# Optical Rotation Approach to Search for the Electric Dipole Moment of the Electron

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## Abstract

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## 1. Introduction

## 2. Theory of the $\mathcal{P}$,$\mathcal{T}$-Odd Faraday Effect

## 3. Application to Transitions in Different Atomic Species

#### 3.1. Ra Atom ($Z=88$)

#### 3.2. Pb Atom ($Z=82$)

- (1)
- The first one is the E1 $6{p}^{2}{(1/2,1/2)}_{0}\to 6p7s{(1/2,1/2)}_{1}$ with the transition wavelength $\lambda =283$ nm. We employ the value for the eEDM enhancement factor of the $6p7s{(1/2,1/2)}_{1}$ state from [18]: $\left(\right)open="("\; close=")">{R}_{d}^{i}+{R}_{d}^{f}$. Assuming $T\sim 300$ K and employing ${\omega}_{0}=6.7\times {10}^{15}$ s${}^{-1}$, according to Equation (19) ${\Gamma}_{D}\approx 3.4\times {10}^{9}$ s${}^{-1}$. The natural line width for the chosen transition is ${\Gamma}_{n}=1.79\times {10}^{8}$ s${}^{-1}$ [49]. For $\mathcal{E}={10}^{5}$ V/cm, ${d}_{e}={10}^{-29}$e cm and $u\approx 4$ from Equation (28) it follows$${\psi}_{\mathrm{max}}\approx 1.6\times {10}^{-14}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}.$$
- (2)
- Now we consider the M1 transition $6{p}^{2}{(1/2,1/2)}_{0}\to 6{p}^{2}{(3/2,1/2)}_{1}$ with $\lambda =1279$ nm. Here we also employ the value for the eEDM enhancement factor of the $6{p}^{2}{(3/2,1/2)}_{1}$ state from [18]: $\left(\right)open="("\; close=")">{R}_{d}^{i}+{R}_{d}^{f}$. Assuming $T\sim 300$ K and employing ${\omega}_{0}=1.5\times {10}^{15}$ s${}^{-1}$, according to Equation (19) ${\Gamma}_{D}\approx 7.7\times {10}^{8}$ s${}^{-1}$. The natural line width for the chosen transition (for the $6{p}^{2}{(3/2,1/2)}_{1}$ metastable state) is ${\Gamma}_{n}=7$ s${}^{-1}$ according to [49]. Let us estimate the value of the collisional broadening ${\Gamma}_{\mathrm{col}}$ according to Equation (4). The characteristic value for the collisional cross-section is ${\sigma}_{\mathrm{col}}\approx 0.5\times {10}^{-14}$ cm${}^{2}$ [41]. Then in terms of density $\rho $ we obtain ${\Gamma}_{\mathrm{col}}=7.6\times {10}^{-11}\rho \left[{\mathrm{cm}}^{-3}\right]$ s${}^{-1}$. So in this case the dimensionless $v=\frac{\Gamma}{2{\Gamma}_{D}}\approx 4.6\times {10}^{-9}+{10}^{-19}\rho \left[{\mathrm{cm}}^{-3}\right]$. It appears that for $\rho >{10}^{11}$ cm${}^{-3}$ the collisional broadening mechanism dominates over the natural broadening one. Since now the maximum rotation angle (Equation (28)) (optimal for the experiment) depends on $\rho $ (${\psi}_{\mathrm{max}}\sim 1/\rho $) and the column density according to Equation (26) is not fixed (the fixed quantity is ${\rho}^{2}l$, i.e., $1/\rho \sim \sqrt{l}$) then let us employ the maximum feasible value for the optical path lengths in our estimates. In [39] path length of 70,000 km for the cavity of the same size as in [38] was reported. If such a large electric field ($\mathcal{E}={10}^{5}$ V/cm) can be implemented in the cavity in a volume of a several centimetres size then the optical path length appears to be $l={10}^{8}$ cm. It corresponds to the optimal number density, according to Equation (26), $\rho \approx 5\times {10}^{14}$ cm${}^{-3}$. Then, for ${d}_{e}={10}^{-29}$e cm and $u\approx 4$ from Equation (28) it follows$${\psi}_{\mathrm{max}}\approx 1.1\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}.$$

