provided by a switchable capacitance bank, installed at demagnetization refrigerator ROTA,39 and can be put in rota- the mixing chamber of the dilution refrigerator. To improve tion together with the cryostat and the measuring equipment. signal to noise ratio, we use a cold preamplifier, thermalized The cryostat is properly balanced and suspended on active vi- to helium bath. bration isolation, and in rotation the heat leak to the sample The measurements are performed at low temperatures remains below 20 pW.35 Vortices are created by increasing an- T < 0.2 Tc, where spin-wave velocities and the Leggett gular velocity Ω from zero to a target value at temperature frequency are temperature-independent. Typically we use around 0.7 Tc, where the mutual friction allows for fast relax- T = 130 350 µK, depending on pressure. The temperature ation of vortex configuration towards an equilibrium array.40 is measured− by a quartz tuning fork thermometer, installed Further cool-down is performed in rotation.

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13 Movshovich, R., Varoquaux, E., Kim, N. & Lee, D.M. (2014). Splitting of the squashing collective mode of superfluid 35 Hosio J.J. et al. Propagation of thermal excitations in a 3He-B by a magnetic field. Phys. Rev. Lett. 61, 1732–1735 cluster of vortices in superfluid 3He-B. Phys. Rev. B 84, (1988). 224501 (2011). 14 Collett, C. A., Pollanen, J., Li, J. I. A., Gannon, W. J. 36 Smith, H., Brinkman, W. F. & Engelsberg, S. Textures & Halperin, W. P. Zeeman splitting and nonlinear field- and NMR in superfluid 3He-B. Phys. Rev. B 15, 199–213 dependence in superfluid 3He. J. Low Temp. Phys. 171, (1977). 214–219 (2013). 37 Cousins, D.J., Fisher, S.N., Gregory, A.I., Pickett, G.R. & 15 Higgs, P.W. Broken Symmetries and the Masses of Gauge Shaw, N.S. Persistent coherent spin precession in superfluid Bosons. Phys. Rev. Lett. 13, 508–509 (1964). 3He-B driven by off-resonant excitation. Phys. Rev. Lett. 16 Anderson, P. W. , gauge invariance, and mass. 82, 4484–4487 (1999). Phys. Rev. 130, 439–442 (1963). 38 Autti, S. et al. Self-trapping of magnon Bose-Einstein con- 17 Pekker D. & Varma, C.M. Amplitude/Higgs modes in con- densates in the and on excited levels: From densed physics. Annu. Rev. Condens. Matter Phys. harmonic to box confinement. Phys. Rev. Lett. 108, 145303 6 269–297 (2015). (2012). 18 Nitta, M. & Takahashi, D.A. Quasi-Nambu-Goldstone 39 Hakonen, P.J. et al., Rotating nuclear demagnetization re- modes in nonrelativistic systems. Phys. Rev. D 91, 025018 frigerator for experiments on superfluid He3. Cryogenics 23, (2015). 243–250 (1983). 19 Schmaltz, M. & Tucker-Smith, D. Little Higgs Review. 40 Eltsov, V.B., H¨anninen, R. & Krusius, M. Quantum turbu- Ann. Rev. Nucl. Part. Sci. 55, 229–270 (2005). lence in superfluids with wall-clamped normal component. 20 Osheroff, D. D. Longitudinal and Transverse Resonance in Proc. Natl. Acad. Sci. USA 111, 4711–4718 (2014). the B Phase of Superfluid 3He. Phys. Rev. Lett. 33, 1009– 1012 (1974). 