# Analysis of an Optical Lattice Methodology for Detection of Atomic Parity Nonconservation

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

**2019**, 100, 050101. The experimental concept is described in more detail and we compute new ab initio data necessary for assessing the plausibility of the approach. Original theoretical data for transition matrix elements on the electric dipole forbidden transition in caesium $6\mathrm{s}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$–$5\mathrm{d}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ are reported, as are a range of electric dipole matrix elements connected to the ground state 6s. The latter is used for an analysis of the wavelength-dependent light shift in Cs. A range of experimental details is presented, combined with a survey of realistic lasers parameters. These are adopted to project the feasibility of the scheme to eventually be capable of delivering data beyond the standard model of particle physics.

## 1. Introduction

## 2. Experimental Methods

#### 2.1. Light Field Geometry

^{+}. The field having a node will address an E2 resonance and since the latter scales with the gradient of the field amplitude, it being driven at a field node maximises the induced E2 light shift.

#### Applying the Fortson Scheme to a Large Sample on Neutral Atoms

#### 2.2. Spectroscopic Detection Scheme

- The detection scheme must isolate the parity-violating component of the total signal.
- A plethora of systematic effects have to be confronted.

#### 2.2.1. RF and Raman Spectroscopy

#### 2.2.2. Isolation of the PNC Signature

## 3. Theoretical Methods

## 4. Results

^{133}Cs, they are necessary for a prediction of the parity violation signature in Equation (12) and thus for a check of the experimental feasibility. The various calculated transition amplitudes will also be essential for a proper estimate of experimental error contributions.

#### 4.1. Theoretical Results

#### 4.1.1. Calculated PNC, E2 and M1 Matrix Elements and Associated Light Shifts

#### 4.1.2. Calculated E1 Matrix Elements and Associated Light Shifts

#### 4.2. Predicted Measured Parity Violation Signature

## 5. Discussion

- uncompensated linear Zeeman effect
- quadratic Zeeman effect
- uncompensated E1 light shifts
- uncompensated E2 light shifts
- polarisation/geometry

#### 5.1. Possible Extensions

#### Other Elements than Cs

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SM | Standard model for elementary particle physics |

PNC | Parity non-conservation |

NSI | Nuclear spin independent |

NSD | Nuclear spin independent |

NAM | Nuclear anapole moment |

E1 | Electric dipole moment |

E2 | Electric quadrupole moment |

M1 | Magnetic dipole moment |

MOT | Magneto-optical trap |

hfs | Hyperfine structure |

RF | Radio-frequency |

DHF | Dirac-Hartree-Fock |

GTO | Gaussian type orbitals |

RCC | Relativistic coupled-cluster |

RCCSD | RCC with a single and double excitations approximation |

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**Figure 1.**Experimental blueprint introduced by [18]. A single ion is held in a tight secular orbit, defining an interaction region (grey central circle in the figure). This is intersected by two standing waves of equal wavelength light, resonant with an E2 transition. One wave (blue in the figure) has an intensity maximum at the site of the ion, maximising the PNC perturbation, whereas the other (red) has a node, thereby providing a local maximum for an E2 transition amplitude. The light wavelength must be large relative to the interaction region.

**Figure 2.**Partial Grotrian diagram, showing the four lowest electronic configurations in Cs. Indicated by the blue arrow is the E2 allowed transition $6\mathrm{s}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$–$5\mathrm{d}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ at ${\lambda}_{\mathrm{s}\mathrm{d}}=689.5$ nm. The thick red arrow shows the wavelength of the optical lattice light (${\lambda}_{\mathrm{ol}}=975.1$ nm, see text). This is well below the D1 and D2 resonances to the 6p configuration.

**Figure 3.**Proposed light field configurations. The red and blue arrows represent cavity enhanced standings waves, aligned at $\pi /4$ with the ${\widehat{\mathrm{e}}}_{x}$ and ${\widehat{\mathrm{e}}}_{z}$ axes. These excite respectively E2 and parity non-conservation (PNC) transition amplitudes (see Equation (2) and Figure 2). They have mutually orthogonal linear polarisations, and are resonant with the 6s–5d transition. The green arrows show optical lattice beams, linearly polarised along ${\widehat{\mathrm{e}}}_{y}$.

**Figure 4.**Left: Two overlapping orthogonal standing waves (red for the E2-field and blue for the PNC field) generate a 2D grid of points optimised for detection of PNC. Full lines represent anti-nodal planes and dotted ones nodal planes. At the ideal detection sites (purple circles), the PNC field has irradiance maxima, whereas the E2 field has a node—with a maximally steep gradient. Right: Optical lattice sites are shown as filled, green concentric circles. All of these will coincide with an idea detection point (purple crosses). The gradual green shading represents the irradiance modulation around the lattice sites.

