One of my most frequent training questions is always "how do you teach a horse to lie down?". My method takes time, but it is completely humane and only requires a lead rope, a dressage whip and some carrots!
Before you teach a horse to lie down, you need to teach them how to bow! To do this, stand on the left side of your horse. Under their stomach, tap the right leg with the handle end of your whip, very lightly. Eventually, they will get annoyed and pick up their foot. When they do so, reward them immediately. After your horse is taught to pick their foot up on command (just takes lots of repetition), you can take a treat and hold it toward the ground. With their leg still being held up, they should ease down into the bow position. You may have to hold their leg up, at first, or continue to tap it, so they hold it up, themselves. Just make sure you are always in a safe position around your horse. Keep doing this over and over and eventually your horse will bow when his right leg is tapped. Give your horse his treat when he is down in the bow position, and verbally praise him, as well. This will make them want to stay down, longer.
Once your horse has mastered the bow, you can move on to lying down. First, take your lead rope and throw over the right side of your horse (as seen in the header photo). Ask your horse to bow, and then tap on their left leg- your horse will, most likely, want to go into the kneeling position to escape the annoyance of the whip. This may take a few times for your horse to understand, so be patient! When your horse is in the kneeling position, pull your lead ever so slightly towards you. this will turn their head, and make them want to lie down. he first time your horse lies down, they will, almost certainly, get up immediately (with this method). It is important that, for the brief period they are down, to reward them immensely. Give them treats- verbally praise them, etc. Make a very big deal out of it. Horses are prey animals, and lying down in broad daylight at the mercy of a "predator" really isn't their thing, so this trick takes immense amounts of trust on your horse's part. Rewarding them and reassuring them is vital.
I hope this helps, and if you need any clarifications or have anymore questions, leave them in the comments section and I will answer as soon as I can!