Electrically-Small Rydberg Sensor for Three-Dimensional Determination of rf k-Vectors

We present an electrically-small Rydberg atom electric field sensor with the ability to extract the three-dimensional $k$-vector of an elliptically polarized radio frequency (rf) field. In most mediums, the $k$-vector (or wave vector) provides the direction of propagation of an electromagnetic wave. Our method uses a field vector measurement at a single point in space and is thus compatible with a sensor volume that is arbitrarily small compared to the carrier wavelength. We measure the $k$-vector of a circularly polarized signal field with average absolute errors in the polar and azimuthal angles of $33\text{ mrad}$ and $43\text{ mrad}$ respectively, and statistical noise of $1.3\text{ mrad}\mathrm{/}\sqrt{\text{Hz}}$ and $1.5\text{ mrad}\mathrm{/}\sqrt{\text{Hz}}$. Additionally, we characterize the performance of the sensor as a function of the ellipticity of the input field and the size of the sensing region. We find that the sensor works over a broad range of ellipticities and validate that an electrically-small sensing region is optimal.