Vortex-mediated relaxation of magnon BEC into light Higgs quasiparticles
S. Autti, P.J. Heikkinen, S.M. Laine, J.T. Mäkinen, E.V. Thuneberg, V.V. Zavjalov, V.B. Eltsov
VVortex-mediated relaxation of magnon BEC into light Higgs quasiparticles
S. Autti , ∗ , P.J. Heikkinen , , S.M. Laine , J.T. Mäkinen , , , E.V. Thuneberg , , V.V. Zavjalov , , and V.B. Eltsov Low Temperature Laboratory, Department of Applied Physics,Aalto University, POB 15100, FI-00076 AALTO, Finland. Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK. *Email: [email protected] Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK. Nano and Molecular Systems Research Unit, University of Oulu, P.O. Box 3000, Oulu FI-90014, Finland Department of Physics, Yale University, New Haven, CT 06520, USA Yale Quantum Institute, Yale University, New Haven, CT 06520, USA QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 AALTO, Finland.
A magnon Bose-Einstein condensate in superfluid He is a fine instrument for studying the sur-rounding macroscopic quantum system. At zero temperature, the BEC is subject to a few, distinctforms of decay into other collective excitations, owing to momentum and energy conservation in aquantum vacuum. We study the vortex-Higgs mechanism: the vortices relax the requirement formomentum conservation, allowing the optical magnons of the BEC to transform into light Higgsquasiparticles. This observation expands the spectrum of possible interactions between magneticquasiparticles in He-B, opens pathways for hunting down elusive phenomena such as the Kelvinwave cascade or bound Majorana fermions, and lays groundwork for building magnon-based quan-tum devices.
One illuminating perspective to the ground state ofa fermionic condensate, such as zero-temperature super-fluid He, is to treat it as a quantum vacuum where mov-ing objects interact with the excitations of the vacuum[1–5]. Various collective excitations, for example mag-netic quasiparticles (magnons), and topological defectssuch as quantised vortices can be manipulated in this ex-tremely pure environment. A Bose-Einstein condensateof optical magnons (magnon BEC), trapped within thesuperfluid, can be instrumented to probe objects in thesystem without influencing them [6–9]. This capacity hasinspired suggestions to use the BEC to detect surface-or vortex-core-bound Majorana fermions [10, 11] or theKelvin wave cascade [12, 13]. Both have so far remainedelusive despite decades of active research. Changes in theBEC ground state frequency as well as the populationdecay rate of the BEC can be devised for such purposes,provided the basic interactions between the excitationsof the quantum vacuum are first thoroughly mastered.On the other hand, macroscopic quantum systems suchas BEC-based time crystals[14–16] provide a promisingbuilding block for quantum technologies, which rely oncontrolled non-destructive manipulation of the system.Such control can be accessed in the superfluid vacuumby coupling the BEC to and decoupling it from availableexcitations selectively.In superfluid He, the spin and orbital angular mo-menta of Cooper pairs are equal to one. In the B phase,the relative spin-orbit symmetry is broken in addition tothe emergence of a coherent phase, as described by a 3 × θ L ≈ ◦ aroundaxis ˆ n . The fermionic thermal excitations of this systemhave energy gap ∆ B which is on the order of k B T c , where k B is the Boltzmann constant and T c the superfluid tran- sition temperature. At temperatures much below T c , thenumber of thermal excitations is reduced exponentially,creating a vacuum void of fermionic quasiparticles.Besides the fermionic quasiparticles, there are threecollective spin-wave modes with a small (or zero) gap,corresponding to the combined oscillations of three spincomponents and three components of spin-orbit rotation[18]. Following Ref. 19, we call these modes opticalmagnons, acoustic magnons, and light Higgs quasipar-ticles. In the absence of a magnetic field, optical andacoustic magnons are gapless, corresponding to the os-cillations of ˆ n . That is, their frequency vanishes in thelong wave length limit. The light Higgs mode correspondsto oscillation of the spin-orbit rotation angle around itsequilibrium value θ L , and has a gap Ω B / π ∼