#### 3.3. Tl Atom ($Z=81$)

#### 3.4. Hg Atom ($Z=80$)

- (1)
- The first one is from the metastable $6s6p{(}^{3}{P}_{1})$ state to the excited $6s7s{(}^{3}{S}_{1})$ state with $\lambda =436$ nm. The population of the lower metastable level can be obtained with the laser pumping [38]. The eEDM enhancement factors were calculated in [42] and $\left(\right)open="("\; close=")">{R}_{d}^{i}+{R}_{d}^{f}$. In [42] ${R}_{d}$ factors are presented for definite hyperfine levels. Here these values are recalculated for the levels $i,J|{M}_{J}|$ and $f,{J}^{\prime}\left|{M}_{{J}^{\prime}}\right|$. The natural line width for the chosen transition is ${\Gamma}_{n}=1.0\times {10}^{8}$ s${}^{-1}$ [49]. Assuming the room temperature $T\sim 300$ K and employing the transition frequency value ${\omega}_{0}=4\times {10}^{15}$ s${}^{-1}$, according to Equation (19) we obtain the value for the Doppler width ${\Gamma}_{D}=5.2\times {10}^{-7}{\omega}_{0}\approx 2\times {10}^{9}$ s${}^{-1}$. For $\mathcal{E}={10}^{5}$ V/cm, ${d}_{e}={10}^{-29}$e cm and $u\approx 4$ from Equation (28) it follows$${\psi}_{\mathrm{max}}\approx 8\times {10}^{-15}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}.$$
- (2)
- The second transition of E1 type is from the ground $6{s}^{2}{(}^{1}{S}_{0})$ to the metastable $6s6p{(}^{3}{P}_{1})$ state with $\lambda =254$ nm. According to [42], $\left(\right)open="("\; close=")">{R}_{d}^{i}+{R}_{d}^{f}$. Employing ${\omega}_{0}=7.4\times {10}^{15}$ s${}^{-1}$, according to Equation (19), ${\Gamma}_{D}=5.2\times {10}^{-7}{\omega}_{0}\approx 3.7\times {10}^{9}$ s${}^{-1}$. The natural line width for the chosen transition is ${\Gamma}_{n}=2.0\times {10}^{7}$ s${}^{-1}$. For $\mathcal{E}={10}^{5}$ V/cm, ${d}_{e}={10}^{-29}$e cm and $u\approx 4$ from Equation (28) it follows$${\psi}_{\mathrm{max}}\approx 8\times {10}^{-14}\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}.$$

#### 3.5. Cs Atom ($Z=55$)

#### 3.6. Xe Atom ($Z=54$)

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) represents the behavior of the maximum value of the $\mathcal{P}$,$\mathcal{T}$-odd Faraday angle $\psi $ (in rad) on the dimensionless detuning u for the E1 transition $6{p}^{2}7{s}^{2}{}^{1}{S}_{0}\to 6{p}^{2}7s7p{}^{3}{P}_{1}$ in Ra atom, (

**b**) represents the behavior of the column density $\rho l$ (in cm${}^{-2}$) on the dimensionless detuning u for the E1 transition $6{p}^{2}7{s}^{2}{}^{1}{S}_{0}\to 6{p}^{2}7s7p{}^{3}{P}_{1}$ in Ra atom.

**Table 1.**The summary of the results of the $\mathcal{P}$,$\mathcal{T}$-odd Faraday optical rotation theoretical simulation for different atomic species. The maximum $\mathcal{P}$,$\mathcal{T}$-odd Faraday rotation angle ${\psi}_{\mathrm{max}}$ (in rad) corresponds to an external electric field $\mathcal{E}={10}^{5}$ V/cm [10] and to the present eEDM bound established in experiments with ThO molecules (${d}_{e}={10}^{-29}$e cm [13]).

