Suppose I put you in front of a light bulb and tell you I need to know what fraction of the time it is on. Easy enough, right? You can sit down and watch it, note the times it turns on and off, and the rest is math. But what if I were to put the light in another room and close the door. What would you do then?
This situation is one that I’ve spent almost a decade considering. How do we know what’s there without looking? You could, of course, open the door and watch directly. But a lot of the time, that’s what we would call a destructive measurement — the act of measuring the light bulb by opening the door spoils another measurement, perhaps the actual experiment that we are doing in that room. There’s nothing wrong with leaving the door open and watching the light, but we can’t do the experiment until it’s closed.
So let’s consider a possible solution, knowing just this. What if we just opened the door for one day out of every four? We could see how long it was on that day and keep a running tally. Then we could do some statistics and figure out a range like “the light is on for 4 hours, plus or minus 3, each day.” We only reduce the experiment’s runtime by 25%, and we can be relatively sure about how often the light is on.
“Ok,” you say, “that’s fine, but what about putting light detectors in the room and running them to a light outside the room you can watch instead?” Now we’re starting to get somewhere. We can’t look directly at the lightbulb, but what about some sort of proxy measurement? You can sit and watch the lights on the outside of the room, record their on/off times, and everything is great, right?
Well, not quite. Close, but not quite. How can you be sure that the lights outside the room are on at the same time the light inside the room is on? Let’s go back to the idea of opening the door once in a while. Except now, while the door is open, you’re watching both of the lights. After you’ve observed for a day and written down when both turned on and off, we’re going to do a bit more math and find out how often they were either both on or both off. This math is called the correlation between the two measurements. Maybe it’s 90%. Now for those next three days, as we watch the outside light, we can say, “there’s a 90% chance the inside light is in the same state as the outside light.”
Measuring the second light’s correlation with the first light provides us with additional data while the door is closed. These data will lead us to a better interpolation of what’s going on in the room. Interpolation is a mathematical process that aims to fill in what’s going on between two (or more) measured points. I’ll write another post about interpolation that gets a bit more into the nitty-gritty.
For context, what I have just described is very close to how we do the magnetic field measurement and analysis on the highly anticipated recently-published Fermilab Muon g-2 experiment. We have a very good magnetometer that can precisely measure the field (the trolley), but we can’t be storing muons while we are using it. So, what we do is use the trolley about once every four days. We also have a suite of other magnetometers that aren’t quite as good, but they can take measurements the whole time, even when storing muons. We use these other magnetometers, the fixed probes, to fill in the gaps (to interpolate) between the trolley runs and better determine the magnetic field.