100 kHz( Ω B is the Leggett frequency). In a magnetic field H ,optical magnons acquire a gap equal to the Larmor fre-quency 2 πf L = ω L = γH , where γ is the gyromagneticratio. The dispersion relations of the three modes areillustrated in Fig. 1(a).The superfluid vacuum can also host topological de-fects [1], in particular quantised vortices. An orderedarray of vortices can be created by rotating the sampleat a constant angular velocity Ω . The density of thevortex array is proportional to Ω . The B-phase vorticeshave a broken-symmetry core [21, 22], where the low-temperature vortex studied in this work has a double-core structure consisting of two tightly-bound sub cores[23–29]. A theoretical description of the interactions ofvortices and collective excitations in superfluid He canbe confidently constructed by expanding on the BCS the-ory with Fermi-liquid effects included [30].In this Letter we study the interaction of a conden-sate of optical magnons with quantised vortices andlight Higgs quasiparticles, which we call the vortex-Higgsmechanism: If a vortex penetrates the magnon BEC, op- a r X i v : . [ c ond - m a t . o t h e r] F e b FIG. 1. ( a ) Spectra of spin waves in He-B. The mass (gap) ω L of optical magnons (blue line) can be tuned using the mag-netic field. Direct conversion of optical magnons into lightHiggs quasiparticles (solid arrow), studied in this Letter, re-quires balancing the momentum mismatch. The parametricconversion of optical magnons into gapless acoustic magnons(red line), and light Higgs quasiparticles (green line) can beobserved when the density and mass of the optical magnons islarge. Processes indicated by dashed arrows were reported inRef. 19. ( b ) Superfluid He in a cylindrical container. A BECof optical magnons (blue blob) is trapped in the middle by thespatial distribution of the orbital order parameter ( ˆ n -vector,small green arrows) and by an axial minimum in the externalmagnetic field H . The coherently precessing magnetisation M (large magenta arrow) in the BEC is parametrised with thetipping angle β M . In constant rotation Ω around the verticalaxis, an array of vortices is created, penetrating the BEC (redvertical rods at the top). For illustrational reasons the vortexrods have been made transparent in the vicinity of the BEC,and drawn only in the upper half of the container. The vor-tex configuration obtained in modulated rotation is sketchedwith the magenta rods at the bottom of the container, basedon Ref. [20]. tical magnons are scattered by the order-parameter dis-tortion that surrounds the vortex. This interaction liftsthe requirement for momentum conservation for inboundand outbound quasiparticles. We show that in such col-lisions the optical magnons in the condensate are con-verted directly into light Higgs quasiparticles. This isseen as zero-temperature relaxation of the BEC with ex-ponential time dependence. We study this conversionin two qualitatively different vortex configurations, or-dered and disordered, and find that the results are ingood agreement with theory.The magnon BEC in superfluid He consists of coher-ent optical magnons [31]. Their magnetisation M pre-cesses around the external magnetic field H and is de-scribed by a macroscopic wave function Ψ . The totalnumber of magnons N ∝ R | Ψ | d V ∝ R β M d V , where β M is the deflection angle of M from the equilibrium di-rection along H , and V is volume. Here we assumed that β M is small, which is satisfied in all the experiments pre-sented in this Letter. The coherently precessing magneti-zation is generated and detected using Nuclear MagneticResonance techniques.The magnon BEC is trapped in the middle of the su-perfluid sample (Fig. 1b) by the combined effect of theorbital order parameter distribution (“texture”), and aprofile of the external magnetic field. The resulting trapis nearly harmonic[32, 33], characterised by the radialand axial trapping frequencies, f r and f z , determinedfrom measurements of the full spectrum of states in thetrap [33]. We concentrate on the ground state magnonBEC, whose precession frequency is f = f L + f r + f z / ∆ν ∝ exp( − ∆ B /k B T ) at temperatures T (cid:28) T c ∼ β M ∝ exp( − t/τ SD ). Here 1 /τ SD is the spin-diffusion relaxation rate. Spin diffusion is proportional tothe thermal quasiparticle density, leading to a linear de-pendence between the thermometer fork resonance widthand the BEC relaxation rate, 1 /τ SD ∝ ∆ν (Fig. 2a) [32].In practice there are also unavoidable losses in the mea-surement circuitry, but this effect can be confidently sub-tracted [32].In the zero-temperature limit intrinsic decay channelsare absent and the condensate lifetime approaches infin-ity [7, 14, 16, 36]. Any (extrapolated) zero-temperaturedissipation in the bulk liquid [36] is therefore an indi-cation of interaction with other collective modes eithervia parametric excitation or direct conversion. The for-mer is allowed assuming the density of optical magnonsis high enough and their mass large