  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 $1.4\times10^{-23}$ $e$ 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 $1\times10^{-28}$ $e$ cm level and a total
systematic uncertainty at the $4\times10^{-29}$ $e$ cm level.
