# Using Comparisons of Clock Frequencies and Sidereal Variation to Probe Lorentz Violation

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## Abstract

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## 1. Introduction

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## 2. Qualitative Outline of Clock-Comparison Experiments

- Frequency measurements can probe atomic energy levels.
- The applied magnetic field defines an orientation of the clock: the “clock axis”.
- In conventional (i.e., Lorentz-symmetric) physics, the clock frequency is independent of the clock axis and the clock velocity.

## 3. Derivation of Nonrelativistic Hamiltonian

#### 3.1. Field Redefinition

#### 3.2. Foldy–Wouthuysen Transformation

- The relativistic Hamiltonian is divided into block-diagonal and off-block-diagonal parts: $m{\gamma}^{0}$ and $m\mathcal{E}$ are block diagonal, while $m\mathcal{P}$ and $m\mathcal{O}$ are off block diagonal.
- In the nonrelativistic limit, the upper two components of $\chi $ satisfy a version of the Schrödinger equation with spin.
- In the nonrelativistic limit, the lower two components of $\chi $ are smaller than the upper components by a factor of $\left|\overrightarrow{p}\right|/m$.
- The off-block-diagonal terms in the Hamiltonian couple the upper and lower components of $\chi $ together. Such terms are unsuppressed by factors of $\left|\overrightarrow{p}\right|/m$, and so, we cannot immediately neglect them.

- The lower components of $\chi $ are still suppressed with respect to the upper components by multiple factors of $\left|\overrightarrow{p}\right|/m$.
- The Hamiltonian is approximately block diagonal. That is, the off-block-diagonal parts are suppressed with respect to the upper-left block by multiple factors of $\left|\overrightarrow{p}\right|/m$.

## 4. Useful Assumptions and Approximations

#### 4.1. Use of Free-Particle Perturbation

#### 4.2. Atomic and Nuclear Models

## 5. Energy-Level Shifts to Atoms

## 6. Relating Theory to Experiments

## 7. Time Variation

#### 7.1. Earth-Based Experiments

#### 7.2. Space-Based Experiments

## 8. Results of Completed Experiments

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basic schematic of a typical atomic clock referenced in this paper. The clock’s frequency is determined by a pair of energy levels of an atom in an applied magnetic field.

**Figure 2.**Qualitative comparison of energy-level shifts to a $F=3/2$ multiplet from the conventional Zeeman effect and from dipole- and quadrupole-type Lorentz-violating operators.

**Figure 3.**Definition of two coordinate frames, one Sun-centered and nonrotating, the other fixed on Earth’s surface. We have drawn the frames as though they had the same origin.

**Figure 4.**Time variation of frequencies in an example $F=3/2$ system. At the top, the variation of conventional frequencies are shown for comparison, followed by variation of dipole-type energy shifts in the center and quadrupole-type energy shifts at the bottom. Effects that are suppressed by factors of $v/c$ have been neglected.

**Table 1.**List of completed clock-comparison experiments to date with very rough bounds accomplished. The first two columns identify the experiment. The third column lists the atomic species involved. The fourth column states which SME tilde-coefficients appear in bounds. In most cases, each bound is placed on a linear combination of these coefficients with weights of order one. The final column gives the order-of-magnitude of the bound placed on the tilde coefficients listed, each of which has units of GeV. For example, the first row states that the experiment of Prestage et al. published in 1985 compared transitions in beryllium and hydrogen atoms. It bounded ${\tilde{c}}_{X}$ and ${\tilde{c}}_{Y}$ associated with the neutron to each be smaller than about ${10}^{-25}$ GeV.

Experiment | Ref. | Atom(s) | Coefficients Bounded | ${\mathbf{log}}_{10}$$\left(\frac{\mathbf{Bound}}{\mathbf{GeV}}\right)$ |
---|---|---|---|---|

Prestage et al. 1985 | [10] | ${}^{9}{\mathrm{Be}}^{+}{,}^{1}\mathrm{H}$ | $n:{\tilde{c}}_{X},{\tilde{c}}_{Y}$ | $-25$ |

Lamoreaux et al. 1986 | [11] | ${}^{201}{\mathrm{Hg},}^{199}\mathrm{Hg}$ | $n:{\tilde{c}}_{-},{\tilde{c}}_{Z}$ | $-27$ |

Chupp et al. 1989 | [12] | ${}^{21}{\mathrm{Ne},}^{3}\mathrm{He}$ | $n:{\tilde{c}}_{-},{\tilde{c}}_{Z}$ | $-27$ |

Berglund et al. 1995 | [13] | ${}^{199}{\mathrm{Hg},}^{133}\mathrm{Cs}$ | $e:{\tilde{b}}_{X},{\tilde{b}}_{Y};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}p:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-27$ |

$e:{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{D,X},{\tilde{g}}_{D,Y}$ | $-22$ | |||

$p:{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{D,X},{\tilde{g}}_{D,Y}$ | $-25$ | |||

$n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-30$ | |||

$n:{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{D,X},{\tilde{g}}_{D,Y}$ | $-28$ | |||

