Description
If you do not want to merge all changes from one branch to another (using git merge), you can cherry-pick some of the commits (either single commits or multiple commits at once). See cherry-pick documentation here or this reply, that elaborates on cherry-picking a range of commits.
As far as I know, there is no direct option to directly merge commits from a certain date, but you can just use the method described here to find the commit created on a certain day and use that commit as the start commit for your cherry pick.
Be aware, however, that cherry-picking does apply the picked commits' changes, but creates new commits. Thus, when you ever merge branch_one and branch_two in the future, you will have multiple commits introducing the same changes.
Example
On branch_two:
git log --after="2013-11-12 00:00" --before="2013-11-12 23:59"
Output: 
commit 5f3be6775d78k28ee0e4f55c05ec897de3f02360
Author: someone <some@one.com>
Date:   Wed Mar 22 17:10:48 2017 +0100
    Commit message
On branch_one:
git cherry-pick 5f3be6775d78k28ee0e4f55c05ec897de3f02360~1..<id of the latest commit on branch_two>
Edit: Note the ~1 behind the first commits' id. This is necessary, since cherry picking multiple commits will 
[...] not cherry-pick A, but rather everything after A up to
  and including B. – J. B. Rainsberger
See here.