Electrically small Rydberg sensor for three-dimensional determination of radio-frequency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi></mml:math>-vectors

We present an electrically small Rydberg atom electric field sensor with the ability to extract the three-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>k</a:mi></a:math>-vector of an elliptically polarized radio-frequency field. In most mediums, the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>k</c:mi></c:math>-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 with the carrier wavelength. We measure the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>k</e:mi></e:math>-vector of a circularly polarized signal field with average absolute errors in the polar and azimuthal angles of 33 and 43 mrad, respectively, and statistical noise of 1.3 and 1.5 mrad/<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msqrt><g:mi>Hz</g:mi></g:msqrt></g:math>, respectively. 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.