the systematic uncertainty, for example, a finite accuracy of the polarization angle flip, errors in the lock-in ampli- fiers, a finite dynamic range of the lock-ins etc. The net contribution of these factors is found to be ≲ 1%. E. D. Commins, B. P. Das, D. DeMille, A. Dilip, S. J. Freedman, J. S. Guzman, G. Gwinner, M. G. Kozlov, S. M. Rochester, and M. Zolotorev. This work has been supported by NSF. A summary of the systematic error budget is presented in Table IV. As pointed out above, a goal of the future measure- ments of the parity-violation effects in ytterbium is ob- serving a difference in the PV effect between different isotopes. The net uncertainty of the PV parameter of a single isotope must be better than 1% based on the the- oretical predictions. To this end, a program of the appa- ratus upgrades and improvements is developed. Besides general improvements of the stability of the system pa- rameters, increase of the signal levels, suppression of the electronics noise etc., the main focus is on elimination of the frequency excursions of the frequency reference, which is a major source of the statistical noise. Improving the statistical uncertainty will contribute to more precise measurement and control of the E-field-imperfection con- tribution to the systematic part of the uncertainty. The latter is another high-priority improvement essential for reaching the goal. In the future apparatus, the referencing of the opti- cal system to the Fabry-P´ erot ´ etalon will be replaced by locking the system to a femtosecond frequency comb that will be available shortly. The impact of the E-field imperfection is planned to be substantially suppressed by redesigning of the interaction region to provide more uniform and controlled electric field distribution. Until now, no scientific or technical obstacles were discovered preventing us to improve the apparatus to the desired level of sensitivity. The authors acknowledge helpful discussions with and important contributions of M. A. Bouchiat, C. J. Bowers, where where ω ( a ) is the energy of state | a 〉 . (Perturbed eigen- states are denoted using an overbar.) The electric-dipole amplitude for the optical transition of interest is In the presence of a strong magnetic field, that is, when Zeeman splitting dominates Stark-shifts, it is use- ful to think of H$_{1}$ ≡ H$_{Stark}$ + H$_{APV}$ as a perturbation to H$_{0}$ ≡ H$_{Atomic}$ + H$_{Zeeman}$ . In this case, the LS -coupled states | 2 S $^{+1}$L$_{J}$ ; M 〉 , such as | $^{3}$D$_{1}$ ; M 〉 and | $^{1}$S$_{0}$ ; 0 〉 , are eigenstates of the unperturbed Hamiltonian H$_{0}$ . Then the first-order perturbation theory can be used to deter- mine the eigenstates of the total Hamiltonian: where H 0 is a scalar operator. Summation over repeated indices is assumed. where d is the atomic electric-dipole operator. Finally, where µ = − gµ$_{B}$ J is the magnetic dipole moment of the atom, g is the Land´ e factor, µ$_{B}$ is the Bohr magneton, and J is the angular-momentum operator. Similarly, where H$_{Atomic}$ is the atomic Hamiltonian and H$_{Zeeman}$ , H$_{Stark}$ , and H$_{APV}$ represent the contributions from the static magnetic field B , the static electric field E , and the parity non-conserving weak interaction, respectively. Here The total Hamiltonian (before including light-atom in- teractions and assuming B is along ˆ z ) can be written as Appendix A: Derivation of transition amplitudes 11 H = H$_{Atomic}$ + H$_{Zeeman}$ + H$_{Stark}$ + H$_{APV}$, (A1) H$_{Zeeman}$ = − µ · B = gµ$_{B}$ J · B = gµ$_{B}$J$_{z}$B, (A2) H$_{Stark}$ = − d · E = − d$_{i}$E$_{i}$, (A3) H$_{APV}$ = iH 0 $_{0}$, (A4) | a 〉 = | a 〉 + ∑ a ′ | a $^{′}$〉〈 a $^{′}$| H$_{1}$ | a 〉 ω ( a ) − ω ( a $^{′}$) , (A5) A (E1) M = 〈 $^{3}$D$_{1}$ ; M | ( − d · E ) | $^{1}$S$_{0}$ 〉 ≡ A (Stark) M + A (APV) M , (A6) A (Stark) M = ∑ a ′ 〈 $^{3}$D$_{1}$ ; M | d · E | a $^{′}$〉〈 a $^{′}$| d · E | $^{1}$S$_{0}$ 〉 ω ( $^{3}$D$_{1}$ ) − ω ( a $^{′}$) + a ′ 〈 $^{3}$D$_{1}$ ; M | d · E | a $^{′}$〉〈 a $^{′}$| d · E | $^{1}$S$_{0}$ 〉 ω ( $^{1}$S$_{0}$ ) − ω ( a $^{′}$) , (A7) IX. ACKNOWLEDGEMENTS VIII. TOWARDS MEASUREMENT OF THE ISOTOPE RATIOS TABLE IV: List of factors contributing to the systematic un- certainty of the PV parameter, ζ/β . Factor Uncertainty (%) ˜ E value: geometry 5 numerical modeling 3 E-field imperfections 5 Phase mixing 0.5 Other 1 Total (in quadrature) 8