enough [19]. Directconversion is ruled out due to momentum conservationunless mediated by boundaries or interaction with topo-logical defects of the superfluid vacuum, such as quantumvortices. The interaction of magnons with a topologicaldefect arises due to the distortion of the order parameterdistribution in the vicinity of the defect. For a vortexthis is quantified by a coupling constant C , which givesthe amplitude of the deviation of the spin-orbit rotationfrom the equilibrium. This deviation decays as one overthe distance from the vortex line. For the double-corevortex, C is nearly equal to the separation of the halfcores. The coupling mechanism is derived in Ref. 37 andapplied for the present case in Supplementary Note.The entire refrigerator used in the experiments can berotated around the axis of the sample container cylin-der. Rotation creates an equilibrium array of vortices,which has a twofold effect on the BEC. First, the globalorbital texture reacts to the vortex array [38], chang-ing the shape of the trap ( f r increases). This affects ∆ν (mHz) τ - ( / s ) Ω = r a d / s . r a d / s . / s . r a d / s . r ad / s T / T c Ω (rad/s) τ - ( T = ) ( / s ) p = . r . r . r f r (Hz) τ - / ∆ ν ( s - H z - ) × -3
18 20 22 24 26
Field (mT) Ω - τ - ( T = ) ( / r ad ) bdac FIG. 2. Vortex-Higgs mechanism in steady rotation: ( a ) Mea-sured relaxation rate at Ω = 0 (red points) is linear in thethermometer fork width ∆ν owing to spin diffusion (red line).Intrinsic fork width has been subtracted from ∆ν shown here.Data measured at Ω > b ) The temperature-dependenceslopes, d τ − ( Ω )d ∆ν , in panel a (points) are proportional to theradial trapping frequency f r (dash line is a linear fit throughzero), as expected for spin diffusion, implying that other re-laxation contributions are temperature-independent. ( c ) Thetemperature-independent relaxation extracted as illustratedin panel a by extrapolating to ∆ν = 0 (coloured dots) is pro-portional to Ω , that is, to the vortex density (dash lines).( d ) The magnetic field dependence of the vortex relaxationin the ordered state, extracted similarly as shown in panel a (large orange circles, Ω = 1 rad s − ), is in good agree-ment with that obtained by modulated rotation (small blackpoints). Dissipation due to losses in the measurement circu-ity and the intrinsic fork width have been subtracted from alldata as explained in Ref. 32 and Fig. 3. Pressure in panels a and b was 0.5 bar. The magnetic field for the 0 . f L = 826 kHz, and for the 4 barand 8 bar data to f L = 833 kHz. Data in panel d was mea-sured at 4 bar pressure. Error bars correspond to uncertaintyin removing the resonant relaxation peaks. the spin diffusion relaxation, which can be written as1 /τ SD = f r D × const . [35] ( f r (cid:29) f z ). Here D is the appli-cable component of the transverse spin diffusion tensor.We find that changes in the measured relaxation 1 /τ areproportional to ∆ν at any given Ω (Fig. 2a), and thatthe slope d τ − ( Ω )d ∆ν is proportional to the measured radialtrapping frequency f r (Fig. 2b). This observation impliesthat the temperature dependence of the relaxation rate1 /τ , contained in D , is not affected by rotation, and anyrelaxation directly related to the vortices is temperature-independent below T = 0 . T c . We emphasise that all the relaxation signals measured were exponential in time,implying that no non-exponential contribution was addedby the vortex array. The second observation is that thezero-temperature ( ∆ν = 0) extrapolation of the relax-ation is proportional to Ω (Fig. 2c). That is, the observedtemperature-independent relaxation is (i) also exponen-tial and (ii) proportional to the density of vortices. Thisis in good agreement with the theoretical expectation forvortex-Higgs mechanism of BEC relaxation, Eq. (S17),We note that peaks in the measured relaxation, as-sociated with the presence of vortices, were observedwith roughly 1 kHz spacing in f L on top of the vortex-Higgs dissipation described above. We account this phe-nomenon for resonant production of standing spin wavemodes in the sample container, mediated by the vortexarray [19]. Both acoustic magnons and light Higgs quasi-particles are viable candidates to explain this observa-tion, but a detailed study is left for a future publication.For simplicity, in what follows we call the peaks “relax-ation peaks”. The peak frequencies were avoided in allmeasurements conducted at stable rotation.As an alternative to constant rotation, the angularvelocity can be modulated. We used linear modula-tion in the range from 1.4 rad s − to 1.8 rad s − with | d Ω/ d t | = 0 .
