The h Bar

So, it seems we should be able to learn about the power spectral density of fluctuations in $\delta \phi$ by looking at the power spectral density in $P$.

Here's one of the big wrinkles:

I can't measure $P$ at an instant of time.

All I can do is projectively measure my quantum spin. When I do that I get either $|0\rangle$ or $|1\rangle$, with probability determined by that expression for $P$.

The data I get is a string of measured results, e.g.:
$\{ |0\rangle, |1\rangle, |1\rangle, \ldots \}$

Each one measurement coming in on some uniform time grid determined by how fast I can repeat the experiment.

@ChrisWhite do you understand the essentials of the problem?

@ChrisWhite I could go on and describe how to make progress on the problem, but these are the essentials.

user54412

@DanielSank I think so. Let me recap: For an instance of this system, you can choose a time $\tau$ and project to $\lvert 0 \rangle$ or $\lvert 1 \rangle$ at $\tau$, where the probability is known to be governed by $\delta \omega$ in the given formula. Is that correct?

DanielSank

Yes. Note that $\tau$ is the procession time. There is another time here which I called $t$ (way up above) denoting time at which you start the instance of the experiment.

user54412

Is the noisy field different (but with the same PSD) in each instance?

DanielSank

7:12 PM

The idea is to do the experiment over and over on a grid of $t$ points ($\tau$ being the same on each) to build a time series of data.

user54412

I see. Fix the delay, and keep repeating in order to get a handle on $\delta \omega$.

DanielSank

@ChrisWhite The noisy field contains rather low frequency fluctuations. It has a roughly $1/f$ power spectrum. The duration of each experiment is roughly $\tau = 500 \, \text{ns}$.

@ChrisWhite Correct.

The total time of the experiment could be say a million repetitions each of duration roughly $\tau$.

In other words, each measurement is spaced by roughly $\tau$, and we have some long string of such measurements.

It's the same physical spin each time so it's sampling the same noisy field. That field surely has a correlation time longer than $\tau$.

So why is this hard:

1) You measuring a noisy variable using what's essentially a 1-bit detector, which is a funny idea that we're not used to.

2) The measurement result is probabilisticly related to the underlying stochastic process, which is weird.

3) The detector is highly nonlinear because the probability distribution for getting the two results involves a $\sin$ function.

Despite all of this weirdness I can reduce the problem to an infinite product that I have no idea how to solve :P

@ChrisWhite I posted a simplified description of this to the signal processing site and after several weeks haven't gotten a single answer.

user54412

@DanielSank Indeed as a theorist I imagine all measurements are infinite-precision reals :P

user54412

@DanielSank Bayesianism to the rescue! Or was it frequentism?

DanielSank

@ChrisWhite Simply not the case here. Quantum two level systems have 1-bit resolution.

In other news I set up a ping-pong scoreboard website for my group.

It's a lot of fun. If anyone else wants to spin up an instance of it let me know and I'll help you set it up.

user54412

7:19 PM

@DanielSank This is interesting. The information you get from a measurement depends on the phase of the $\sin$ argument. With the wrong combination of $\tau$ and $\delta \omega$, you learn nothing at all.

DanielSank

@ChrisWhite Correct!

Yes indeed. You've definitely grasped one of the odder aspects of the problem.