\section*{Supplementary Note}


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 of the container and other vortices. We follow closely the theory presented in Ref.~\onlinecite{laine2018}, although with slightly different notation.

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 pointing along $\hat{{\bm z}}$. We also assume that the unperturbed magnetisation precesses uniformly about $\hat{{\bm z}}$ with a small tipping angle $\beta_{\bm  M}$, as in the experiment presented in the main text of this Article, and work in lowest non-trivial order in $\beta_{\bm  M}$.

Outside the immediate core region of the vortex, the order parameter of the system is proportional to a rotation matrix,
\begin{equation}
\mathsf{A} \propto \mathsf{R} \left( \bm \theta_0 \right) \mathsf{R} \left( \bm \theta_1 + \bm \theta_2  \right).
\end{equation}
Three factors affect the direction and the angle of the rotation. One is the magnon condensate, which fixes the vector $\bm \theta_0=\theta_L\hat{{\bm n}}$, where the rotation angle $\theta_L =  \arccos(-1 / 4)$ and the rotation axis $\hat{{\bm n}} = \hat{{\bm z}} + \sqrt{2/5} \beta_{\bm  M} \mathsf{R}(\hat{{\bm z}}\omega_\mathrm{L} t) \cdot \hat{{\bm y}}$.  The vortex modifies this by the asymptotic form
\begin{equation}\label{e.asymp}
\bm \theta_1 = \frac{C \cos \varphi}{r} \left( \frac{\sin \varphi}{1+c} \hat{{\bm r}} + \cos \varphi \bm{\hat{\varphi}} \right).
\end{equation}
Here $(r, \varphi)$ are the standard polar coordinates so that $x=r \cos \varphi  $ and $y=r \sin \varphi $, and parameter $c \sim 1$ depends on temperature and pressure. The coefficient $C$ is approximately equal to the distance between the sub cores of the vortex, and also depends on temperature and pressure. Finally, there is a contribution $\bm \theta_2$ stemming from the interaction between the condensate and the vortex. It satisfies a wave-like equation
\begin{equation}\label{eq:eom}
\begin{split}
&\bm{\ddot{\theta}}_2 - \omega_\mathrm{L} \hat{{\bm z}} \times \bm{\dot{\theta}}_2 + \Omega_B^2 \bm{\hat{z}} \left(\hat{{\bm z}} \cdot \bm \theta_2 \right) - v^2 \left[ (1+c) \nabla^2 \bm{\theta}_2 - c \nabla \left( \nabla \cdot \bm{\theta}_2 \right) \right] = \bm \rho
\end{split}
\end{equation}
together with the boundary condition $\lim_{r \to 0} \bm \theta_2 = 0$ at the origin and a causal boundary condition at $r \to \infty$. Here, $v = 2 \xi_D \Omega_B / \sqrt{15}$, $\xi_D$ is the dipole length,
\begin{align}
\bm \rho (\bm r, t) &= \Re \left\{ e^{-i \omega_\mathrm{L} t} \tilde{\rho}(r, \varphi) \right\} \hat{{\bm z}}, \label{eq:rho_vector} \\
\tilde{\rho}(r, \varphi) &= -\sqrt{\frac{2}{5}} \frac{\beta C \Omega_B^2}{r} \cos \varphi \frac{2 + c + c e^{2 i \varphi}}{2(1 + c)},
\end{align}
and $\Re\{\cdot\}$ 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 $\bm \theta_2$ by
\begin{equation}
\delta {\bm  M} = \frac{\chi}{\mu_0 \gamma} \mathsf{R}_z(\theta_L) \cdot \bm{\dot{\theta}}_2,
\end{equation}
where $\chi$ is the B-phase susceptibility, $\mu_0$ is the vacuum permeability, and $\gamma$ is the gyromagnetic ratio of $^3$He.