03 rad s − . In the steady state the vor-tex number is expected to remain constant but the vor-tex array is distorted. We find experimentally that thisremoves the resonance peaks found at constant rota-tion. The measured zero-temperature relaxation withand without modulation, the latter avoiding the relax-ation peaks, are shown in Fig. 2d. The two vortex con-figurations yield the same BEC relaxation rate. Thisobservation allows us to probe the vortex-Higgs mecha-nism at arbitrary magnetic fields, avoiding the relaxationpeaks altogether.We can further characterise the vortex-Higgs mecha-nism by varying the coupling between magnons and thevortices. The coupling constant C can be controlled withthe pressure and the magnetic field, as derived in theSupplementary Note. We compare the experiment withthe theoretical model in Fig. 3. The data is fitted usingthe coupling constant C (other parameters were takenfrom Ref. 39). We find very good agreement in themagnetic field as well as the pressure dependence with C = 7 . R ( R is a characteristic length scale of the or-der of the coherence length in the superfluid as definedin the Supplementary Note). This value is close to thetheoretical value C/R = 5 . − .
17 18 19 20 21 22 23 24 25 26 27
Field (mT) Ω - τ - ( T = )( / r ad ) p = r p = . r p = . r p = r p = r p = r Pressure (bar) C ( µ m ) FIG. 3. Vortex-Higgs mechanism as a function of magneticfield and pressure: Measured BEC-relaxation field depen-dence at different pressures (coloured dots) is in good agree-ment with the theoretical expectation (dash lines) for vortex-mediated conversion of BEC magnons into light Higgs quasi-particles. Theory lines correspond to Eq. S17, fitted to thedata using parameter C . All measurements were carried outat T = 0 . T c . Spin diffusion dissipation and radiation losseshave been subtracted based on measured trapping frequen-cies f r and f z [35]. This correction is about 5 % for the 4 bardata, and less than 1% for the 29 bar data. The inset shows C vs. pressure, in good agreement with theoretical expectation C = ˜ CR , with fitted ˜ C = 7 . The observations presented above imply that thevortex-Higgs mechanism opens an otherwise-unavailablerelaxation channel for the magnon BEC, correspondingto zero-temperature conversion of optical magnons of theBEC into light Higgs quasiparticles. This connection ismediated by the double-core vortex of low-temperaturesuperfluid He, and it is robust against changes in vortexorientation and order. The role of the vortices, acting viathe textural distortion that surrounds them, is to bridgethe mismatch of momentum between the two species ofquasiparticles. The measured dissipation is proportionalto vortex density, and reacts to changes in the core di-mensions of the B phase vortices and the order parame-ter distribution surrounding them as expected: the mea-sured magnetic field and pressure dependencies as wellas the temperature independence of the measured relax-ation follow the theory developed in the SupplementaryNote. These observations add the vortex-Higgs mecha-nism to the set of confirmed interaction channels of mag-netic quasiparticles in superfluid He.It remains an interesting task for the future to confirmfurther predictions of the vortex-Higgs mechanism. Theconversion of optical magnons into light Higgs quasipar- ticles is only possible via this mechanism assuming theoptical magnons have a mass ( ∝ ω L ) larger than that oflight Higgs quasiparticles ( ∝ Ω B ). The former is con-trolled by the magnetic field, and the latter by pressure.Probing the region where ω L < Ω B , and thus the vortex-Higgs mechanism is disabled, requires developing a spec-trometer capable of measuring at a sufficiently low mag-netic field (<8 mT). Such setup could be used to con-clusively identify the source of the resonant relaxationpeaks as well. On the other hand, our results providebasis for looking for relaxation contributions beyond thevortex-Higgs mechanism, for example those originatingfrom vortex-core- or surface-bound fermionic quasipar-ticles [10, 11, 40–42]. For such studies one should alsooperate below the cut-off frequency Ω B , allowing the de-tection of smaller relaxation contributions.The magnon BEC also makes a sophisticated new toolfor probing other emergent phenomena, such as