## 1. Introduction

Extensions of the Standard Model of particle physics are necessary to explain the presence of dark matter and to explain the asymmetry between matter and antimatter in the universe. The Standard Model predicts that the electric dipole moment of the electron,

${d}_{e}$, is along the direction of its spin, and is probably of order 10

${}^{-40}\phantom{\rule{3.33333pt}{0ex}}e$ cm [

1,

2,

3], whereas most extensions (for example, supersymmetric theories [

4,

5]) predict a much larger value for

${d}_{e}$.

The two most precise measurements of

${d}_{e}$ are by the ACME collaboration [

6] (who use a beam of metastable

${}^{232}$Th

${}^{16}$O, and measure

${d}_{e}=-$2.1(4.5) × 10

${}^{-29}\phantom{\rule{3.33333pt}{0ex}}e\phantom{\rule{3.33333pt}{0ex}}$cm), and the JILA collaboration [

7] (who use trapped metastable

${}^{180}$Hf

${}^{19}$F

${}^{+}$ ions and measure

${d}_{e}=$ 0.9(7.8) × 10

${}^{-29}$ $e\phantom{\rule{3.33333pt}{0ex}}$cm). The results of these measurements are both consistent with zero and their weighted average gives

${d}_{e}=-$1.4(3.9) × 10

${}^{-29}\phantom{\rule{3.33333pt}{0ex}}e$ cm, which sets a 90% confidence interval of

$|{d}_{e}|<$ 7 × 10

${}^{-29}\phantom{\rule{3.33333pt}{0ex}}e$ cm. A stronger limit (or a nonzero measurement of

${d}_{e}$) is necessary to guide Standard Model extensions.

Measurements of

${d}_{e}$ are performed by watching electron spins precess within a magnetic field, and measuring any change in this precession rate due to the presence of an electric field. The angle through which they precess is given by

where

$g{\mu}_{B}$ is the magnetic moment of the electron,

B is the applied magnetic field,

${E}_{\mathrm{eff}}$ is the effective electric field that the electron experiences inside the molecule or molecular ion, and

T is the time that the electron is precessing. The ± signs correspond to the cases for which the electric and magnetic fields are oriented in parallel and antiparallel directions.

The precision of

${d}_{e}$ from such a measurement, assuming the measurement is statistically limited by shot-noise, is

where

N is the total (integrated) number of electrons whose precession is detected. The value of

${E}_{\mathrm{eff}}$ is between 10 and 100 GV/cm for most polar molecules (and polar molecular ions) used for electron electric-dipole-moment studies, and therefore greatly improved measurements can only be obtained by large improvements in

N or in

T. The ACME experiment [

6] uses an

N of order 10

${}^{10}$, and a

T of order 1 ms. The JILA measurement [

7,

8] uses an

N of order 10

${}^{6}$, and a

T of approaching 1 s.

Improvements in

N and in

T should be possible for both the molecular ion and the neutral molecule measurements. For molecular ion experiments, the number of trapped ions seems to be practically limited to be less than approximately 10

${}^{5}$ (

$N<{10}^{11}$ for a month of one-second-long measurements) due to interactions between co-trapped ions. Laser cooling and trapping experiments of neutral polar molecules are planned by several groups (e.g., with YbF [

9] and TlF [

10]), but, even assuming the same performance from a molecular trap as an atom trap, the maximum molecule number is unlikely to exceed 10

${}^{8}$ molecules per trap cycle (

$N<{10}^{14}$ for a month of one-second-long measurements). Beam experiments with a

T of approaching 1 s could potentially reach

N of 10

${}^{10}$, if the considerable experimental difficulties of maintaining beam collimation and field control over tens of meters can be achieved.

We propose here a method that will allow for a much larger

N, while maintaining a large value of

T. The method involves embedding polar molecules in an inert-gas matrix. We refer to this method as EDM

${}^{\mathbf{3}}$ (Electric Dipole Measurements using Molecules within a Matrix). The method exploits the fact that the inert-gas matrices are transparent, and that the influence of the matrix on the molecule is small enough to still allow for state preparation and detection techniques similar to those used in molecular beam measurements of

${d}_{e}$. The number of embedded polar molecules could range from 10

${}^{12}$ to 10

${}^{16}$, or more. We expect that a

T of approaching 1 s should be possible given that second-long coherence times were demonstrated for cesium atomic spins in solid-helium crystals [

11,

12] and 0.1 s has been demonstrated for rubidium atoms in an argon matrix [

13]. With a measurement cycle time of one second and a one-month-long measurement,

N would be 10

${}^{18}$ to 10

${}^{22}$. This would lead to a statistical measurement precision of

$\delta {d}_{e}$ = ∼10

${}^{-35}$ to ∼10

${}^{-37}e$ cm, which represents an improvement of between seven and nine orders of magnitude when compared to the current uncertainty on

${d}_{e}$.

The EDM

${}^{\mathbf{3}}$ method requires the embedded polar molecules to be oriented within the inert-gas matrix. Only recently [

14] has it been demonstrated that polar molecules can be fully oriented in an inert-gas matrix by the application of electric fields of between 1 and 3 MV/cm, using the novel technique [

15] of ice-film nanocapacitors. Previous to this achievement, it was thought [

16] that a completely oriented sample would not be feasible.

Furthermore, molecules trapped in a matrix can have their rotations inhibited by the forces between the ions that make up the molecule and the polarizable inert-gas atoms that are their nearest neighbors in the matrix. Theoretical understanding of this inhibition of rotations (which can lock molecules into librator states instead of their usual rotational states) has been studied using Devonshire octahedral model potentials [

17,

18]. The model describes the motion of the trapped molecules in terms of hindered rotations and librations, and shows that the rotational spectrum includes trapped librator states for strong coupling. Here, strong coupling occurs for large values of the ratio of the potential barrier that hinders rotation (caused by the inert-gas–molecule interactions) divided by the rotational constant of the molecule. The librator states may allow the polar molecules to remain oriented, even after the applied electric field is turned off.