Atom | Transition | Wavelength | Linewidth | Column Density | Rotation Angle |
---|---|---|---|---|---|

$\mathit{\lambda}$, nm | ${\mathbf{\Gamma}}_{\mathit{n}}$, s${}^{-1}$ | $\mathit{\rho}\mathit{l}$, cm${}^{-2}$ | ${\mathit{\psi}}_{\mathbf{max}}$, rad | ||

Ra | $6{p}^{2}7{s}^{2}{}^{1}{S}_{0}\to 6{p}^{2}7s7p{}^{3}{P}_{1}$ (M1) | 714 | $2.37\times {10}^{6}$ | $5.0\times {10}^{16}$ | $2.6\times {10}^{-12}$ |

Pb | $6{p}^{2}{(1/2,1/2)}_{0}\to 6p7s{(1/2,1/2)}_{1}$ (E1) | 283 | $1.79\times {10}^{8}$ | $3.7\times {10}^{14}$ | $1.6\times {10}^{-14}$ |

Pb | $6{p}^{2}{(1/2,1/2)}_{0}\to 6{p}^{2}{(3/2,1/2)}_{1}$ (M1) | 1279 | 7 | $5.0\times {10}^{22}$ | $1.1\times {10}^{-11}$ |

Tl | $6{p}_{1/2}\to 6{p}_{3/2}$ (M1) | 1283 | 4 | $6.6\times {10}^{22}$ | $2.0\times {10}^{-11}$ |

Hg | $6s6p{(}^{3}{P}_{1})\to 6s7s{(}^{3}{S}_{1})$ (E1) | 436 | $1.0\times {10}^{8}$ | $1.5\times {10}^{14}$ | $8.0\times {10}^{-15}$ |

Hg | $6{s}^{2}{(}^{1}{S}_{0})\to 6s6p{(}^{3}{P}_{1})$ (E1) | 254 | $2.0\times {10}^{7}$ | $4.2\times {10}^{16}$ | $8.0\times {10}^{-14}$ |

Cs | $6{s}_{1/2}\to 6{p}_{1/2}$ (E1) | 895 | $3.23\times {10}^{7}$ | $1.9\times {10}^{14}$ | $9.2\times {10}^{-14}$ |

Xe | ${(}^{2}{P}_{3/2}^{0})6s{[3/2]}_{2}^{0}\to {(}^{2}{P}_{3/2}^{0})6p{[1/2]}_{1}$ (E1) | 980 | $2.6\times {10}^{7}$ | $2.2\times {10}^{14}$ | $9.0\times {10}^{-15}$ |

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**MDPI and ACS Style**

Chubukov, D.V.; Skripnikov, L.V.; Kutuzov, V.N.; Chekhovskoi, S.D.; Labzowsky, L.N.
Optical Rotation Approach to Search for the Electric Dipole Moment of the Electron. *Atoms* **2019**, *7*, 56.
https://doi.org/10.3390/atoms7020056

**AMA Style**

Chubukov DV, Skripnikov LV, Kutuzov VN, Chekhovskoi SD, Labzowsky LN.
Optical Rotation Approach to Search for the Electric Dipole Moment of the Electron. *Atoms*. 2019; 7(2):56.
https://doi.org/10.3390/atoms7020056

**Chicago/Turabian Style**

Chubukov, Dmitry V., Leonid V. Skripnikov, Vasily N. Kutuzov, Sergey D. Chekhovskoi, and Leonti N. Labzowsky.
2019. "Optical Rotation Approach to Search for the Electric Dipole Moment of the Electron" *Atoms* 7, no. 2: 56.
https://doi.org/10.3390/atoms7020056