vortexdynamics. In particular, zero-temperature vortex turbu-lence is believed to be terminated by the Kelvin wavecascade, but it remains a long-standing challenge to con-firm and explore this effect experimentally [12, 13, 43, 44].On the other hand, surfaces and vortex cores of super-fluid He host elusive Majorana bound states [10, 11],with a characteristic zero-temperature dissipation signa-ture [45], which remains to be conclusively evidenced.Both these realms can now be explored using a magnonBEC as an instrument. For example, one way of detect-ing the Majorana quasiparticles relies on changing therelative magnetic field orientation and measuring the re-lated magnetic relaxation [45]. Our work provides a solidbasis for measuring any delicate field dependence of theBEC relaxation rate. Finally, applying the BEC to buildquantum devices [16] — eventually perhaps even at roomtemperature [15, 46–49] — is an exciting new avenue forresearch that our work and other recent advances haveenabled. Quantum vortices could be used to manipulatethe state and features of such a device via the vortex-Higgs mechanism with minimal disturbance and couplingto the external world.This work has been supported by the EuropeanUnion’s Horizon 2020 research and innovation pro-gramme (grant no. 694248). The experimental work wascarried out in the Low Temperature Laboratory, whichis a part of the OtaNano research infrastructure of AaltoUniversity and of the EU H2020 European MicrokelvinPlatform (grant No. 824109). S.A. and V.V.Z. werefunded by UK EPSRC (grant no.EP/P024203/1). S.A.acknowledges support from the Jenny and Antti Wihurifoundation, P.J.H. that from the Väisälä foundation ofthe Finnish Academy of Science and Letters, and S.M.L.that from both of the above. E.V.T. acknowledges sup-port by the Academy of Finland Centre of Excellenceprogram (project 312057). [1] G. E. Volovik,
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In order to derive a description of the vortex-Higgs mechanism, let us study the interaction between one double-core vortex and coherently precessing magnetisation. To make the calculations tractable, we ignore the presence ofthe container and other vortices. We follow closely the theory presented in Ref. 37, although with slightly differentnotation.The geometry of the problem is as follows. The vortex line is aligned along the z axis, and is oriented in the x - y plane so that the two sub cores lie along the y axis. We assume that there is an external static magnetic field pointingalong ˆ z . We also assume that the unperturbed magnetisation precesses uniformly about ˆ z with a small tipping angle β M , as in the experiment presented in the main text of this Article, and work in lowest non-trivial order in β M .Outside the immediate core region of the vortex, the order parameter of the system is proportional to a rotationmatrix, A ∝ R ( θ ) R ( θ + θ ) . (S1)Three factors affect the direction and the angle of the rotation. One is the magnon condensate, which fixes the vector θ = θ L ˆ n , where the rotation angle θ L = arccos( − /
4) and the rotation axis ˆ n = ˆ z + p / β M R ( ˆ z ω L t ) · ˆ y . Thevortex modifies this by the asymptotic form θ = C cos ϕr (cid:18) sin ϕ c ˆ r + cos ϕ ˆ ϕ (cid:19) . (S2)Here ( r, ϕ ) are the standard polar coordinates so that x = r cos ϕ and y = r sin ϕ , and parameter c ∼ C is approximately equal to the distance between the sub cores ofthe vortex, and also depends on temperature and pressure. Finally, there is a contribution θ stemming from theinteraction between the condensate and the vortex. It satisfies a wave-like equation ¨ θ − ω L ˆ z × ˙ θ + Ω B ˆ z ( ˆ z · θ ) − v (cid:2) (1 + c ) ∇ θ − c ∇ ( ∇ · θ ) (cid:3) = ρ (S3)together with the boundary condition lim r → θ = 0 at the origin and a causal boundary condition at r → ∞ . Here, v = 2 ξ D Ω B / √ ξ D is the dipole length, ρ ( r , t ) = < (cid:8) e − iω L t ˜ ρ ( r, ϕ ) (cid:9) ˆ z , (S4)˜ ρ ( r, ϕ ) = − r βCΩ B r cos ϕ c + ce iϕ c ) , (S5)and <{·} denotes the real part of a complex number.The deviation of the magnetisation from the coherently precessing one is related to the time derivative of θ by δ M = χµ γ R z ( θ L ) · ˙ θ , (S6)where χ is the B-phase susceptibility, µ is the vacuum permeability, and γ is the gyromagnetic ratio of He.
