Coherent Magnon Optics in a Ferromagnetic Spinor Bose-Einstein Condensate

We measure the dispersion relation, gap, and magnetic moment of a magnon in the ferromagnetic $F=1$ spinor Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$. From the dispersion relation we measure an average effective mass $1.033(2{)}_{\text{stat}}(10{)}_{\mathrm{sys}}$ times the atomic mass, as determined by interfering standing and running coherent magnon waves within the dense and trapped condensed gas. The measured mass is higher than theoretical predictions of mean-field and beyond-mean-field Beliaev theory for a bulk spinor Bose gas with $s$-wave contact interactions. We observe a magnon energy gap of $h\times 2.5(1{)}_{\text{stat}}(2{)}_{\mathrm{sys}}\mathrm{Hz}$, which is consistent with the predicted effect of magnetic dipole-dipole interactions. These dipolar interactions may also account for the high magnon mass. The effective magnetic moment of $-1.04(2{)}_{\text{stat}}(8{)}_{\mathrm{sys}}$ times the atomic magnetic moment is consistent with mean-field theory.

Figure 1

Magnon interferometry. (a) We initialize the system with a standing wave of magnons, in which the magnetization vector is periodically tilted in space (top row). For small tilt angles $\theta$, the magnon density is proportional to the atomic density in the $|m_F=0⟩$ hyperfine state, shown in images. Here, $k=2\pi /(15.4\mu \mathrm{m})$, and the scale bar is $50\mu \mathrm{m}$. (b) From this initial state, the spatial modulation in the magnon density (open black circles) oscillates in time at twice the recoil frequency, $2[\omega (k)-\omega (0)]$ (fit, solid red line). (c, bottom) The dispersion relation versus $k$. Least-squares fit of the contrast oscillation frequency from single (filled black circles) or several values of $\theta$ (open blue squares) are corrected for frequency shifts due to the finite magnon populations (such a frequency shift is shown in the inset, along with a linear fit) and due to the trap curvature. The fitted quadratic dispersion relation (black dashed line) lies below the free-atom dispersion relation (solid red line). (c, top) Fractional deviation of data from the free-atom result. Data are fit to quadratic ($k^2$) (black dashed line) and power law ($k^{\alpha }$) models for $\alpha =2.02(1{)}_{\text{stat}}$ (orange dot-dashed line). The error bars show $1\sigma$ statistical uncertainty and do not include the $\sim 1\%$ systematic uncertainty.

Figure 2

The density-dependent energy shift $\mathrm{\Delta }(n)$ of uniform ($k=0$) magnon excitations is measured by observing the Larmor precession of a ferromagnetic spinor condensate whose magnetization is tipped uniformly by a variable angle $\theta$ from the direction of an applied (in plane) magnetic field. The spatially dependent Larmor precession frequency is deduced from measurements of one component of the transverse magnetization after $\tau =100$ to 300 ms of precession. (a) The frequency shift varies quadratically in position, with a curvature $f_{xx}=d^{2}f/d{x}^2$ (b) that varies with $\cos \theta$. We identify the curvature at $\theta =90°$ with the spatially inhomogeneous magnetic field. (c) A pixel-by-pixel analysis of the $\theta$-dependent frequency shift (left) shows a magnon energy gap map with the same geometry as the in-plane density of the condensate (right). The field of view is 125 by $265\mu \mathrm{m}$. (d) A comparison of the gap and in-plane density of each pixel shows the expected linear trend. Without magnetic dipole-dipole interactions, the gap would be exactly zero.

Figure 3

(a) Magnons are created as a small wave packet (blue) within the larger condensate (gray). A magnetic field gradient accelerates the magnons, here to the lower left. The field gradient is independently measured with a Ramsey sequence involving a third hyperfine state [26]. The region of interest box is 52 by $80\mu \mathrm{m}$. (b) Fits of position with time for three values of the magnetic field gradient. The measurement noise is heteroscedastic in time due to random center-of-mass motion of the condensate. (c) The measured acceleration versus gradient (dashed black line) yields a magnetic moment consistent with the free-atom, $s$-wave mean-field theory (solid red line).