**Figure 5.**Detailed energy level scheme of the spectral line $6\mathrm{s}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$–$5\mathrm{d}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ including all involved Zeeman levels (${M}_{F}$). The degeneracy is broken, as shown in Equation (9). Our spectroscopic detection scheme is illustrated by the arrows. The blue ones correspond to radio-frequency (RF)-spectroscopy of the two level splittings $\hslash \phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{RF},\mathrm{a}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(+4)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(+3)$ and $\hslash \phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{RF},\mathrm{b}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(-4)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(-3)$, and the par of green arrows the Raman spectroscopy of $\hslash \phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{Raman}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(+1)\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mathsf{\Delta}E(-1)$.

**Table 1.**Lists of ${\eta}_{0}$, $\zeta $ and number of Gaussian type orbitals (GTOs) (${N}_{k}$) used to define the basis functions for different symmetries in the construction of single particle orbitals in the present calculations. In the bottom row, we also give the number of active orbitals (${N}_{v}$) allowed to participate in the estimation of electron correlation effects, using the RCCSD method.

$\mathbf{s}$ | $\mathbf{p}$ | $\mathbf{d}$ | $\mathbf{f}$ | $\mathbf{g}$ | $\mathbf{h}$ | $\mathbf{i}$ | |
---|---|---|---|---|---|---|---|

${\eta}_{0}$ | $0.00009$ | $0.0008$ | $0.001$ | $0.004$ | $0.005$ | $0.005$ | $0.005$ |

$\zeta $ | 2.15 | 2.15 | 2.15 | 2.25 | 2.35 | 2.35 | 2.35 |

${N}_{k}$ | 40 | 39 | 38 | 37 | 36 | 35 | 33 |

$Nv$ | 1–19 | 2–19 | 3–19 | 4–17 | 5–15 | 6–13 | 7–13 |

**Table 2.**Hypothesised laser parameters for the sd- and ol-lasers. P is the total laser power, ${Q}_{\mathrm{ef}}$ a cavity enhancement factor, w the beam diameter at the interaction region, ${\mathcal{E}}_{\mathrm{int}}$ the field amplitude at the detection sites and the final column shows the maximum reduction in amplitude at the edge of the interaction region. To estimate the numbers for the sd-laser, we have used [22]. For the ol-laser, the amplitude refers to the total one, taking into account all lattice beams (see (7)).

$\mathit{\lambda}$ (nm) | P (W) | ${\mathit{Q}}_{\mathbf{ef}}$ | w (mm) | Mode | ${\mathcal{E}}_{\mathbf{int}}$(V/m) | $\mathsf{\Delta}{\mathcal{E}}_{\mathbf{max}}/{\mathcal{E}}_{\mathbf{int}}$ | |
---|---|---|---|---|---|---|---|

sd-laser | 689.5 | 3 | 100 | 0.5 | flat-top | ≃$3\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{6}$ | 0.015 |

ol-laser | 975.1 | 5 | 1 | 1.5 | Gaussian | ≃$1\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{5}$ | 0.03 |

**Table 3.**Contributions to the overall $6\mathrm{s}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$–$5\mathrm{d}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{D}}_{1/2}$ transition amplitude from the M1, E2 and PNC interactions, computed with the methods described in Section 3. The M1 and E2 amplitudes are given in a.u., and the PNC amplitude is given in $-\mathrm{i}e{a}_{0}[{Q}_{\mathrm{W}}/N]\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-11}$, with the weak charge ${Q}_{\mathrm{W}}$ and a neutron number $N=78$ of ${}^{133}$Cs atom.

Method | ${\mathit{A}}_{\mathbf{M}}1$ | ${\mathit{A}}_{\mathbf{E}}2$ | ${\mathit{A}}_{\mathbf{PNC}}$ |
---|---|---|---|

DHF | ∼0 | 43.85 | 2.396 |

RCCSD | $2.54\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-4}$ | 33.61 | 3.210 |

**Table 4.**Calculated energy shifts for relevant $\mathsf{\Delta}M\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0$ transitions on the spectral line $6\mathrm{s}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$, $F\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$–$5\mathrm{d}{\phantom{\rule{0.166667em}{0ex}}}^{2}{\mathrm{D}}_{1/2},F\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}5$, using the calculated transition amplitudes from Table 3 and the hypothesised electric field amplitude of Table 2.

${\mathit{W}}_{\mathbf{E}2}/\mathit{h}$ | ${\mathit{W}}_{\mathbf{PNC}}/\mathit{h}$ | |
---|---|---|

${M}_{F}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$–${M}_{F}^{\prime}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1$ | $-11.02$ MHz | $-0.544$ Hz |

${M}_{F}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3$–${M}_{F}^{\prime}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3$ | $-27.01$ MHz | $-0.445$ Hz |

${M}_{F}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$–${M}_{F}^{\prime}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ | $-27.01$ MHz | $-0.333$ Hz |

**Table 5.**Calculated E1 matrix elements (in a.u.) from the Dirac–Hartree–Fock (DHF) and RCCSD methods. We have also quoted extracted available experimental values from measurements of lifetimes and Stark shifts for states of Cs, and from another recent calculation.

