[T]ruly to enjoy bodily warmth, some small part of you must be cold, for there is no quality in this world that is not what it is merely by contrast.
—Herman Melville,Moby Dick
Figure2.5.1shows some example data collected using the scheme described in Section 2.1. As derived above, this measurement scheme determines the accumulated phase due to the energy shift between the twoMJ levels in a givenN state. This energy shift is given by (see
Figure 2.5.1: Average fluorescence signal profile from a molecule pulse v. time since ablation. Each pulse of molecules is ≈ 2 ms wide. Each trace on which the analysis was performed is an average of 25 molecule beam pulses and takes 0.5 s to acquire. The inset shows a zoom-in on the fluorescence signal over a 50 µs interval, revealing the 100 kHz chopping of the probe laser between xˆand yˆpolarization, used to measureSX and SY, respectively. An
asymmetry value A (see Eq.2.11) can be calculated from each adjacent pair of polarization chopping bins, and 20–30 such consecutive asymmetries are averaged together to make a measurement. These averaged asymmetries are used to construct parity sums and extract physical quantities as described in the text. Figure reproduced from reference [35], licensed under Creative Commons.
Eq. (2.4)):
∆U(N, ⃗E, ⃗B) ≡ U(MJ = +1,N, ⃗E, ⃗B)−U(MJ =−1,N, ⃗E, ⃗B) (2.15)
= −2gH,J=1µBBB˜−2deEeffNE˜ (2.16)
In order to isolate de from the Zeeman interaction term (and various other terms we have so
far ignored), we perform a number of “switches.” For example, we can repeat the mea- surement with both B˜ = ±1 and take the sum of the measurements, ∆U(N, ⃗E, ⃗B) + ∆U(N, ⃗E,−B⃗) = −4deEeffNE˜ to eliminate terms that switch sign with B˜—as the eEDM
netic field interaction,∆U(N, ⃗E, ⃗B)−∆U(N, ⃗E,−B⃗) = 4gH,J=1µBB. In other words, since
the spin precession in the magnetic field is “B˜-odd” (reverses whenB˜ is reversed), and the electron EDM precession is “B˜-even,” we can distinguish these effects by taking sums or differences of precession phases measured with opposite orientations of B˜. Notice that we can also separate the spin and EDM precession by reversingN orE˜ since the two terms also have opposite parity under reversal of those quantities.
In a real experiment, a number of uncontrolled effects are present, including background fields, correlated fields (e.g. magnetic fields from leakage currents which reverse synchronously withE˜), motional fields, geometric phases, and many more [10,11,115]. Despite the best ex- perimental efforts, many of these effects cause energy shifts larger than the eEDM; however, we can isolate the eEDM from these effects using its unique “NE˜B˜ = − −+” parity–i.e. odd parity under molecular dipole or electric field reversal and even parity under magnetic field reversal. Table 2.5.1 shows a sample of the types of effects that are separated from the eEDM “parity channel” by means of switches.
Table 2.5.1: Parity of energy shifts of selected effects in the ACME measurement. The difference between the g-factors of the two N-states of H is ∆g [30], and the subscript nr denotes the non-reversing component of an applied field. Products of terms denote corre- lations between those terms. The terms with +− − parity are higher-order and negligibly small. Table reproduced from reference [35], licensed under Creative Commons.
NE˜B˜ Parity Quantities
+ + + Spin precession in background (non-reversing) magnetic field Bnr,
Pump/probe relative polarization offset
+ +− Electron spin precession in applied magnetic field +−+ Leakage currents Bleak
−+ + ∆gBnr, ∆gBleakEnr
+− − —
−+− Electric-field-dependent g-factors [30]
− −+ Electron EDM
− − − ∆gEnr
If we perform 8 repeated experiments, with each of the23 = 8combinations of±N,±E,˜ ±B˜, we can take sums and differences to compute the 8 different possible parities under N,E,˜ B˜
reversals, as shown in Table 2.5.1. Apart from higher-order terms, such as cross-terms be- tween background electric and magnetic fields, the eEDM is the only term expected to have
NE˜B˜ =− −+parity.8 This technique of isolation by parity is how EDM experiments can perform sensitive measurements of the electron EDM with achievable levels of control of experimental parameters. We also perform a number of auxiliary switches to check for other systematic dependences of theNE˜B˜ =− −+signal, such as rotating the polarization angle of the pump and probe lasers and interchanging the positive and negative field plate voltage leads. The results of these tests are shown in Fig. 2.7.3and discussed further in Section 2.7. In order to avoid introducing our psychological biases into the analysis, the mean value of the NE˜B˜ = − −+ eEDM channel was not revealed until the entire run was complete and we had devised a procedure for determining the systematic uncertainty. The “blind” that concealed the eEDM value was a randomly generated number drawn from a Gaussian distribution centered on zero with a width of 1×10−27ecm, determined by the YbF ex- periment eEDM limit [103]. This number, in appropriate units, was automatically added to the NE˜B˜ =− −+ parity sum channel in the lowest levels of the analysis code so that researchers could only look at the eEDM value plus the unknown offset—never the offset or the eEDM channel alone. Significantly, this blinding method did not inhibit observation of eEDM channel correlations or statistical fluctuations. In order to measure and control sys- tematic errors, it was necessary to analyze such variations in the eEDM channel; therefore, we were only blind to the critical result: the eEDM’s consistency or inconsistency with zero. The above summary is intended to give a sense for the general approach to data analysis in ACME or any EDM experiment. A thorough treatment of the ACME analysis routine and experimental switching scheme is a vast subject that is beyond the scope of this thesis. Exhaustive treatments can be found in references [11, 105, 144, 181]. Three independent
8To tell the whole truth, the electron-nucleonCP-violating coupling is also expected to have this behavior [10, 11]. We have ignored this term so far in the discussion because its physical interpretation is not as intuitive as the EDM, but we will touch on it briefly in Section2.8.
We also reserve our examination of terms that incidentally displayNE˜B˜ parity because of experimental imperfections (most of which display the “higher-order” character referenced above) for the discussion on systematic errors in Section2.7.
analysis routines were run on the ACME Gen. I data. The three codes were developed by Nick Hutzler [105], Ben Spaun [181] and Brendon O’Leary [144]. The results of these analyses agreed well within uncertainty, and the final results reported in our limit paper [10] and discussed below were constructed from the average of the three.