\subsection*{Spin wave modes}
In order to gain more insight into the equation of motion (\ref{eq:eom}), let us consider the eigenmodes $\bm \theta_2(\bm r, t) = {\bm A} e^{i({\bm k} \cdot {\bm r} - \omega t)}$ of the corresponding homogeneous equation. Since the source term $\bm \rho$ in Eq.\ (\ref{eq:eom}) is independent of $z$, we are only interested in modes with ${\bm  k} \cdot \hat{{\bm z}} = 0$. Substituting the ansatz into the equation yields
\begin{equation}
-\omega^2 {\bm A} + i \omega_\mathrm{L} \omega \hat{{\bm z}} \times {\bm A} + \Omega_B^2 \hat{{\bm z}}(\hat{{\bm z}} \cdot {\bm A}) + v^2 \left[ (1 + c) k^2 {\bm A} - c {\bm  k}({\bm  k} \cdot {\bm A}) \right] = \bm 0.
\end{equation}
There are two transverse eigenmodes (${\bm A} \cdot  \hat{{\bm z}} = 0$) with spectra
\begin{equation}
\omega_\pm(k) = \sqrt{ \frac12\left[(2 + c) v^2 k^2 + \omega_\mathrm{L}^2 \pm \sqrt{c^2 v^4 k^4 + 2 (2 + c) v^2 k^2 \omega_\mathrm{L}^2 + \omega_\mathrm{L}^4}\right]}.
\end{equation}
Since $\omega_+(0) = \omega_\mathrm{L}$ and $\omega_-(0) = 0$, these modes correspond to optical magnons and acoustic magnons, respectively (cf. Fig.\ \ref{3He_modes}).
Furthermore, there is a longitudinal mode (${\bm A} \parallel \hat{{\bm z}}$) with spectrum
\begin{equation}
\omega_z(k) = \sqrt{\Omega_B^2 + (1 + c) v^2 k^2}.
\end{equation}
Since $\omega_z(0) = \Omega_B$, this corresponds to the light Higgs mode (cf. Fig.\ \ref{3He_modes}).

The above analysis helps us to understand the physics behind Eq.\ (\ref{eq:eom}) qualitatively. Since $\bm \rho \parallel \hat{{\bm z}}$, the vortex can only excite light Higgs modes. Furthermore, since $\bm \rho \propto e^{-i \omega_\mathrm{L} t}$, the vortex can only excite modes with $\omega(k) = \omega_\mathrm{L}$. Due to the fact that the minimum of $\omega_z(k)$ is $\Omega_B$, no excitations are produced when $\omega_\mathrm{L} < \Omega_B$. Conversely, when $\omega_\mathrm{L} \geq \Omega_B$, the vortex excites light Higgs modes, transferring energy away from the condensate, and thus dissipating the coherently precessing magnetisation towards the equilibrium.


\subsection*{The vortex-Higgs mechanism}
Let us then solve Eq.\ (\ref{eq:eom}) for $\bm \theta_2$. Due to the form of $\bm \rho$ [see Eq.\ (\ref{eq:rho_vector})], we can write the solution as
\begin{equation}
\bm \theta_2(\bm r, t) = \Re \left\{ e^{-i \omega_\mathrm{L} t} \tilde{\theta}_2(r, \varphi) \right\} \hat{{\bm z}},
\end{equation}
where $\tilde{\theta_2}$ satisfies the inhomogeneous Helmholtz equation
\begin{equation}
\nabla^2 \tilde{\theta}_2 + \frac{\omega_\mathrm{L}^2 - \Omega_B^2}{(1 + c) v^2} \tilde{\theta}_2 = - \frac{\tilde{\rho}}{(1 + c) v^2}.
\end{equation}
One way to solve this is to make use of the Fourier transform, as was done in Sec.\ IV in Ref.\ \onlinecite{laine2018}. As a result, we obtain
\begin{equation}
\begin{split}
\tilde{\theta}_2(r, \varphi) = -\frac{1}{\sqrt{10}} \frac{\beta C \Omega_B^2}{(1 + c)^2 v^2} \Bigg\{& \left[ (1 + c) e^{i \varphi} + \left(1 + \frac{c}{2} \right) e^{-i \varphi} \right] \left[ -\frac{1}{r K^2} + \frac{i \pi}{2 K} H_1^{(1)}(K r)  \right] \\
&+ \frac{c}{2} e^{3 i \varphi} \left[ -\frac{1}{r K^2} -\frac{8}{r^3 K^4} + \frac{i \pi}{2 K} H_3^{(1)}(K r)  \right] \Bigg\},
\end{split}
\end{equation}
where $K^2 = (\omega_\mathrm{L}^2 - \Omega_B^2) / (1 + c) v^2$ and $H_n^{(1)}(x)$ are Hankel functions of the first kind.

