Susan, we certainly sympathize with your situation. Almost all of us have had incompatibility problems with our cats from time to time. It's easy to interpret the sort of behavior that Chilli is exhibiting as bullying, but they're cats, not people. We have to look at their behavior that way and not interpret it in terms of human behaviors. There is no such thing as "bad" behavior in a cat. There is only the behaviors that humans desire from them and the behaviors we dislike.
The first step toward resolving this conflict is to stop blaming Chilli. Something in his world is causing him to react this way. He's just as unhappy as Polly, if not more so. It can often seem like the aggressor is the cause of the problems in a situation like this, but every relationship is a two-way street.
Our question is always, "How were the cats introduced?" A poor introduction of putting them together and letting them work it out on their own can certainly create conflict. For more on introductions, see this post.
The main key to a happy multi-cat household is for the caregiver to provide adequate resources. By that, we mean safe places to eat, eliminate, and rest without challenge. Cats are highly territorial, so if there is a deficiency of resources, there are bound to be conflicts. To keep your multi-cat household settled, you'll need to make sure the cats have what they need. You need to provide at least one more litter box than there are cats, and the litter boxes must be in different rooms. You should also provide each cat with their own set of food and water bowls that are far enough apart that the cats can't see one another when eating. You can also help them by adding cat trees to insure that there are plenty of prime real estate spots for each cat to claim. Vertical spaces and window spaces are both especially important. You'll also benefit from having several good scratching posts.
With Chilli and Polly, resources may indeed be a factor, but based on your description of them, Susan, we suspect that there's also a personality mismatch at work. As the younger cat, Chilli has probably been more playful and may have actually gotten his feelings hurt when Polly wouldn't indulge him at playtime. In situations like yours, the newcomer cat will often bond with the resident cat more than with the resident humans. If Chilli was rebuffed by Polly early on, he may now be holding a grudge against her.
So how do we change this behavior? First, let's talk about what NOT to do. Do not, under any circumstances, punish or reprimand either cat. All that will teach the cat is that you are not to be trusted. He or she is simply doing what a cat does in the given situation. There should never be any punishment. Never.
So what DO you do? You must realize that there is no quick fix. Imagine having an angry divorced couple living with you under the same roof and you'll have some idea of how long it may take to get the two cats to tolerate one another. That's right--The goal isn't to have them become besties. It's to have them tolerate one another and live under the same roof in harmony.
First, sort out any issues with resources, as listed above. Make sure both cats have plenty of perches in different spots in your home. Make sure their feeding areas and litter boxes are separated. Make sure your family doesn't try to pet both cats at one time. Separate them and have one family member play with one while another family member plays with the other.
It's especially important that Chilli get lots of extra playtime to help him work out some of that young cat energy. If you notice that Chilli is asking Polly to play, that's your cue to intervene with one of Chilli's favorite toys. Redirect his play behaviors to the humans in the family and he'll be much happier. And don't forget that Polly needs playtime too. She just may not ask for it so vociferously. Don't try to play with them at the same time, but if they end up playing together harmoniously, don't discourage it.
If, after a month or so of this behavior rehab program you still see very little in the way of results, you may need to step back and ask yourself if you need to resort to the third cat approach. Many times when there's an energy conflict between two housemates, a third cat of the proper disposition can help dispel any conflicts by bridging the needs of the other two.
If Chilli is indeed a high-energy, high-valiance cat and Polly is a low-energy, low-valiance cat (valiance is just a fancy word for courage), adopting a new kitten younger than both who is of medium valiance can actually help the entire family of cats to get along by fulfilling key roles for both. The ASPCA has implemented a feline personality evaluation for shelters that they call Feline-ality. Their goal is to have shelters adopt out more animals by matching a cat's personality to the new caregiver's needs and desires. Here's a quick video about the program that was prepared for member shelters.