Guide to the ACME EDM Experiment: Differential Measurements to the Extreme

In our overview of the ACME experiment, we outlined the broad method of the experiment: we measure the deformation in the shape of the electron by performing spin precession on the molecule ThO. In the post about atomic structure, we outlined the basics of energy levels in a molecule, as well as that of ThO itself. We also touched briefly on the concept of a differential measurement. Today we’ll expand our discussion on differential measurements and look at how they are crucial in making ACME such a precise experiment.

Basics: One Variable

A differential measurement is useful in the following situation: when we want to measure a certain quantity, but the device we’re measuring it with can only give back that quantity plus some constant. In this case, we’re measuring a shift in the molecule’s energy H caused by the electron EDM:

(1)   \begin{align*} H = H' + d_e E_{eff}. \end{align*}

When we measure the spin precession on our molecules, we are measuring their total energy, so we will get H along with d_e E_{eff}, although we only want to know d_e E_{eff}. Luckily, we have control over E_{eff}: we can reverse its sign from positive to negative. To separate the two, we perform the measurement twice, once with positive sign, and once with negative sign:

(2)   \begin{align*} H_1 = H' + d_e E_{eff}\\ \nonumber H_2 = H' - d_e E_{eff} \end{align*}

and then we take the difference of the two results, and divide it by 2:

(3)   \begin{align*} \frac{\Delta E}{2} &= \frac{1}{2} H_1 - H_2 \\ \nonumber &= \frac{1}{2} (H' + d_e E_{eff} - (H' - d_e E_{eff}))\\ \nonumber &= \frac{1}{2} (2 d_e E_{eff}) \\ \nonumber &= d_e E_{eff}. \end{align*}

Because we know the numerical value of E_{eff}, we can figure out the electron EDM d_e from this method.

Two spin precession measurements using different directions of E_{eff}. (The magnetic field B and other switch parameters are not shown for simplicity, as reflected in the discussion.)

The above is an example of a bare-bones differential measurement of a quantity, H, with just one variable to reverse, E_{eff}. We can more elegantly express the two equations in Equation 2 by expressing E as

(4)   \begin{equation*} H = H' + \tilde{\mathcal{E}} d_e E_{eff}, \end{equation*}

where we specify that \tilde{\mathcal{E}} takes the values +1 or -1. When \tilde{\mathcal{E}}=+1, we set the voltage knob on our electric field generator to 200 volts. When \tilde{\mathcal{E}}=-1, we set it to -200 volts. Thus we say that there is a single switch in the experiment, \tilde{\mathcal{E}}. We call \tilde{\mathcal{E}} the parity of the switch E.

Two Variables

In the ACME experiment there is actually more than just one switch. Besides \tilde{\mathcal{E}}, there is another switch, the direction of the magnetic field B, which we denote \tilde{\mathcal{B}}. Just like \tilde{\mathcal{E}}, \tilde{\mathcal{B}} can independently take the value of +1 or -1. When \tilde{\mathcal{B}} = +1, the magnetic field in the lab is pointing West, while when \tilde{\mathcal{B}}=-1, it’s pointing East. The energy we are measuring has the form

(5)   \begin{equation*} H = \tilde{\mathcal{B}} g_1 \mu_B B + \tilde{\mathcal{E}} d_e E_{eff}, \end{equation*}

where g_1 and \mu_B are constants that we just look up or measure through other means.

As there are 2 switches, there are 4 different possible configurations the experiment can take, one for each possible value of \tilde{\mathcal{N}} and \tilde{\mathcal{B}}:

    \begin{align*} \{\tilde{\mathcal{N}},\tilde{\mathcal{B}}\} = {+1,+1}\\ \{\tilde{\mathcal{N}},\tilde{\mathcal{B}}\} = {+1,-1}\\ \{\tilde{\mathcal{N}},\tilde{\mathcal{B}}\} = {-1,+1}\\ \{\tilde{\mathcal{N}},\tilde{\mathcal{B}}\} = {-1,-1} \end{align*}