21 Bunkov, Yu. M. & Volovik, G. E. Spin superfluidity Acknowledgements and magnon BEC. International Series of Monographs on Physics 156, Volume 1, 253–311 (2013). 22 Dobrescu, B.A. & Frugiuele, C. Hidden GeV-scale interac- We thank M. Krusius and V.S L’vov for useful discussions. tions of . Phys. Rev. Lett. 113, 061801 (2014). This work has been supported in part by the EU 7th Frame- 23 Weinberg, S. Approximate symmetries and pseudo- work Programme (FP7/2007-2013, Grant No. 228464 Mi- Goldstone bosons. Phys. Rev. Lett. 29, 1698–1701 (1972). crokelvin), by the Academy of Finland (project no. 284594), 24 Volovik, G.E. & Zubkov, M.A. Scalar excitation with and by the facilities of the Cryohall infrastructure of Aalto Leggett frequency in 3He-B and the 125 GeV Higgs parti- University. P.J.H. acknowledges financial support from the cle in top condensation models as pseudo-Goldstone V¨ais¨al¨aFoundation of the Finnish Academy of Science and bosons. Phys. Rev. D 92, 055004 (2015). Letters, and S.A. that from the Finnish Cultural Foundation. 25 Fisher, S. N., Pickett, G. R., Skyba, P. & Suramlishvili, N. Decay of persistent precessing domains in 3He-B at very low temperatures. Phys. Rev. B 86, 024506 (2012). Author contributions 26 Heikkinen, P.J., Autti, S., Eltsov, V.B., Haley, R.P. & Za- vjalov, V.V. Microkelvin thermometry with Bose-Einstein The experiments were conducted by S.A., P.J.H., V.V.Z., condensates of magnons and applications to studies of the and V.B.E., the theoretical analysis was carried out by G.E.V 3 AB interface in superfluid He. J. Low Temp. Phys. 175, and V.V.Z., the paper was written by V.V.Z., G.E.V. and 681–705 (2014). V.B.E., with contribution from all the authors. 27 Suhl, H. The theory of ferromagnetic resonance at high signal powers. J. Phys. Chem. 1, 209–227 (1957). 28 3 Thuneberg, E.V. Hydrostatic theory of superfluid He-B. Conflicting financial interests J. Low Temp. Phys. 122, 657–682 (2001). 29 Zavjalov, V.V., Autti, S., Eltsov, V.B. & Heikkinen, P.J. Measurements of the anisotropic mass of magnons confined The authors declare no competing financial interests. in a harmonic trap in superfluid 3He-B JETP Letters 101, 802-807 (2015) 30 Kondo, Y. et al. Direct observation of the nonaxisymmet- ric vortex in superfluid 3He-B. Phys. Rev. Lett. 67, 81–84 (1991). 31 Silaev, M.A., Thuneberg, E.V. & Fogelstr¨om, M. Lif- shitz transition in the double-core vortex in 3He-B. arXiv:1505.02136 (2015). 32 Aad, G. et al. (ATLAS Collaboration) Search for Higgs boson pair production in the γγbb− final state using pp collision data at √s = 8 TeV from the ATLAS detector. Phys. Rev. Lett. 114, 081802 (2015). 33 Ueda M., Fundamentals and New Frontiers of Bose- Einstein Condensation. World Scientific, 2010. 34 Chernodub, M.N. Superconducting properties of vacuum in strong magnetic field, Int. J. Mod. Phys. D 23, 1430009 7

