Figure 1

A sketch of the trapping potential Eq. (2) which is formed in the cylindrically symmetric “flare-out” texture of the orbital anisotropy axis $\mathbf{L}$ (thin lines) in a shallow minimum of the vertical magnetic field $\mathbf{H}$ (right). The arrows represent the magnetization $\mathbf{M}$, which precesses coherently with constant phase angle in the condensate droplet (dark blue), in spite of the inhomogeneity in the texture and in the magnetic field.Reuse & Permissions

Figure 2

Numerical calculations of the magnon wave function and the order-parameter texture for multimagnon bubbles. The deflection angles of the magnetization, ${\beta }_M$, and of the orbital anisotropy axis, ${\beta }_L$, are plotted as a function of radius for different populations $\mathcal{N}$ of the ground-state, $m=n=0$. $\mathcal{N}$ increases from bottom to top in the ${\beta }_M$ plot and from left to right in the plot of ${\beta }_L$. The red curves correspond to the experimental regime in Fig. 3 (starting with $M_{\perp }/M_{\mathrm{HPD}}=5\times {10}^{-4}$ and then continuing from ${10}^{-3}$ to $5\times {10}^{-3}$ with a step of ${10}^{-3}$). The blue curves illustrate extrapolations to the asymptotic regime $f\to f_L(0)$ ($M_{\perp }/M_{\mathrm{HPD}}$ varies from ${10}^{-2}$ to $5\times {10}^{-2}$ with a step of $5\times {10}^{-3}$). When the magnon occupation increases, the magnon wave function suppresses the orbital texture ${\beta }_L$ and the potential well transforms towards a box with impenetrable walls. Fits to the wave function of the condensate in a box are shown with broken lines in the ${\beta }_M$ plot. The effective radius of the box obtained from such fits is shown in the inset as a function of $\mathcal{N}$. The slope from Eq. (5) with $p=1/5$ is shown for comparison. Calculations for the excited radial states (ignoring relaxation to the ground state) present a similar picture. The difference is that the wave function (and thus ${\beta }_M$) crosses zero an appropriate number of times and the radius of localization is larger than in the ground state, as shown in the inset for $m=2$. The reason is the larger amplitude of zero-point motion in Eq. (6), so that from (5) we have $R_{2,0}/R_{0,0}=({\lambda }_2/{\lambda }_0{)}^{1/5}$.Reuse & Permissions

Figure 3

Formation of the magnon condensate in the ground state of the trap in Fig. 1 in cw NMR measurement. The condensate magnetization $M_{\perp }$ precessing in the transverse plane is plotted on the vertical axis, normalized to that when the homogeneously precessing domain (HPD) fills the volume within the detector coil [23]. The arrows indicate the sweep direction of the applied rf frequency $f=\omega /(2\pi )$. $M_{\perp }$ grows when the frequency is swept down. Only a tiny response is obtained on sweeping in the opposite direction. During the downward frequency sweep the condensate is destroyed (vertical lines), when energy dissipation exceeds the rf pumping. This point depends on the applied rf excitation amplitude, marked at each vertical line. The lower green line represents the result of calculations from Fig. 2. These calculations have no fitting parameters: the vertical difference from the measurements can be attributed to the experimental uncertainty in determining the normalization for $M_{\perp }$. Inset: The experimental curve measured with the largest excitation and compared to the numerical curve from the main panel, replotted with logarithmic axes to demonstrate the asymptotic limit for large magnon numbers: $M_{\perp }\propto [f-f_L(0){]}^{-1.75}$, which corresponds to the condensate in a box.Reuse & Permissions

Figure 4

Formation of magnon condensates at different excited levels ($m$, $n$) of the magnetic trap in Fig. 1. When the frequency of the rf excitation field is swept down and crosses one of the levels ($m$, $n$) at $f_{mn}$, the magnon condensate starts to grow according to Eq. (7). Only modes with even m and n are excited. The three examples of measured spectra are for different values of rotation velocity $\Omega$ in the vortex-free state and for different depths $\Delta H/H$ of the axial field minimum in the trap. In the topmost case the depth of the field minimum is twice that of the two bottom examples. As a result, the smaller axial level spacing increases, as the axial oscillator frequency ${\omega }_z/2\pi \mathrm{\text{:}}22\to 27\mathrm{Hz}$, but the larger radial level spacing remains unchanged. Similarly, when $\Omega$ is increased from $0.6\mathrm{rad}/\mathrm{s}$ (bottom spectrum) to $0.8\mathrm{rad}/\mathrm{s}$ (two top most spectra), the radial level separation increases, as ${\omega }_r/2\pi \mathrm{\text{:}}170\to 240\mathrm{Hz}$. The equidistant vertical tick marks refer to the level positions in the absence of magnons and have been fitted to the harmonic trap relation in Eq. (4).Reuse & Permissions

Figure 5

Decay of the magnon condensate, after rf pumping is switched off at $t=0$. The amplitude of the Fourier transform of the signal from the NMR pick-up coil is shown in the main panel. Magnons are pumped to the (2, 0) level at $t<0$, but the ground state is simultaneously populated owing to the decay of magnons from the excited state. At $t>0$ both states decay [24] and the frequency of precession increases as the trap responds to the decreasing magnon population (Fig. 3). The inset shows the amplitudes of the two peaks and fits to exponential decay with time constant $\tau$. The measurements have been performed at $0.14T_c$ in equilibrium rotation at $0.9\mathrm{rad}/\mathrm{s}$ with rectilinear vortices.Reuse & Permissions