What is the proper event for input="text" - ajax

Question

I have an input field which is start_time and end_time and these fields are formatted with the data-format HH:mm:ss PP.
What I want to do is when the user is done entering the data, then I'll do some AJAX to calculate the time.
But, I'm a little bit confused because I don't know what is the proper event for the input fields. I tried keydown,keyup,focus but still doesn't work.
Here is the input field start_time:
<div class='input-append' id='datetimepicker2'>
<input data-format='HH:mm:ss PP' type='text' name="start_time" id="start_time" value="<?php echo set_value('start_time')?>"/>
<span class='add-on'>
<i data-date-icon='icon-calendar' data-time-icon='icon-time'>
</i>
</span>
</div>
</div>
<script type='text/javascript'>
$(function() {
$('#datetimepicker2').datetimepicker({
language: 'en',
pick12HourFormat: true,
pickDate: false
});
});
</script>
And for the end_time:
<div class='input-append' id='datetimepicker3'>
<input data-format='HH:mm:ss PP' type='text' name="end_time" id="end_time" value="<?php echo set_value('end_time') ?>"/>
<span class='add-on'>
<i data-date-icon='icon-calendar' data-time-icon='icon-time'>
</i>
</span>
</div>
</div><script type='text/javascript'>
$(function() {
$('#datetimepicker3').datetimepicker({
language: 'en',
pick12HourFormat: true,
pickDate: false
});
});
</script>

Answer 1

Go for onblur, sothat when the user finish entering the data in input field and gets focus out of the field the event gets triggered.
edited according to comments below
I guess you are using bootstrap datepicker in your code. If not so, please mention the datpicker or the js library that you are using.
In the case of bootstrap date picker yu can look at the changedate event for your task which will get triggered each time a change is made on your datepicker value. Also you can check the changeYear && changeMonth events also which are available.
here is a demo fiddle using the code you provided which will trigger an alert whenever a change is made on the corresponding datepicker. You can replace it with whatever code you want to execute on change.
sample code snippet
$('#datetimepicker2').datetimepicker({
language: 'en',
pick12HourFormat: true,
pickDate: false
}).on('changeDate', function (e) {
alert("first one changed"); // replace with your code block
});;