Transition | DHF | RCCSD | Roberts et al. [43] | Experiment |
---|---|---|---|---|

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 6\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 5.278 | 4.549 | 4.512 | 4.5097(74) [44] |

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 0.372 | 0.302 | 0.2724 | 0.2825(20) [45] |

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 0.132 | 0.092 | ||

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 0.069 | 0.040 | ||

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 6\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 7.426 | 6.397 | 6.351 | 6.3403(64) [44] |

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 0.694 | 0.610 | 0.5659 | 0.57417(57) [46] |

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 0.282 | 0.234 | ||

$6\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 0.159 | 0.124 | ||

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 6\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 4.413 | 4.252 | 4.249(5) [47] | |

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 11.012 | 10.297 | 10.308(15) [48] | |

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 0.934 | 0.950 | ||

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 0.394 | 0.389 | ||

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 6\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 6.671 | 6.501 | 6.4890(50) [47] | |

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 15.349 | 14.298 | ||

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 1.622 | 1.670 | ||

$7\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 0.726 | 0.736 | ||

$8\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 18.719 | 17.821 | ||

$8\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 1.997 | 2.040 | ||

$8\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 25.977 | 24.618 | ||

$8\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 3.289 | 3.395 | ||

$9\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | 29.818 | 28.755 | ||

$9\mathrm{s}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{S}}_{1/2}\to 9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | 41.423 | 39.773 | ||

$7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 4.039 | 2.052 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 0.989 | 0.634 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 8.058 | 4.945 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 0.489 | 0.336 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 2.166 | 1.521 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{1/2}\to 7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 13.186 | 8.823 | ||

$7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 1.688 | 0.809 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 0.428 | 0.255 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 3.336 | 1.945 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 0.212 | 0.134 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 0.925 | 0.610 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}$ | 5.453 | 3.379 | ||

$7\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 5.024 | 1.852 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 1.277 | 0.588 | ||

$8\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 10.011 | 4.637 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 0.633 | 0.301 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 2.778 | 1.500 | ||

$9\mathrm{p}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{P}}_{3/2}\to 7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}$ | 16.107 | 8.107 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 10.660 | 10.355 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 25.583 | 23.811 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 4.722 | 4.224 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 9.607 | 9.674 | ||

$7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{3/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 46.630 | 45.090 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 2.840 | 2.855 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 6.843 | 6.704 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 1.261 | 1.185 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 2.568 | 2.762 | ||

$7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{5/2}$ | 12.471 | 12.418 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{7/2}$ | 12.703 | 12.772 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 4\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{7/2}$ | 30.602 | 29.978 | ||

$5\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{7/2}$ | 5.642 | 5.301 | ||

$6\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{7/2}$ | 11.490 | 12.257 | ||

$7\mathrm{d}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{D}}_{5/2}\to 5\mathrm{f}{\phantom{\rule{3.33333pt}{0ex}}}^{2}{\mathrm{F}}_{7/2}$ | 55.769 | 55.531 |

**Table 6.**Light shift of the fine structure level $6\mathrm{s}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$, when irradiated by an optical lattice light field at ${\lambda}_{\mathrm{ol}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}975.1$ nm. Beam parameters are as in Table 2 and the the values of transition matrix elements from Table 3.

Contribution | Total | ||
---|---|---|---|

Level | from | MHz | kHz |

$6\mathrm{s}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{S}}_{1/2}$ | $6\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | $-267.31$ | |

$6\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | $-169.10$ | ||

$7\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | $-0.82$ | ||

$7\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | $-0.88$ | ||

$8\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | $-0.13$ | ||

$8\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | $-0.29$ | ||

$9\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{1/2}$ | $-1.88$ | ||

$9\mathrm{p}{\phantom{\rule{0.222222em}{0ex}}}^{2}{\mathrm{P}}_{3/2}$ | $-1.64$ | ||

$-438.53$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kastberg, A.; Sahoo, B.K.; Aoki, T.; Sakemi, Y.; Das, B.P.
Analysis of an Optical Lattice Methodology for Detection of Atomic Parity Nonconservation. *Symmetry* **2020**, *12*, 974.
https://doi.org/10.3390/sym12060974

**AMA Style**

Kastberg A, Sahoo BK, Aoki T, Sakemi Y, Das BP.
Analysis of an Optical Lattice Methodology for Detection of Atomic Parity Nonconservation. *Symmetry*. 2020; 12(6):974.
https://doi.org/10.3390/sym12060974

**Chicago/Turabian Style**

Kastberg, Anders, Bijaya Kumar Sahoo, Takatoshi Aoki, Yasuhiro Sakemi, and Bhanu Pratap Das.
2020. "Analysis of an Optical Lattice Methodology for Detection of Atomic Parity Nonconservation" *Symmetry* 12, no. 6: 974.
https://doi.org/10.3390/sym12060974