SUPPLEMENTARY NOTE 1 equation (8) describes a longitudinal wave, the light Higgs mode with a “relativistic” spectrum Spin waves in 3He-B 2 2 ωHiggs = ΩB +(C k) . (10) q Spin waves in3He-B correspond to motions of the rotation The first line of Supplementary equation (8) describes two matrix Raj . The matrix can be represented by means of the modes of transverse waves, optical and acoustic magnons, rotation axis ˆn and the rotation angle θ as with spectra of the form:

2 2 Raj = cos θ δaj + (1 cos θ) nanj sin θ eajknk. (3) ωL ωL 2 ωL ωL 2 − − ωopt = + +(c k) , ωac = + +(c k) . 2 r 2 − 2 r 2 The motion is affected by the energy of the spin-orbit in-     (11) teraction Fso and the gradient energy F : In Supplementary equations (10) and (11) the effects of ∇ anisotropy and of the spin-orbit interaction are omitted for 2 χB ΩB simplicity. Fso = (R R + R R ), (4) 15γ2 jj kk jk kj 1 2 F = ∆ (K1G1 + K2G2 + K3G3), (5) 3 ∇ 2 Little Higgs field for spin waves in He-B where Let us introduce a vector field G1 = j Rak j Rak, ∇ ∇ n = nˆ sin θ/2 . (12) G2 = j Rak kRaj , ∇ ∇ The spin-orbit interaction (6) provides a “Mexican Hat” po- G3 = j Raj kRak, ∇ ∇ tential for the n-field 3 χB is the spin susceptibility of the He-B, γ the gyromag- 2 2 2 3 Fso = Λ( n n0) . (13) netic ratio for the He atom, ΩB the Leggett frequency, ∆ | | − 2 32 χB 2 the superfluid gap, and K1, K2 and K3 are parameters of the where n0 = 5/8 and parameter Λ = 15 γ2 ΩB. gradient energy. In the terminology of particle physics the n-vector serves The spin-orbit interaction energy has a simple form in as the “little Higgs” field. In the vacuum states the ampli- terms of ˆn and θ with a minimum at θ = arccos( 1/4): tude of the field is fixed, n = n0, while they are degener- − ate with respect to the orientation| | of nˆ. The broken SU(2) 2 χBΩB 2 symmetry leads to two Nambu-Goldstone modes (propagat- Fso = (cos θ + 1/4) + const. (6) 15γ2 ing oscillations of the orientation of nˆ), and one light Higgs mode (propagating oscillations of the amplitude n around The equation of small spin oscillations near the equilibrium | | 0 2 0 value S =(χB/γ) H is n ). These three modes comprising the little Higgs field are similar to the bosonic sector of Standard Model, where also

S¨c = [S˙ γH]c (7) the SU(2) symmetry is instrumental. This low energy sector 2×2 of Standard Model contains the NG modes (the gauge bosons) ∆ γ 2 0 0 + K Sc K′ j Rcj Rak k Sa and one “light Higgs” (the 125 GeV Higgs boson). Our two χB ∇ − ∇ ∇ Nambu-Goldstone spin-wave modes correspond to the doublet 2  0  ΩB ˆn (S S )n ˆc, of the W-bosons. The spectrum of the spin wave modes in − · − 3He-B splits in magnetic field into acoustic and optical modes. where K = 2K1 + K2 + K3 and K′ = K2 + K3. The similar splitting is discussed for the spectrum of the W- In a texture with ˆn H or in a high magnetic field bosons in magnetic field (see e.g. Ref.3). Moreover, in strong 2 2 k ωL/ΩB 1 one can separate transverse and longitudinal os- magnetic fields the Bose condensation of the W-bosons is ex- ≫ cillations. In the case of short wavelengths (when the spin pected, which is similar to the Bose condensation of optical changes on a much shorter distance than the texture) one can magnons. write the quasiclassical spectra for plane waves:

2 2 2 2 ˆ 2 1 2 2 c k +(c c )(k l) + ΩB sin βn = ω(ω ωL), SUPPLEMENTARY NOTE 2 ⊥ k − ⊥ · 2 − 2 2 2 2 ˆ 2 2 2 2 C k +(C C )(k l) + ΩB cos βn = ω , (8) ⊥ k − ⊥ · Resonance condition for acoustic magnons where βn is an angle between ˆn and H, the orbital anisotropy ˆ 0 axis lj = Raj Sa and the spin wave velocities are introduced Let’s consider an excitation of acoustic magnons with fre- as quency NωL. Here N = 1/2 corresponds to the parametric 2 2 2 2 2 γ ∆ 2 γ ∆ c = (K K′/2), c = K, exitation, N = 1, 2 . . . to the exitation of acoustic magnons ⊥ χB − k χB with frequency ωL, 2 ωL,.... The resonances observed in the 2 2 2 2 (9) 2 γ ∆ 2 γ ∆ experiments correspond to standing waves in the cylindrical C = K, C = (K K′). cell with radius R. For the short spin waves the resonances ⊥ χB k χB − can be treated in the quasiclassical approximation: The spin wave velocities are anisotropic, they have different values if the wave propagates in the direction of ˆl or in the R perpendicular direction. The second line of Supplementary 2 kr dr = nπ , (14) Z0 8