How do we extract E_{eff}? On first glance, it might seem that we have to perform the experiment 4 times, once for each of the following configurations. We shall obtain the following measured values:

(6)   \begin{align*} H_1 = g_1 \mu_B B + d_e E_{eff}\\ H_2 = - g_1 \mu_B B + d_e E_{eff}\\ H_3 = g_1 \mu_B B - d_e E_{eff}\\ H_4 = - g_1 \mu_B B - d_e E_{eff}\\ \end{align*}

We can see that by calculating H_1+H_2 = 2 d_e E_{eff}, we are already able to isolate E_{eff}. There is no need to do the experiment 4 times, only twice (as before):

A series of two spin precession measurements with different values of \tilde{\mathcal{B}}, necessary to extract the electron EDM term.

Three Variables

 

In reality the situation is even more complicated: there is a third switch, called \tilde{\mathcal{N}}. This switch exists because it is possible to perform the spin precession measurement in two different atomic states. (This is a property of the omega-doublet structure of the H-state of ThO, which we discussed briefly in the post about atomic structure.) The actual form of the energy H we are measuring is

(7)   \begin{equation*} H = D_1 E \tilde{\mathcal{N}} + \tilde{\mathcal{B}} g_1 \mu_B B + \eta \mu_B E B \tilde{\mathcal{N}}\tilde{\mathcal{B}}+\tilde{\mathcal{N}}\tilde{\mathcal{E}} d_e E_{eff}, \end{equation*}

a mess which also intimidated me when I first saw it. (One can just barely identify the d_e term in there, only now that it is dependent on the product of two switches, \tilde{\mathcal{N}} and \tilde{\mathcal{E}}.)

Equation 7 is just the terms that we know of in a perfect system. There could be sources of noise or error that show up in H in a weird way. With three switches, there are many different possibilities of terms in H (eight, to be exact):

(8)   \begin{align*} H = & D_1 E \tilde{\mathcal{N}} + \tilde{\mathcal{B}} g_1 \mu_B B + \eta \mu_B E B \tilde{\mathcal{N}}\tilde{\mathcal{B}}+\tilde{\mathcal{N}}\tilde{\mathcal{E}} d_e E_{eff} \\ \nonumber +& k_1 \tilde{\mathcal{E}} + k_2 \tilde{\mathcal{E}} \tilde{\mathcal{B}} + k_3 \tilde{\mathcal{N}} \tilde{\mathcal{E}} \tilde{\mathcal{B}} + k_4, \end{align*}

where we have included every possible product of the the three switches. (Notice the presence of the term k_4, a constant which doesn’t depend on the switches at all.) If this were the case (and we don’t know for sure), we would have to take data more than just twice: we would have to take peculiar combinations of \tilde{\mathcal{N}}, \tilde{\mathcal{E}}, \tilde{\mathcal{B}} to isolate out the term containing d_e.

The Solution: Take Every Possible Combination

Because of this, to be safe, in ACME we take every possible combination of parities of the three switches \tilde{\mathcal{N}}, \tilde{\mathcal{E}}, \tilde{\mathcal{B}}, and sort out the data afterwards. Since each switch can take on two possible values (+1 or -1), this results in 8 different configurations. So if we were to graph the settings in our apparatus, it would look something like this, a sort of fractal:

Parameter settings on ACME experiment during 1 block of data, consisting of 8 measurements in total. 

We call the above one “block” of data. After finishing the 8 measurements, we’re left with 8 numbers which we can denote by H_1, H_2, …, H_8, each with its own set of associated parameter switches:

\tilde{\mathcal{N}} \tilde{\mathcal{E}} \tilde{\mathcal{B}} Value
+1 +1 +1 H_1
+1 +1 -1 H_2
+1 -1 +1 H_3
etc.

When we have all the data like this, we can sum the 8 values in different ways (subtracting some, adding others) in order to pick out the term in Equation 8 which we want. These are called parity sums. The full general expression for a parity sum is quite abstract and so we shall not quote it in full here (and it’s not that physically interesting, only some tedious math). What’s important to know is that there is a specific combination of the different H_i‘s which add up together to pick out the EDM term which contains d_e. We call this the EDM parity sum or EDM channel, and we like to call denote it by the symbol \phi^{\tilde{\mathcal{N}}\tilde{\mathcal{E}}}.1 This becomes one data point for the experiment. Thus you can say that to get one data point, we actually need to take 8 measurements and add them up in a very specific way.