To calculate the energy transferred to the light Higgs mode, we integrate the energy flux density vector $\bm \Sigma$ over a cylindrical surface of radius $r$ centred at the vortex axis. Since $\bm \theta_2 \parallel \hat{{\bm z}}$, $\bm \Sigma$ takes a simple form
\begin{equation}
\bm \Sigma = - \frac{\chi v^2}{\mu_0 \gamma^2} (1 + c) \dot{\theta}_2 \bm \nabla \theta_2,
\end{equation}
where we denote $\theta_2 = \bm \theta_2 \cdot \hat{{\bm z}}$. Plugging in the solution for $\bm \theta_2$, we find that the time-averaged energy flux through the surface, per vortex length, is given by
\begin{equation}
P(r) = \frac{\chi \Omega_B^4}{\mu_0 \gamma^2} \frac{\pi^2}{40} \frac{\omega_\mathrm{L} \mathcal{H}(\omega_\mathrm{L} - \Omega_B)}{\omega_\mathrm{L}^2 - \Omega_B^2} \beta^2 C^2 \left\{\frac{3 c^2 + 6c + 4}{(1 + c)^2} -2 \frac{2 + c}{1 + c} J_0(K r) - \frac{2 c^2}{(1 + c)^2} \frac{J_1(K r)}{K r} \right\},
\end{equation}
where $\mathcal{H}(x)$ 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 $P \equiv \lim_{r \to \infty} P(r)$ and find
\begin{equation}\label{eq:P}
P = \frac{\chi \Omega_B^4}{\mu_0 \gamma^2} \frac{\pi^2}{40} \frac{\omega_\mathrm{L} \mathcal{H}(\omega_\mathrm{L} - \Omega_B)}{\omega_\mathrm{L}^2 - \Omega_B^2} \beta^2 C^2 \frac{3 c^2 + 6c + 4}{(1 + c)^2}.
\end{equation}
This analysis ignores the finite dimensions of the magnon BEC in the $z$ direction and, consequently, all the power is emitted perpendicular to the vortex line. The power is maximum in the $x$ direction (perpendicular to the line connecting the vortex sub-cores, see Fig.~4 in Ref.~\onlinecite{laine2018}).

\subsection*{Vortex-Higgs relaxation time}
Let us consider what happens to the uniformly precessing magnetisation if each vortex line dissipates energy at the rate given by Eq.\ (\ref{eq:P}).

In a rotating container, the areal density of vortices is given by $n_v = 2 \Omega / \kappa$, where $\Omega$ is the angular velocity of the container, $\kappa = \pi \hbar / m_0$ the circulation quantum, and $m_0$ the mass of a $^3$He atom. One vortex thus occupies an area $A_v = 1/n_v = \kappa / 2 \Omega$. The amount of energy stored in the uniformly precessing magnetisation per vortex length is therefore given by $\mathcal{E} A_v$, where $\mathcal{E} = \chi \omega_\mathrm{L}^2 \beta_{\bm  M}^2 / 2 \mu_0 \gamma^2 +{\rm  const.}$ is the energy density. Thus, we must have $\dot{\mathcal{E}} A_v = -P$. This yields
\begin{equation}\label{exp_relax}
\dot{\beta}_{\bm  M} = - \tau^{-1} \beta_{\bm  M},
\end{equation}
where
\begin{equation}\label{final_relax}
\frac{1}{\tau} = \frac{\pi^2}{20} \Omega \frac{\Omega_B^4}{\kappa} \frac{\mathcal{H}(\omega_\mathrm{L} - \Omega_B)}{\omega_\mathrm{L} (\omega_\mathrm{L}^2 - \Omega_B^2)} \frac{3 c^2 + 6c + 4}{(1 + c)^2} C^2.
\end{equation}
We see that $\beta_{\bm  M}$ relaxes exponentially towards the equilibrium, with relaxation time $\tau$.

