json_decode returns a PHP data structure. If you want to serialise that to a JavaScript data structure you have to pass it through json_encode (and then actually echo the string that it returns).

Note that json_encode outputs a JavaScript data structure that is safe for injecting into a <script> element in an HTML document. Not all JSON is safe to do that with (PHP adds additional escape sequences, and will transform plain strings, numbers, null values, etc (which aren't legal JSON on their own).

Note that there is also no point in creating a new array and assigning it to arr if you are going to immediately assign something else to arr.

Also note that '$json_object' will give you a string starting with the $ character and then the name of the variable. Single quoted string in PHP are not interpolated.

var arr;
arr = <?php echo json_encode(json_decode($json_object,TRUE)); ?>;

Also note that this JSON:

{"product_id":"62","product_quantity":"65"}

Will transform in to a PHP associative array or a JavaScript object (which is not an array).

So given this PHP:

<?php
    $json_object = '{"product_id":"62","product_quantity":"65"}';
?>
<script>
    var arr;
    arr = <?php echo json_encode(json_decode($json_object,TRUE)); ?>;
    alert(arr.product_id);
</script>

You get this output:

<script>
    var arr;
    arr = {"product_id":"62","product_quantity":"65"};
    alert(arr.product_id);
</script>

Which alerts 62 when run.

The standard JavaScript way to do this would be to use JSON.parse:

var myArray = JSON.parse(someJSONString);

For compatibility with older browsers that lack a built-in JSON object, jQuery has its own method:

var myArray = jQuery.parseJSON(someJSONString);

Such method is deprecated as of jQuery/3.0.

The problem that you're running into is that the output of json_encode is not meant to be used directly as a string in JavaScript.

json_encode outputs a usable JavaScript object:

<?php
$foo = new stdClass();
$foo->test='hello world';
$bar = new stdClass();
$bar->foo = json_encode($foo);
$encoded_string = json_encode($bar);
?>
var a = <?php $encoded_string ?>;
console.log(a.foo); // produces '{"test":"hello world"}'

If you want to needlessly parse the JSON output from a string value, you simply need to double encode $encoded_string:

<?php
$foo = new stdClass();
$foo->test='hello world';
$bar = new stdClass();
$bar->foo = json_encode($foo);
$encoded_string = json_encode(json_encode($bar));
?>
var aStr = <?php $encoded_string ?>;
var a = JSON.parse(aStr);
console.log(a.foo); //same as before

Of course, you should avoid using server side languages to generate JavaScript code, instead set up the data as either a data-* attribute or as a JSON source that can be requested with AJAX.

When the data is requested from the server (or from the attribute) it will be as a properly escaped JavaScript string, which is where JSON.parse will be necessary to parse the object.