Benefits of Extreme Differential Measurement: Systematic Rejection

In Equation 8 above, the d_e term contains the product \tilde{\mathcal{N}}\tilde{\mathcal{E}}. It’s important to realize what happens when we do parity sums to pick out the term containing d_e. Yes, we obtain the quantity we want (d_e), but we also subtract out any contributions to H which are not related to \tilde{\mathcal{N}}\tilde{\mathcal{E}}. This include constants not related with any switches. This means if there is a source of noise or imperfection in the system which changes the spin precession process somehow (we call these systematic errors), it will not affect the measured EDM value unless it is connected with \tilde{\mathcal{N}}\tilde{\mathcal{E}}. 

This is a benefit which is felt in the day-to-day running of the ACME experiment. Say one day one of our detectors (we have 8 of them) stops functioning such that the overall signal we get is reduced by 1/8th, and we’re unaware of that. This does not mean that we would wrongly think that the measured EDM value would shift – thanks to the data-taking scheme above, we would still identify the right value of the EDM. The same goes for random electronic vibrations from random sources in the lab or near it (and at the level of precision we’re looking at, even small disturbances can be picked up by our detectors). We’re not worried about them until there is evidence that their behavior is connected to our switching of \tilde{\mathcal{N}}\tilde{\mathcal{E}}.

In practice, the data-taking structure illustrated by the fractal-like diagram above is not the end of the story: there are even more switches than just the three that we’ve identified so far. Indeed, the 3 switches we denote above are grouped into one block of 8 measurements that constitute one measurement. But there are a total of 5 more switches that we regularly go through in our routine data-taking runs (this means 2^8 = 256 different unique parameter combinations!) It’s not that important to know what they are right now. There are even more settings that we also vary more sporadically. This is the resulting data acquisition structure graph that we actually put in our paper:

The full, fractal-like data acquisition structure diagram which was included in our long paper about the ACME I experiment.

One can see here not just the switches but the timescales required. It takes 30 minutes just to get through all the 8 “regular” switches, which we call one “superblock” of data.

Most of these adjustments to the experiment are not expected to actually contribute to H in a substantial way, but we cycle through all of them anyway in order to be safe. If it’s truly the case that a switch \tilde{\mathcal{X}} shouldn’t affect the experiment, then \phi^{\tilde{\mathcal{N}}\tilde{\mathcal{E}}} should be the same whether \tilde{\mathcal{X}}=+1 or \tilde{\mathcal{X}}=-1. In principle, \tilde{\mathcal{X}} could be absolutely anything, from things which might plausibly affect the experiment (changing the magnitude of the magnetic fields B we apply, for example), to things more dubious (turning the lights on or off during the experiment), to those which smack of magical voodoo (two people in the room while running the experiment versus one person only). The main reason why we have a finite number of switches is simply practicality and “common sense”.

Conclusions

I know that this post is pretty abstract, and many might just gloss over the assortment of various curly letters. What I’ve hoped to convey is that the ACME experiment has a lot of switches and knobs, and when we take data we play around with all of these knobs in every possible configuration, looking to see any effects they make on the quantity we measure, d_e, the EDM of the electron. Due to the data acquisition structure we impose on the experiment, many sources of noise and systemic errors are canceled out. In order for them to disturb our measured values and confound us, they have to act in a very specific way – to switch together with a particular combination of switches that we have (\tilde{\mathcal{N}} and \tilde{\mathcal{E}}). There are not many things which behave that way, which is why the ACME experiment is very resilient against these disturbances. The large number of switches we implement in our system also points to its thoroughness and rigor: at this level of precision, we want to be absolutely sure that what we measure is really true.

 

  1. The \phi is because we are really measuring the phase of the spin precession. ↩

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