Bragg spectroscopy of vortex lattices in Bose-Einstein condensates

We have measured the velocity field of a vortex lattice within a sodium Bose-Einstein condensate using Bragg scattering. The phase gradient of the macroscopic wave function was mapped into the spatial structure of the diffracted atom cloud, allowing for the single shot measurement of the rotation parameters. A combination of spectral and spatial information yields a complete description of the superfluid flow, coarse grained over the lattice structure, including direct and independent measurements of the rate and sense of rotation. Signatures of the microscopic quantum rotation have also been observed.

Figure 1

(Color online) Bragg scattering from vortex lattices. (a) Atoms in a TOP trap are made to rotate about the $z$ axis by phase control of the transverse fields produced in coil pairs $x$ and $y$. This procedure forms a lattice of quantized vortices (b), which is imaged in TOF after $40\mathrm{ms}$. The field of view is $0.6\mathrm{mm}$. The Bragg beam containing frequencies $\omega$ and $\omega +\delta$ is applied along the $x$ direction and retroreflected as shown in (a).

Figure 2

(Color online) Vortices probed by Bragg scattering. Outcoupled atoms (to the far right and left of each image) showed no particular structure for nonrotating clouds (a); whereas, from vortices (b) and (c), the outcoupled atoms were tilted according to the direction of rotation. Also in (a), the higher atomic density causes collisional-induced heating and makes the cloud look more diffuse. Images (a)–(c) were taken at $10\mathrm{ms}$ TOF. The tilt angle increases with respect to TOF, as shown in (e). Each pair of Bragg frequencies is resonant with a thin strip of atoms parallel to the $x$ axis, as illustrated in (d). All images were taken at $\delta =102\mathrm{kHz}$.

Figure 3

(Color online) Mapping the velocity field of a vortex lattice. Measuring the center-of-mass locations of the right and left diffracted atoms [$(X_R,Y_R)$ and $(X_L,Y_L)$, respectively, as defined in Fig. 2] in the $x\text{−}y$ plane allows us to reconstruct the velocity profile of the rotating cloud. For negative detuning, as in (a), $\delta -{\delta }_0=-10\mathrm{kHz}$; therefore, $Y_L>0$ and $Y_R<0$. For positive detuning, the opposite is the case, as in (b) where $\delta -{\delta }_0=+10\mathrm{kHz}$. The difference $X_R-X_L$ is mapped as a function of frequency in (c), while $Y_R-Y_L$ is plotted in (d). Finally, in (e), we have plotted $Y_R-Y_L$ vs $X_R-X_L$ for the data in (c) and (d). The data in (e) are a direct map of the position ($Y$ coordinate) vs velocity ($X$ coordinate) distribution within the vortex lattice, and shows a linear relationship consistent with rigid-body rotation. The lines are fits to the data.

Figure 4

(Color online) Spectroscopy of the vortex lattice. The number of Bragg diffracted atoms is measured as a function of the frequency $\delta$, for samples with (solid circles) and without (squares) vortices. A significant broadening of the spectrum is observed in the presence of vortices due to the broad velocity distribution caused by the rigid-body rotation. The solid curve represents the best Lorentzian fit to the data, while the dashed curve refers to the model described in the text.

Figure 5

(Color online) Microscopic structures seen in Bragg scattered clouds. A $T_B=20\mu \mathrm{s}$ pulse $\delta =102\mathrm{kHz}$ was used in (a), in a single vortex condensate (at the left side of the image). The sequences (b)–(e) show detailed images of outcoupled clouds for rapidly rotating (full lattice) condensates. Experimental parameters are $T_B=75\mu \mathrm{s}$, $\delta =102\mathrm{kHz}$ in (b); $T_B=60\mu \mathrm{s}$, $\delta =102\mathrm{kHz}$ in (c); $T_B=60\mu \mathrm{s}$, $\delta =122\mathrm{kHz}$ in (d); and $T_B=50\mu \mathrm{s}$, $\delta =115\mathrm{kHz}$ in (e). For all images, $\mathrm{TOF}=20\mathrm{ms}$. An interesting combination of the quantum (microscopic structures) and classical (rigid-body tilting) superfluid flow is shown in (e).