content_image/440770.jpg content_image/440768.jpg content_image/440767.jpg content_image/440764.jpg content_image/440763.jpg content_image/440769.jpg content_image/440766.jpg content_image/440765.jpg content_image/440762.jpg content_image/440774.jpg content_image/440771.jpg content_image/440772.jpg
and 

<img src='content_image/440770.jpg'>

The Stark amplitude can be written as 

where T$_{ij}$ = E$_{i}$ E j and 

<img src='content_image/440768.jpg'>

Let T k q and U k q represent the irreducible spherical com- ponents of the tensors T$_{ij}$ and U$_{ij}$ . Then T$_{ij}$U$_{ij}$ = ( − 1) $^{q}$T k − $_{q}$U k q and Eq. (A7) becomes 

<img src='content_image/440767.jpg'>

Here β is the vector Stark transition polarizability and defined by 

To derive Eq. (A11), we used 〈 0 , 0; k, q | 1 , M 〉 = δ$_{k}$$_{1}$δ$_{qM}$ and T 1 − q = ∑ q 1 ,q 2 〈 1 , q$_{1}$ ; 1 , q$_{2}$ | 1 , − q 〉 E$_{q}$$_{1}$ E q 2 = ( $_{i/}$√$_{2)(}$$_{E}$ × E )$_{−}$$_{q}$. For the parity-violating contribution to the E1 transi- 

tion amplitude we can likwise write 

where 

<img src='content_image/440764.jpg'>

Let E 1 q and W 1 q represent the spherical components of the vectors E i and W$_{i}$ , respectively. Then E $_{i}$W$_{i}$ = ( − 1) $^{q}$E 1 − $_{q}$W 1 q and we have 

<img src='content_image/440763.jpg'>

Here ζ is given by 

<img src='content_image/440769.jpg'>

<img src='content_image/440766.jpg'>

<img src='content_image/440765.jpg'>

<img src='content_image/440762.jpg'>

## Appendix B: Characterization of the PBC mirrors 

The finesse of the cavity is measured using the cavity- ring-down method [20]. The laser beam is sent through a Pockels cell (Cleveland Crystals Inc. QX 1020 Q-Switch) and a polarizer before entering the cavity. The polarizer is aligned with the laser polarization so that the light is transmitted when there is no voltage applied to the Pock- els cell. A high-voltage pulse generator is used to send a fast step signal (30-ns wavefront) to the Pockels cell which rotates the polarization of the light so that it is not transmitted through the polarizer. The laser frequency is locked to the resonance frequency of the cavity, and then the Pockels cell is switched into the non-transmitting state, causing a fast interruption of the laser power. The subsequent decay of the light inside the cavity is moni- tored with a fast photodiode (50-MHz bandwidth) mea- suring the power transmitted through the back mirror of the cavity. The signal is fit to an exponential decay. The decay time is related to the finesse of the cavity ( F ) by 

where c is the speed of light, L is the cavity length, and τ is the intensity decay time. An example of the PBC transmission signal and its fit are shown in Fig. 10. 

<img src='content_image/440774.jpg'>

Following the analysis discussed in [21], if we denote the transmission of mirrors 1 and 2 by T$_{1}$ and T$_{2}$ , re- spectively, and the absorption+scatter loss per mirror as l$_{1}$$_{,}$$_{2}$ = ( A + S )$_{1}$$_{,}$$_{2}$, then the total cavity losses L = T$_{1}$ + T$_{2}$ + l$_{1}$ + l$_{2}$ determine the finesse F : 

Information on the transmission of the mirrors discrim- inated from the A+S losses can be obtained using the 

<img src='content_image/440771.jpg'>

<img src='content_image/440772.jpg'>

