Using update field does not work for date (it works for string).
So the solution I’ve put in place is :
$this->birthdate = date( "Ymd", $this->birthdate );
update_post_meta ($this->ID, 'birthdate', $this->birthdate);
update_post_meta ($this->ID, '_birthdate', 'field_53e11dd3841f6'); // CONSTANT KEY