The relaxation rate (\ref{final_relax}) has been derived assuming a constant $\beta_{\bm M}$. It remains valid for a non-uniform distribution $\beta_{\bm M}({\bm r})$ as long as the variation of $\beta_{\bm M}$ within each vortex unit cell can be neglected. Moreover, because the radiation is effectively generated within a dipole length from the vortex axis \cite{laine2018}, it may be sufficient that $\beta_{\bm M}$ is nearly constant within this region, which at practical rotation speeds is smaller than vortex unit cell.

We see that the relaxation rate $\tau^{-1}$ (\ref{final_relax}) depends linearly on the number of vortices, as it is proportional to the  angular velocity $\Omega$ of the rotation. The relaxation rate depends on the magnetic field via the Larmor frequency $\omega_\mathrm{L}=\gamma H$. It vanishes at $\omega_\mathrm{L}<\Omega_B$ because of the energy gap of the light Higgs quasiparticles [Fig.\ 1(a)], as expressed mathematically in Eq.~(\ref{final_relax}) by the step function $\mathcal{H}(\omega_\mathrm{L} - \Omega_B)$.  For a quantitative evaluation of $\tau^{-1}$ we need  values of $\Omega_B$, $C$ and $c$, which are functions of pressure. The Leggett frequency $\Omega_B$ can be extracted from NMR experiments as discussed in Ref.\ \cite{thuneberg2001}.

The evaluation of parameters $C$ and $c$ is based on a version of the BCS theory extended to include Fermi-liquid corrections (also know as the quasiclassical theory, or the Fermi-liquid theory of superfluidity) \cite{serene1983quasiclassical}. 
The order-parameters rotation amplitude $C$ (\ref{e.asymp}) can be calculated by solving numerically the full form of the order parameter in the vortex core including all 18 real degrees of freedom at each $x$ and $y$ in the plane perpendicular to the vortex axis. At low temperatures this has been done in Ref.~\onlinecite{silaev2015} (further details of this work will be published later). This calculation includes the effect of $F_1^s$ but neglects other Fermi-liquid interaction parameters, such as $F_2^s$, $F_1^a$ and higher. It also uses the weak coupling approximation, that is, all strong-coupling effects are neglected. The result is that $C \approx 6R_0$, and the ratio $C/R_0 $ is nearly independent of pressure. Here
\begin{eqnarray}\label{e.R0}
R_0=\frac{\hbar p_F}{2\pi m_0T_c}=\left(1+\frac13F_1^s\right)\xi_0
\end{eqnarray}
is the length scale that characterises the distance between the half cores. It differs by the effect of the  Fermi liquid parameter $F_1^s$
 from the more standard coherence length $\xi_0=\hbar v_F/2\pi T_c$, 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 distance between the half cores in the double-core vortex.

The remaining parameter $c$ characterizes the anisotropy of the spin wave velocity.  In the weak coupling approximation in the limit $T/T_c\rightarrow 0$  \cite{serene1983quasiclassical}, 
\begin{equation} c=1+\frac{\frac{1}{3}F_1^{\rm a}-\frac{1}{7}F_3^{\rm a}}
{ 2(1+\frac{1}{7}F_3^{\rm a})}.
\label{e.3.6}\end{equation}
Measurements indicate $F_1^{\rm a}$ varies between -0.55 and -1.0 \cite{zavjalov2015measurements}. Assuming negligible $F_3^{\rm a}$, we find that the effect of deviation of $c$ from unity on $\tau^{-1}$ (\ref{final_relax}) is approximately one per cent. 




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