There are several advantages of measuring

${d}_{e}$ in a matrix, as opposed to in a solid such as gadolinium–iron garnet [

19,

20,

21]. The spacing between inert-gas atoms in the matrix is large, leading to only small perturbations on the embedded molecules to be studied. Since the matrix is transparent, standard spectroscopy techniques can be used in studying the embedded molecules. In addition, the matrix allows for the large

${E}_{\mathrm{eff}}$ available in polar molecules, and, in addition, one can change the species and density of the embedded molecules. With EDM

${}^{\mathbf{3}}$,

${d}_{e}$ can be measured using the individual molecules (using methods similar to those used for trapped ions or molecular beams), rather than observing the bulk properties of the solid. That is, EDM

${}^{\mathbf{3}}$ gives both the advantage of measuring

${d}_{e}$ with large sample sizes (as in solid-state measurements), and the advantage of applying precise, shot-noise-limited spectroscopic techniques (as used in the most precise measurements of

${d}_{e}$ [

6,

7,

22]).

A previous suggestion [

23] of using molecules in inert-gas matrices for measurements of

${d}_{e}$ involved detection of a very small magnetization induced by the presence of

${d}_{e}$ in the

${E}_{\mathrm{eff}}$ of the molecule, assuming that the molecules are oriented by an external field. Here, we suggest a more standard approach to measuring

${d}_{e}$ using techniques similar to those used for molecular-beam measurements of

${d}_{e}$ (see, for example, References [

22,

24]). Other authors have explored [

25] the possibility of using atoms within an inert-gas matrix to measure

${d}_{e}$. Using atoms instead of molecules has the advantage of less complicated, and more easily calculated, quantum states, but has the drawback of the much smaller

${E}_{\mathrm{eff}}$ (of approximately 100 times the applied field [

26]). Nonetheless, the possibility of using atoms may have to be reevaluated in light of the much larger applied electric fields recently demonstrated in Reference [

14], although these fields will still lead to an

${E}_{\mathrm{eff}}$ of approximately 100 MV/cm, compared to the approximately 100 GV/cm fields obtained with polar molecules.

Some details of the proposed EDM

${}^{\mathbf{3}}$ method are outlined in the following section. However, many of the steps will need further development and experimental verification to prove the viability and strength of the EDM

${}^{\mathbf{3}}$ method for determining

${d}_{e}$. The methods described are also well-suited to nuclear electric dipole measurements, as nuclear spin coherence times of well over 1 s can be obtained at low temperatures in bulk material [

27], and we would expect even longer times within inert-gas matrices.

## 3. Potential Advantages of the EDM${}^{\mathbf{3}}$ Method

The main advantage of the EDM

${}^{\mathbf{3}}$ method is the large number of molecules that can be embedded into the matrix, and the long interaction times possible for these trapped molecules, along with the large effective electric field that is available with polar molecules. That is, EDM

${}^{\mathbf{3}}$ can allow for large

N,

T and

${E}_{\mathrm{eff}}$ in Equation (

2), allowing for a large improvement in the statistical uncertainty of a

${d}_{e}$ measurement.

Other potential advantages of EDM${}^{\mathbf{3}}$ involve the control of systematic effects. Because the crystal is built out of an inert gas, it can be produced with ultrahigh purity compared to ordinary solid-state hosts. Additionally, the measurement can easily be repeated with different inert gases, and with different distances between the polar molecules, in order to study systematic effects. Similarly, the substrate on which the gas is frozen, which could be an important contributor to decoherence and systematic effects, could be varied. The substrate (and any impurities in the substrate) will likely have to be within micrometers of the polar molecules. Magnetic Johnson noise from conducting substrates could be important (for this, low temperatures and resistive substrates would be better), and impurities (including nuclear spins) in the substrate can lead to decoherence. A material such as isotopically pure silicon or diamond might be necessary in order to obtain a mechanically and magnetically clean substrate. Finally, we propose to also use a second molecule (possibly co-located in the same inert-gas matrix and measured simultaneously) as a further test against systematics.

Another advantage of EDM${}^{\mathbf{3}}$ is the small volume in which the molecules are contained. The small volume will allow for efficient collection and detection of fluorescence. It will also allow for easier magnetic shielding, and could allow for the magnetic coils producing the fields to be mechanically rotated to reverse the field (in addition to reversing the field by reversing the current). The thin film envisioned for the measurement would minimize the effect of magnetic fields caused by leakage currents. These fields, if they have a component along $\widehat{z}$, can mimic the effect of an electron electric dipole moment. Here, this effect should be minimal, since only a small amount of charge is needed to create the fields, and, even if it did leak across the sample, it would be expected to take an approximately direct path across the film. Thus, any potential leakage current would be small, and the migrating charges would be expected to travel almost parallel to the z-direction, leading to an insignificant component of magnetic field along $\widehat{z}$.

The possibility of having the precession take place without an applied electric field could be a powerful weapon against electric-field induced systematic effects, as well as providing orientation reversal without electric field reversal (even for molecules that do not have

$\mathsf{\Omega}$-doublets [

49,

50]), simply by separately detecting the interspersed

$+\widehat{z}$ and

$-\widehat{z}$ oriented molecules (by a small tuning of the detection laser).

Although the prospects for the control of systematic effects seem hopeful, clearly, further very careful studies of possible systematic effects will have to be performed when implementing the EDM${}^{\mathbf{3}}$ method.