APS/123-QED Analysis of two-pathway coherent control for precision measurement of weak optical transitions D. Antypas 1 and D. S. Elliott 1 , 2 $^{1}$Department of Physics, $^{2}$School of Electrical and Computer Engineering Purdue University, West Lafayette, IN 47907 (Dated: October 28, 2018) We present a new technique based on two-pathway optical coherent control for the sensitive detection and precise measurement of highly-forbidden transitions in atomic systems. Specifically, we show that ( ω , 2 ω ) coherent control can be applied to the measurement of the magnetic dipole and electroweak parity nonconserving amplitudes in atomic cesium, with the principal advantage of reduced systematic errors related to field reversals often encountered in previous measurements of these effects. We present a complete analysis in one specific geometry, and discuss prospects for improved laboratory determinations of these weak transition amplitudes. PACS numbers: 11.30.Er, 32.90.+a, 32.80.Qk Highly-forbidden optical transitions have been mea- sured with great precision using the interference between the forbidden transition and a much stronger transition between the same two states. Key examples of these techniques are found in measurements of magnetic dipole transitions [1–8], electric quadrupole transitions [9, 10], and parity nonconserving (PNC) electroweak interac- tions [11–19]. Still, pushing the precision of these mea- surements to greater limits, as well as extending them to a variety of other atomic systems, is an active pur- suit, in that it will allow further tests of the Standard model and of nucleon-nucleon interactions within the nu- cleus [20, 21]. Currently a number of measurements of PNC interactions in various systems are underway, em- ploying a variety of different techniques, but each based on an interference between interactions driven by a sin- gle laser frequency component [22–24]. Systematic effects associated with imperfect reversal of the various dc elec- tric fields, magnetic fields, and optical polarization are often the limiting factor of these measurements. In this letter, we report a new technique based upon coherent control concepts for measuring these extremely weak in- teractions, and show that it eliminates the requirement for field reversals as the primary means of shifting the phase between terms, thereby reducing these associated errors. For concreteness, we consider measurement of the PNC-induced electric dipole moment using the 6 s $^{2}$S$_{1}$$_{/}$$_{2}$ → n $^{′}$s $^{2}$S$_{1}$$_{/}$$_{2}$ transition in atomic atomic cesium, but we expect the principles we discuss to be generally applicable to other cases as well. For n ′ = 7, this tran- sition was used extensively by Wieman et al. [18] and Bouchiat et al. [19] for the most precise measurements of PNC amplitudes ( E$_{PNC}$ ) to date. As an L = 0 → L = 0 transition, it is electric dipole and electric quadrupole forbidden. Stark-induced electric dipole (St), magnetic dipole (M1), and PNC-induced electric dipole interac- tions are each linear in the applied optical field ε ω 1 , where where ˆ k = k ω 1 / | k ω 1 | , and k ω 1 is the propagation vector of the ω$_{1}$ beam. Additionally, the electroweak interaction between the nucleus and the electrons, as well as higher order PNC effects, lead to a very slight mixing between even- and odd-parity eigenstates of the atom, making an To first order, the 6 s $^{2}$S$_{1}$$_{/}$$_{2}$ → n $^{′}$s $^{2}$S$_{1}$$_{/}$$_{2}$ transition is magnetic dipole forbidden, but spin-orbit interactions and relativistic effects relax this somewhat, and magnetic dipole moments M = 〈 n $^{′}$S | µ$_{z}$/c | 6 S 〉 , where µ$_{z}$ is the z - component of the magnetic dipole operator, have been determined for n ′ = 7 [5, 6] and n ′ = 8 [7]. The ampli- tude of the magnetic dipole transition is given by where α and β are the scalar and vector polarizabilities, ε ω 1 and φ ω 1 are the electric field amplitude and phase of the optical field, and the C F $^{′}$,m′ F,m are constants derived from the Clebsch-Gordon coefficients [16]. The polariz- abilities are given in Ref. [25]. The phase term in Eq. (1), e iφ $^{$^{ω}$1}$, which can be omitted for single-beam excitation, must be retained in our analysis. A$_{St}$ ( F, m ; F $^{′}$, m$^{′}$) = (1) e iφ $^{ω}$1 {[ α E · ε ω 1 δ$_{F,F}$ ′ + iβ ( E × ε ω 1 )$_{z}$ C F $^{′}$,m′ F,m ] δ$_{m,m}$ ′ + β [ ± i ( E × ε ω 1 )$_{x}$ − ( E × ε ω 1 )$_{y}$ ] C F $^{′}$,m′ F,m δ$_{m,m}$ ′ ± 1 } , the frequency ω$_{1}$ is resonant with one of the hyperfine components the 6 s → n $^{′}$s transition. The Stark-induced amplitude for transitions from state | 6 s $^{2}$S$_{1}$$_{/}$$_{2}$ , F, m 〉 to the state | n $^{′}$s $^{2}$S$_{1}$$_{/}$$_{2}$ , F $^{′}$, m$^{′}$〉 , where F and m ( F ′ and m $^{′}$) represent the total angular momen- tum and its projection in the ˆ z -direction of the | 6 s 〉 ( | n $^{′}$s 〉 ) state, is, in the notation of Gilbert and Wie- man [16], of the form I. INTRODUCTION A$_{M}$$_{1}$ ( F, m ; F $^{′}$, m$^{′}$) = e iφ $^{ω}$1 { ( ˆ k × ε ω 1 )$_{z}$ δ$_{m,m}$ ′ (2) + [ ± ( ˆ k × ε ω 1 )$_{x}$ + i ( ˆ k × ε ω 1 )$_{y}$ ] δ$_{m,m}$ ′ ± 1 } MC F $^{′}$,m′ F,m , arXiv:1001.2465v1 [physics.atom-ph] 14 Jan 2010