Spin wave modes
In order to gain more insight into the equation of motion (S3), let us consider the eigenmodes θ ( r , t ) = A e i ( k · r − ωt ) of the corresponding homogeneous equation. Since the source term ρ in Eq. (S3) is independent of z , we are onlyinterested in modes with k · ˆ z = 0. Substituting the ansatz into the equation yields − ω A + iω L ω ˆ z × A + Ω B ˆ z ( ˆ z · A ) + v (cid:2) (1 + c ) k A − c k ( k · A ) (cid:3) = . (S7)There are two transverse eigenmodes ( A · ˆ z = 0) with spectra ω ± ( k ) = s (cid:20) (2 + c ) v k + ω ± q c v k + 2(2 + c ) v k ω + ω (cid:21) . (S8)Since ω + (0) = ω L and ω − (0) = 0, these modes correspond to optical magnons and acoustic magnons, respectively (cf.Fig. 1). Furthermore, there is a longitudinal mode ( A k ˆ z ) with spectrum ω z ( k ) = q Ω B + (1 + c ) v k . (S9)Since ω z (0) = Ω B , this corresponds to the light Higgs mode (cf. Fig. 1).The above analysis helps us to understand the physics behind Eq. (S3) qualitatively. Since ρ k ˆ z , the vortex canonly excite light Higgs modes. Furthermore, since ρ ∝ e − iω L t , the vortex can only excite modes with ω ( k ) = ω L .Due to the fact that the minimum of ω z ( k ) is Ω B , no excitations are produced when ω L < Ω B . Conversely, when ω L ≥ Ω B , the vortex excites light Higgs modes, transferring energy away from the condensate, and thus dissipatingthe coherently precessing magnetisation towards the equilibrium. The vortex-Higgs mechanism
Let us then solve Eq. (S3) for θ . Due to the form of ρ [see Eq. (S4)], we can write the solution as θ ( r , t ) = < (cid:8) e − iω L t ˜ θ ( r, ϕ ) (cid:9) ˆ z , (S10)where ˜ θ satisfies the inhomogeneous Helmholtz equation ∇ ˜ θ + ω − Ω B (1 + c ) v ˜ θ = − ˜ ρ (1 + c ) v . (S11)One way to solve this is to make use of the Fourier transform, as was done in Sec. IV in Ref. 37. As a result, weobtain ˜ θ ( r, ϕ ) = − √ βCΩ B (1 + c ) v ( h (1 + c ) e iϕ + (cid:16) c (cid:17) e − iϕ i (cid:20) − rK + iπ K H (1)1 ( Kr ) (cid:21) + c e iϕ (cid:20) − rK − r K + iπ K H (1)3 ( Kr ) (cid:21) ) , (S12)where K = ( ω − Ω B ) / (1 + c ) v and H (1) n ( x ) are Hankel functions of the first kind.To calculate the energy transferred to the light Higgs mode, we integrate the energy flux density vector Σ over acylindrical surface of radius r centred at the vortex axis. Since θ k ˆ z , Σ takes a simple form Σ = − χv µ γ (1 + c ) ˙ θ ∇ θ , (S13)where we denote θ = θ · ˆ z . Plugging in the solution for θ , we find that the time-averaged energy flux through thesurface, per vortex length, is given by P ( r ) = χΩ B µ γ π ω L H ( ω L − Ω B ) ω − Ω B β C (cid:26) c + 6 c + 4(1 + c ) − c c J ( Kr ) − c (1 + c ) J ( Kr ) Kr (cid:27) , (S14)where H ( x ) is the Heaviside step function and J n ( x ) are Bessel functions of the first kind. To calculate the total rate(per vortex length) at which energy is transferred to the light Higgs mode, we take the limit P ≡ lim r →∞ P ( r ) andfind P = χΩ B µ γ π ω L H ( ω L − Ω B ) ω − Ω B β C c + 6 c + 4(1 + c ) . (S15)This analysis ignores the finite dimensions of the magnon BEC in the z direction and, consequently, all the poweris emitted perpendicular to the vortex line. The power is maximum in the x direction (perpendicular to the lineconnecting the vortex sub-cores, see Fig. 4 in Ref. 37). Vortex-Higgs relaxation time
