Millimeter-wave precision spectroscopy of d-d transitions in potassium Rydberg states

We measure two-photon millimeter-wave transitions between nd$_j$ and (n+1)d$_j$ Rydberg states for 30 $\le$ n $\le$ 35 in $^{39}$K to an accuracy of 5$\times$ 10$^{-8}$ to determine high-n d-state quantum defects and absolute energy levels. $^{39}$K atoms are magneto-optically trapped and cooled to 2-3 mK, and excited from ground state 4s$_{1/2}$ to nd$_{3/2}$ or nd$_{5/2}$ by frequency-stabilized 405 nm and 980 nm external-cavity diode lasers in succession. The magnetic-field insensitive nd$_j\rightarrow $ (n+1)d$_j$ $\Delta$m $=0$ transitions are driven by a 16 $\mu$s-long pulse of mm-waves before the atoms are selectively ionized for detection. The (n+1)d population is measured as a function of mm-wave frequency. Static electric fields in the MOT are nulled in three dimensions to eliminate DC Stark shifts. The two-photon transitions exhibit small but measurable AC Stark shifts in the resonance frequencies. We determine the field-free intervals both by extrapolating a sequence of measurements made as a function of mm-wave power to zero and directly without extrapolation by applying Ramsey's separated oscillating fields method. Our results give quantum defects for the high-n states that are an order of magnitude more accurate than earlier measurements of these quantities.