Inverse problem approach to evaluate quadratic Zeeman effect for an atomic fountain frequency standard KRISS-F1

The evaluation of the quadratic Zeeman shift in atomic fountain frequency standards requires information on the magnetic field distribution along the drift region. Plotting Zeeman frequencies against various launching heights provides the temporally averaged magnetic field values seen by the flying atoms. Using those data, we were able to deduce a reasonable field map by taking the approach of solving an inverse problem. We employed a regularization method after establishing a linear set of equations. The analysis enabled us to evaluate the quadratic Zeeman shift and its uncertainty against various launching heights. Our method encompasses and generalizes the deconvolution method. This paper covers the mathematical tricks to solve the inverse problem and its application to the atomic fountain frequency standard KRISS-F1.