Star I'd listen to all the avdice here as it's all on the money. You look like you have a good amount of size on you, if you were to train say back or legs on 40 grams of carbs, by your 3rd set you'd feel like you'd rather be dead. Carb cycling is something to try, but if you thrive on structure, just limit your carbs to 150-200 grams when cutting, while monitoring your overall caloric intake and keeping it 500 below maintenance. You want your protein high (obviously), but at the end of the day it's still cals burned versus cals ingested (and BMR too). As Ulter said, these extreme atkins type diets are simply not suited for weight training. Just eat moderate amounts of clean carbs and if you want a little edge, no carbs after 4-5 pm.
Just to give you an idea, many guys I've competed against and who I know from here and there actually use insulin all throughout their pre-contest cycles up to a few days out from the show (I personally do not). So this obviously means they're eating a good amount of carbs as well. Biochemically, excess insulin is the exact opposite of what you want for fat loss, but obiously (in addition to other compounds) it is not the enemy if your diet is on and most importantly, you know your body. Another personal example I can share (I know you probably could care less but i'll tell it anyway lol), is that a while ago I started drinking a carb drink during my most intense workouts (for me that's back day and leg day). It's 440 cals of just pure carbs, 110 g if I'm not mistaken. For years I never touched these things as I thought they were just total crap and would make you fat because of all teh carbs. However, this mix only has 30 grams of simple sugars, which for me at this stage and size is pretty much nothing. The remaining 80 grams are more complex. Long story short, my 20+ set back and leg workouts are much more manageable now, and I haven't gained an ounce of fat. Basically the harder you can train, the more you will turn your body into a fat burning furnace for hours and hours post-workout.
Sooooo, the moral of the story is that carbs are not the enemy, even in a cut. Just don't go all Jay Cutler on us and "diet" on 600 grams of carbs..