Bear et al. 2000 | [14] | ${}^{129}{\mathrm{Xe},}^{3}\mathrm{He}$ | $n:{\tilde{b}}_{X},{\tilde{b}}_{X}$ | $-31$ |

Phillips et al. 2001 | [15] | ${}^{1}\mathrm{H}$ | $p:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-27$ |

Humphrey et al. 2003 | [16] | ${}^{1}\mathrm{H}$ | $e:{\tilde{b}}_{X},{\tilde{b}}_{Y};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}p:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-27$ |

Canè et al. 2004 | [17] | ${}^{129}{\mathrm{Xe},}^{3}\mathrm{He}$ | $n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-31$ |

$n:{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{D,X},{\tilde{g}}_{D,Y}$ | $-29$ | |||

$n:{\tilde{b}}_{T},{\tilde{d}}_{-},{\tilde{d}}_{+},{\tilde{d}}_{Q},{\tilde{d}}_{YZ},{\tilde{g}}_{c},{\tilde{g}}_{T}$ | $-27$ | |||

$n:{\tilde{d}}_{XY},{\tilde{H}}_{XT},{\tilde{H}}_{YT},{\tilde{H}}_{ZT}$ | $-27$ | |||

Wolf et al. 2006 | [18] | ${}^{133}\mathrm{Cs}$ | $p:{\tilde{c}}_{Q}$ | $-23$ |

$p:{\tilde{c}}_{-},{\tilde{c}}_{X},{\tilde{c}}_{Y},{\tilde{c}}_{Z}$ | $-25$ | |||

$p:{\tilde{c}}_{TX},{\tilde{c}}_{TY},{\tilde{c}}_{TZ};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n:{\tilde{c}}_{-}$ | $-21$ | |||

$p:{\tilde{c}}_{TT}$ | $-16$ | |||

$n:{\tilde{c}}_{Q}$ | $-20$ | |||

$n:{\tilde{c}}_{X},{\tilde{c}}_{Y},{\tilde{c}}_{Z}$ | $-22$ | |||

$n:{\tilde{c}}_{TX},{\tilde{c}}_{TY}$ | $-18$ | |||

$n:{\tilde{c}}_{TZ}$ | $-19$ | |||

$n:{\tilde{c}}_{TT}$ | $-13$ | |||

Kornack et al. 2008 | [19] | ${\mathrm{K},}^{3}\mathrm{He}$ | $e:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-28$ |

$p:{\tilde{b}}_{X},{\tilde{b}}_{Y};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-31$ | |||

Brown et al. 2010 | [20] | ${\mathrm{K},}^{3}\mathrm{He}$ | $p:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-32$ |

$n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-33$ | |||

Gemmel et al. 2010 | [21] | ${}^{3}{\mathrm{He},}^{129}\mathrm{Xe}$ | $n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-32$ |

Smiciklas et al. 2011 | [22,23] | ${}^{21}\mathrm{Ne},\mathrm{Rb}$ | $p:{\tilde{c}}_{X},{\tilde{c}}_{Y},{\tilde{c}}_{Z},{\tilde{c}}_{-}$ | $-29$ |

[22] | ${}^{21}\mathrm{Ne},\mathrm{Rb}$ | $n:{\tilde{c}}_{X},{\tilde{c}}_{Y},{\tilde{c}}_{Z},{\tilde{c}}_{-}$ | $-29$ | |

Peck et al. 2012 | [24] | ${}^{199}{\mathrm{Hg},}^{133}\mathrm{Cs}$ | $p:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-30$ |

Hohensee et al. 2013 | [25] | Dy | $e:{\tilde{c}}_{-},{\tilde{c}}_{X},{\tilde{c}}_{Y},{\tilde{c}}_{Z}$ | $-17$ |

$e:{\tilde{c}}_{TX},{\tilde{c}}_{TY},{\tilde{c}}_{TZ}$ | $-14$ | |||

$e:{\tilde{c}}_{TT}$ | $-8$ | |||

$p:{\tilde{b}}_{Z},{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{DX},{\tilde{g}}_{DY}$ | $-28$ | |||

$n:{\tilde{b}}_{Z},{\tilde{d}}_{X},{\tilde{d}}_{Y},{\tilde{g}}_{DX},{\tilde{g}}_{DY}$ | $-29$ | |||

$n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-31$ | |||

Allmendinger et al. 2014 | [26] | ${}^{3}{\mathrm{He},}^{129}\mathrm{Xe}$ | $n:{\tilde{b}}_{X},{\tilde{b}}_{Y}$ | $-34$ |

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Lane, C.D.
Using Comparisons of Clock Frequencies and Sidereal Variation to Probe Lorentz Violation. *Symmetry* **2017**, *9*, 245.
https://doi.org/10.3390/sym9100245

**AMA Style**

Lane CD.
Using Comparisons of Clock Frequencies and Sidereal Variation to Probe Lorentz Violation. *Symmetry*. 2017; 9(10):245.
https://doi.org/10.3390/sym9100245

**Chicago/Turabian Style**

Lane, Charles D.
2017. "Using Comparisons of Clock Frequencies and Sidereal Variation to Probe Lorentz Violation" *Symmetry* 9, no. 10: 245.
https://doi.org/10.3390/sym9100245