Let us consider what happens to the uniformly precessing magnetisation if each vortex line dissipates energy at therate given by Eq. (S15).In a rotating container, the areal density of vortices is given by n v = 2 Ω/κ , where Ω is the angular velocity ofthe container, κ = π (cid:126) /m the circulation quantum, and m the mass of a He atom. One vortex thus occupies anarea A v = 1 /n v = κ/ Ω . The amount of energy stored in the uniformly precessing magnetisation per vortex lengthis therefore given by E A v , where E = χω β M / µ γ + const . is the energy density. Thus, we must have ˙ E A v = − P .This yields ˙ β M = − τ − β M , (S16)where 1 τ = π Ω Ω B κ H ( ω L − Ω B ) ω L ( ω − Ω B ) 3 c + 6 c + 4(1 + c ) C . (S17)We see that β M relaxes exponentially towards the equilibrium, with relaxation time τ .The relaxation rate (S17) has been derived assuming a constant β M . It remains valid for a non-uniform distribution β M ( r ) as long as the variation of β M within each vortex unit cell can be neglected. Moreover, because the radiationis effectively generated within a dipole length from the vortex axis [37], it may be sufficient that β M is nearly constantwithin this region, which at practical rotation speeds is smaller than vortex unit cell.We see that the relaxation rate τ − (S17) depends linearly on the number of vortices, as it is proportional to theangular velocity Ω of the rotation. The relaxation rate depends on the magnetic field via the Larmor frequency ω L = γH . It vanishes at ω L < Ω B because of the energy gap of the light Higgs quasiparticles [Fig. 1(a)], as expressedmathematically in Eq. (S17) by the step function H ( ω L − Ω B ). For a quantitative evaluation of τ − we need valuesof Ω B , C and c , which are functions of pressure. The Leggett frequency Ω B can be extracted from NMR experimentsas discussed in Ref. [39].The evaluation of parameters C and c is based on a version of the BCS theory extended to include Fermi-liquidcorrections (also know as the quasiclassical theory, or the Fermi-liquid theory of superfluidity) [30]. The order-parameters rotation amplitude C (S2) can be calculated by solving numerically the full form of the order parameterin the vortex core including all 18 real degrees of freedom at each x and y in the plane perpendicular to the vortexaxis. At low temperatures this has been done in Ref. 26 (further details of this work will be published later). Thiscalculation includes the effect of F s but neglects other Fermi-liquid interaction parameters, such as F s , F a and higher.It also uses the weak coupling approximation, that is, all strong-coupling effects are neglected. The result is that C ≈ R , and the ratio C/R is nearly independent of pressure. Here R = (cid:126) p F πm T c = (cid:18) F s (cid:19) ξ (S18)is the length scale that characterises the distance between the half cores. It differs by the effect of the Fermi liquidparameter F s from the more standard coherence length ξ = (cid:126) v F / πT c , which characterises the size of a half core.Here p F and v F are the Fermi momentum and Fermi velocity. Note that C is approximately equal to the distancebetween the half cores in the double-core vortex.The remaining parameter c characterizes the anisotropy of the spin wave velocity. In the weak coupling approxi-mation in the limit T /T c → c = 1 + F a1 − F a3 F a3 ) . (S19)Measurements indicate F a1 varies between -0.55 and -1.0 [33]. Assuming negligible F a3 , we find that the effect ofdeviation of c from unity on τ −1