Condensing Magnons in a Degenerate Ferromagnetic Spinor Bose Gas

We observe the quasicondensation of magnon excitations within an $F=1$ ${}^{87}\mathrm{Rb}$ spinor Bose-Einstein condensed gas. Magnons are pumped into a ferromagnetically ordered gas, allowed to equilibrate to a nondegenerate distribution, and then cooled evaporatively at near-constant net longitudinal magnetization, whereupon they condense. The critical magnon number, spatial distribution, and momentum distribution indicate that magnons condense in a potential that is uniform within the volume of the ferromagnetic condensate. The macroscopic transverse magnetization produced by the degenerate magnon gas remains inhomogeneous within the $\sim 10\mathrm{s}$ equilibration time accessed in our experiment, and includes signatures of Mermin-Ho spin textures that appear as phase singularities in the magnon quasicondensate wave function.

Figure 1

Magnons in a flat potential. (a) The effective potential $V_{\mathrm{eff}}$ for magnon excitations is the sum of the harmonic trap potential (red) and the repulsive potential from the inhomogeneous ferromagnetic condensate (blue, maximum potential of ${\mu }_{-1}$), shown, schematically, in one dimension. (b) The column density of a low temperature nondegenerate magnon gas (bottom column density image) is large within the volume of the $m_F=-1$ condensate (top column density image). Line densities (integrated within black rectangles) are fitted to predicted critical density in a harmonic trap (red), constant density within condensate volume (green, best fit for magnon distribution), or Thomas-Fermi condensate distribution (blue, best fit for the $m_F=-1$ condensate). (c) Images and integrated line profiles (purple) of magnon momentum distribution below (left), at (center), and above (right) critical number for magnon condensation. Profiles are fitted to Bose distributions $\propto g_{\alpha }[z(p)]$ with $z(p)$ defined in the text and $\alpha =\{1,1.5,2\}$ (black, blue, red lines, respectively), and the distribution expected in $V_{\mathrm{eff}}$ (green).

Figure 2

The measured critical magnon number $N_{\mathrm{mag},c}$ (black markers) is compared to the expected value (blue, red, purple points, respectively) for magnons in a harmonic, box (size defined by condensate volume), and $V_{\mathrm{eff}}$. Shaded lines are guides to the eye. Error bars on model predictions account for statistical uncertainties in experimental parameters. Inset: The magnon condensate number, shown for $k_B{T}_f/{\mu }_{-1}=0.7$, is the integrated residual above the fits to the noncondensed magnon momentum distributions using one of four fit functions: $g_{\alpha }[z(p)]$ with $\alpha =1$ (squares), 1.5 (triangles), or 2 (circles), or the predicted distribution in the effective potential (diamonds). A linear fit shows the magnon condensate number increasing linearly above $N_{\mathrm{mag},c}$; error bars are statistical. Extracted measured critical magnon numbers from the four fitting functions are shown in the main plot (diamonds, squares, triangles, circles with a decrease in gray scale).

Figure 3

(a) Inhomogeneous transverse magnetization measured for four experimental repetitions, in coldest samples and with $\sim 10\%$ magnon fraction, with hue and brightness representing orientation and amplitude, as indicated by color wheel. Identified phase singularities are marked by white circles. The inset highlights one such singularity, with the dashed white line showing a path along which the magnon order parameter phase winds by $2\pi$. (b) Spatial Fourier power spectral density (PSD) of transverse magnetization averaged over 20 experimental repetitions, for magnon fraction increasing from 7% (below critical value) to 10% (above critical value). White dashed circle marks $2\pi /k_r$. Red dashed circle represents the imaging resolution.