Background: Octupole-deformed nuclei, such as that of $^{225}$Ra, are expected to amplify observable atomic electric dipole moments (EDMs) that arise from time-reversal and parity-violating interactions in the nuclear medium. In 2015, we reported the first proof-of-principle measurement of the $^{225}$Ra atomic EDM. Purpose: This work reports on the first of several experimental upgrades to improve the statistical sensitivity of our $^{225}$Ra EDM measurements by orders of magnitude and evaluates systematic effects that contribute to current and future levels of experimental sensitivity. Method: Laser-cooled and trapped $^{225}$Ra atoms are held between two high voltage electrodes in an ultra high vacuum chamber at the center of a magnetically shielded environment. We observe Larmor precession in a uniform magnetic field using nuclear-spin-dependent laser light scattering and look for a phase shift proportional to the applied electric field, which indicates the existence of an EDM. The main improvement to our measurement technique is an order of magnitude increase in spin precession time, which is enabled by an improved vacuum system and a reduction in trap-induced heating. Results: We have measured the $^{225}$Ra atomic EDM to be less than $1.4times10^{-23}$ $e$ cm (95% confidence upper limit), which is a factor of 36 improvement over our previous result. Conclusions: Our evaluation of systematic effects shows that this measurement is completely limited by statistical uncertainty. Combining this measurement technique with planned experimental upgrades we project a statistical sensitivity at the $1times10^{-28}$ $e$ cm level and a total systematic uncertainty at the $4times10^{-29}$ $e$ cm level.
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