You can automatically save the time of randomization, or any other status change, using the @CALCDATE action tag. This tag lets you calculate a new date that is offset from an existing date. In this case, the existing date is “now” and the offset is zero. The following text can be entered as the Action Tags/Field Annotation for a text box field type. The variable randomized_arm should be replaced with the randomization field as configured on the project, and randomized_date should be replaced with the variable name that is tracking the datetime of randomization.

@READONLY
@CALCDATE(if([randomized_arm] != '' and [randomized_date] = '', 'now', [randomized_date]), 0, 'd')