where kr is the classical trajectory along the cell diameter and equation (7) can be rewritten in a form of a Schr¨odinger n is integer quantum number. This quantization corresponds equation for magnon quasiparticles, where complex value 1 + to the wave modes in cylinder with high radial and small s = √2 (Sx + iSy) plays role of the wave function and pre- azimuthal quantum numbers. cession frequency ω plays role of the energy. Effect of texture The effect of the spin-orbit interaction on the spectrum of on the gradient terms is neglected here because it adds only short-wave acoustic magnons can be neglected, but anisotropy a small correction to the total gradient energy. of wave velocity is important. The ratio of the velocities for k ˆl and k ˆl is c /c 4/3. Substituting the transversek magnon⊥ spectrumk (8)⊥ ≈ without the spin-orbit term p into (14) and taking into account that ω = NωL we get

1 πn c 2 c2 2 ωL = (15) c 2 2 2 ΩB 2 N(1 + N) 2R ⊥ ( x + y) k z + sin βn + ωL s+ = ω s+ "− ωL ∇ ∇ − ωL ∇ 2ωL # where c is a harmonic meanp velocity in the non-uniform tex- (18) ture: Non-uniform potential for magnons is formed by the order parameter texture and the magnetic field (βn and ωL param- R 1 2 2 2 2 1/2 eters). 1/c = c +(c c ) sin βl(r) − dr, (16) R ⊥ k − ⊥ Z0  ˆ and βl is an angle between l and H. 2 ΩB 2 The distance between the resonances is: U = sin β + ωL. (19) 2γH n 1 ∂ωL 1 c δfN = = . (17) 2π ∂n N(1 + N) 4R

Note that the spin wave spectrap (8) have been obtained with the assumption of zero coupling between transverse and 2 2 In our setup the potential has a quadratic minimum in the longitudinal modes (ω /Ω 1 or βn 1). In our experi- B center of the sample: in the flare-out texture angle βn near ment this condition is approximately≫ valid≪ for directly excited the sample axis is linear, βn = βn′ r and magnetic field of the magnons with ω > 2ΩB. We use the same approximation longitudinal coil has also quadratic profile near the center. also for parametrically excited magnons with ω ΩB. This is probably the reason why the agreement of the≈ experimen- tal data with Eq. (17) is much better for the directly excited We use pulsed NMR to populate a few lowest levels in this magnons. harmonic trap. If the number of magnons in the system is small enough, interaction between the levels is negligible and the excited states can be resolved in the measurements in- dependently. If the magnon population is above a certain SUPPLEMENTARY NOTE 3 threshold, they collapse to the ground state and form a Bose- Einstein condensate. Trap for magnon quasiparticles From the spectra of the magnon levels in the trap we can In the case of optical magnons with ω ωL, localized find values of spin-wave velocities c and c . This work is in the center of the cell, where ˆn is almost≈ parallel to H, presented in Ref. 4. ⊥ k

3 ∗ Electronic address: vladislav.zavyalov@aalto.fi ing He-B. J. Low Temp. Phys. 52, 559–591 (1983). 3 Chernodub, M.N. Superconducting properties of vacuum in strong magnetic field, Int. J. Mod. Phys. D 23, 1430009 SUPPLEMENTARY REFERENCES (2014). 4 Zavjalov, V.V., Autti, S., Eltsov, V.B. & Heikkinen, P.J. Measurements of the anisotropic mass of magnons confined 1 Thuneberg, E.V. Hydrostatic theory of superfluid 3He-B. J. in a harmonic trap in superfluid 3He-B. JETP Letters 101, Low Temp. Phys. 122, 657–682 (2001). 802–807 (2015). 2 Theodorakis, S. & Fetter, A.L. Vortices and NMR in rotat-