Propagation of self-localized $Q$-ball solitons in the ${}^3\mathrm{He}$ universe

In relativistic quantum field theories, compact objects of interacting bosons can become stable owing to conservation of an additive quantum number $Q$. Discovering such $Q$ balls propagating in the universe would confirm supersymmetric extensions of the standard model and may shed light on the mysteries of dark matter, but no unambiguous experimental evidence exists. We have created long-lived $Q$-ball solitons in superfluid ${}^3\mathrm{He}$, where the role of the $Q$ ball is played by a Bose-Einstein condensate of magnon quasiparticles. The principal qualitative attribute of a $Q$ ball is observed experimentally: its propagation in space together with the self-created potential trap. Additionally, we show that this system allows for a quantitatively accurate representation of the $Q$-ball Hamiltonian. Our $Q$ ball belongs to the class of the Friedberg-Lee-Sirlin $Q$ balls with an additional neutral field $\zeta$, which is provided by the orbital part of the Nambu-Goldstone mode. Multiple $Q$ balls can be created in the experiment, and we have observed collisions between them. This set of features makes the magnon condensates in superfluid ${}^3\mathrm{He}$ a versatile platform for studies of $Q$-ball dynamics and interactions in three spatial dimensions.

Figure 1

Experimental setup: top part of the sample container, magnon condensate with precessing magnetization $\mathbf{M}$ (red blob) in a potential trap (thick solid lines), and corresponding wave functions for a small number of magnons (red dash-dotted lines). In the radial direction the potential minimum is formed by a combination of magnetic and textural energies, $U_H$ and $U_{\mathrm{text}}$. In the axial direction the minimum is formed by the magnetic energy alone. Green arrows illustrate the spatial distribution of vector $\widehat{\mathbf{l}}$, which is uniform in the $\widehat{\mathbf{z}}$ direction in the absence of magnons. The pinch coil position defines $z=0$, which corresponds also to the common axis of the NMR pickup coils. The potentials and wave functions are calculated for $T=0.15T_{\mathrm{c}}, p=0.5$ bar, and pinch coil current of 3 A.

Figure 2

Frequency and amplitude of a magnon $Q$ ball during its decay: The $Q$ ball is created by an rf pulse at $t=0$ ($T=0.15T_{\mathrm{c}}, T_{\mathrm{c}}\approx 1\mathrm{mK}$). (a) The signal recorded from the NMR pickup coils. (b) Windowed Fourier transform of the signal showing the magnon BEC as a sharp peak whose frequency shift $\mathrm{\Delta }\nu$ and amplitude $A$ change in time. (c) and (d) Measured (blue lines) and calculated (red dashed lines) dependencies, $A\left(t\right)$ and $A\left(\mathrm{\Delta }\nu \right)$. Thin black lines in (c) are exponential fits to the measured decay. The initial decay time in the simulation is $\tau =1.7\mathrm{s}$. The red square and green circle mark the peripheral and central $Q$ balls illustrated in Fig. 4.

Figure 3

(a) $Q$-ball decays and (b) corresponding frequency-amplitude dependencies at different temperatures and with different initial amplitudes at $p=0$ bar. The measured decay signals are scaled in time according to ${\tau }_{\mathrm{cen}}$ to the units of the green ($0.16T_{\mathrm{c}}$) line, revealing that the decay paths are identical in both $A\left(t\right)$ and $A\left(\mathrm{\Delta }\nu \right)$. The signals begin with a nondecaying part of constant amplitude, where the condensate is supported by rf pumping ($t<0$ in Fig. 2). In this plot $t=0$ corresponds to turning off pumping for the magenta ($0.17T_{\mathrm{c}}$) line, and other lines have been shifted in time in order to allow a comparison of the decay processes. ${\tau }_{\mathrm{cen}}$ stands for the time constant of exponential decay of the central $Q$ ball (tail of the signal), and ${\tau }_{\mathrm{per}}$ stands for that of the peripheral one (beginning part of the signal). Thin black lines are exponential fits to the green ($0.16T_{\mathrm{c}}$) line.

Figure 4

Propagating magnon $Q$ ball: (a) The peripheral $Q$ ball, as marked in Fig. 2 by a square, is plotted in terms of ${\beta }_M$ at $z=0$ along the direction of $Q$-ball movement, labeled $x^{\prime }$. The thick dash-dotted line is used where the condensate frequency (level indicated by the dotted blue line) is above the total potential $U$ (thick solid blue line), and the thin dashed line is used where it is below. The magnetic potential $U_H$ is drawn with a thick dashed black line. (b) While propagating from the periphery to the axis, the $Q$-ball frequency crosses $\mathrm{\Delta }\nu =0$. (c) The $Q$-ball state at the axis, marked in Fig. 2 by a circle. (d) and (f) The parts of the peripheral and central $Q$-ball wave functions (compact red and green blobs) that correspond to $Q$-ball frequency being above the total potential (lighter blue surface) are plotted in the $z=0$ plane to illustrate the broken azimuthal symmetry of the peripheral state. The magnetic part of the potential is shown by a darker gray surface. (e) The top view of the sample container, the NMR coils, and the two $Q$-ball states [as plotted in (d) and (f)] reveals the time evolution of the $Q$-ball size. The peripheral $Q$ ball is plotted traveling to one of the four degenerate directions with respect to the NMR coils.

Figure 5

Coexistence of central and peripheral $Q$ balls: a true $Q$ ball on the periphery of the sample container in the broken-symmetry trap and a central $Q$ ball on the sample container axis (1 rad/s rotation, $T=0.13T_c$). During the decay both eigenenergies (frequencies) increase, and when they become close enough, the condensates merge. This process of merging of two $Q$ balls into a single one is demonstrated in two different experiments conducted under similar conditions. The panel on the left shows a straightforward merger, whereas on the right the peripheral $Q$ ball goes through a metastable state: Just before the merger its energy is higher than that of the central $Q$ ball. The metastability of the $Q$ balls obtained by interaction of charged and neutral fields is discussed in Ref. [3]. Close to $t=0$ some exited levels in the central trap are visible at higher frequencies.

Figure 6

$Q$ ball in ${}^3\mathrm{He}-B$ as a MIT hadron bag. (a) The BEC of magnons (magenta circles) in the limit of large $N_{\mathrm{M}}$ in a box cavity, which is void of neutral field and filled with the charge field $\mathrm{\Phi }\left(\mathbf{r}\right)$. (b) Simplified MIT bag model of hadrons, where the quarks forming a hadron are confined within a blob of a false-vacuum void of the actual vacuum of the QCD field.

Figure 7

Illustration of a conventional (left) and an unconventional (right) spontaneous breaking of global continuous symmetry: In the conventional situation, the potential acquires the form of a Mexican hat but remains axisymmetric. The trapped particle(s) becomes localized in one of the degenerate points in the valley of the potential, thus breaking the $U\left(1\right)$ symmetry. For the magnon $Q$ ball the situation is different: The Mexican hat potential itself is unstable towards breaking of axial symmetry due to the self-trapping effect. The $\zeta$ field conforms to the movements of the magnon $Q$-ball soliton ($\mathrm{\Phi }$ field).