and The Stark amplitude can be written as where T$_{ij}$ = E$_{i}$ E j and 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 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}$. For the parity-violating contribution to the E1 transi- tion amplitude we can likwise write where Let E 1 q and W 1 q represent the spherical components of the vectors E i and W$_{i}$ , respectively. Then E $_{i}$W$_{i}$ = ( − 1) $^{q}$E 1 − $_{q}$W 1 q and we have Here ζ is given by Information on the transmission of the mirrors discrim- inated from the A+S losses can be obtained using the FIG. 10: (color online) Application of the cavity-ring-down method for the determination of the finesse of PBC. 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. Appendix B: Characterization of the PBC mirrors 12 ζ ≡ 1 √$_{3}$ ( $^{3}$D$_{1}$ || W $^{1}$| | $^{1}$S$_{0}$ ) . (A16) A (APV) M = i ( − 1) $^{q}$E 1 − q 〈 $^{3}$D$_{1}$ ; M | W 1 q | $^{1}$S$_{0}$ 〉 = i ( − 1) $^{q}$E 1 − q ( $^{3}$D$_{1}$ || W $^{1}$| | $^{1}$S$_{0}$ ) √$_{3}$ 〈 0 , 0; 1 , q | 1 , M 〉 = iζ ( − 1) $^{q}$E 1 − q 〈 0 , 0; 1 , q | 1 , M 〉 . (A15) W$_{i}$ = ∑ a ′ H 0 | a $^{′}$〉〈 a $^{′}$| d$_{i}$ ω ( $^{3}$D$_{1}$ ) − ω ( a $^{′}$) − ∑ a ′ d$_{i}$ | a $^{′}$〉〈 a $^{′}$| H 0 ω ( $^{1}$S$_{0}$ ) − ω ( a $^{′}$) . (A14) A (APV) M = i E i 〈 $^{3}$D$_{1}$ ; M | W$_{i}$ | $^{1}$S$_{0}$ 〉 , (A13) β ≡ 1 √$_{6}$ ( $^{1}$S$_{0}$ || U $^{1}$| | $^{3}$D$_{1}$ ) . (A12) A (Stark) M = ( − 1) $^{q}$T k − q 〈 $^{3}$D$_{1}$ ; M | U k q | $^{1}$S$_{0}$ 〉 = ( − 1) $^{q}$T k − q ( $^{3}$D$_{1}$ || U $^{k}$| | $^{1}$S$_{0}$ ) √$_{3}$ 〈 0 , 0; k, q | 1 , M 〉 = iβ ( − 1) $^{q}$( E × E )$_{−}$$_{q}$ 〈 0 , 0; 1 , q | 1 , M 〉 . (A11) U$_{ij}$ = ∑ a ′ d$_{i}$ | a $^{′}$〉〈 a $^{′}$| d$_{j}$ ω ( $^{3}$D$_{1}$ ) − ω ( a $^{′}$) + d$_{j}$ | a $^{′}$〉〈 a $^{′}$| d$_{i}$ ω ( $^{1}$S$_{0}$ ) − ω ( a $^{′}$) . (A10) A (Stark) M = T$_{ij}$ 〈 $^{3}$D$_{1}$ ; M | U$_{ij}$ | $^{1}$S$_{0}$ 〉 , (A9) A (APV) M = ∑ a ′ 〈 $^{3}$D$_{1}$ ; M | iH 0 | a $^{′}$〉〈 a $^{′}$| d · E | $^{1}$S$_{0}$ 〉 ω ( $^{3}$D$_{1}$ ) − ω ( a $^{′}$) − a ′ 〈 $^{3}$D$_{1}$ ; M | d · E | a $^{′}$〉〈 a $^{′}$| iH 0 | $^{1}$S$_{0}$ 〉 ω ( $^{1}$S$_{0}$ ) − ω ( a $^{′}$) . (A8) F = πc L τ, F = 2 π T$_{1}$ + T$_{2}$ + l$_{1}$ + l$_{2}$ . (B1) Following the analysis discussed in [21], if we denote the transmission of mirrors 1 and 2 by T$_{1}$ and T$_{2}$ , re- spectively, and the absorption+scatter loss per mirror as l$_{1}$$_{,}$$_{2}$ = ( A + S )$_{1}$$_{,}$$_{2}$, then the total cavity losses L = T$_{1}$ + T$_{2}$ + l$_{1}$ + l$_{2}$ determine the finesse F :