# Precision Measurements

## Electric dipole transition moments

Using a variety of different techniques, we have been carrying out a series of precision measurements of electric dipole (E1) transition moments in atomic cesium. These measurements are critical to PNC studies such as those described elsewhere on this site, for two reasons. First, the weak-force induced E1 transition moment E_{PNC} for the 6s ^{2}S_{1/2} → 7s ^{2}S_{1/2} transition is calculated by theorists as a sum over states expression of products of these electric dipole moments and weak interaction moments. So these moments are needed for the direct calculation of E_{PNC}. Second, these measurements of electric dipole moments serve as a validity check for the high-power theoretical models that theorists devise for the electronic structure of the cesium atom during the course of their calculations. Not all the terms needed in their intensive calculations can be checked experimentally, but if there is agreement between experimental and theoretical results for those that can be checked, we gain confidence in each of the quantities calculated.

In the figure to the right, we show a representation of the level of agreement between the state-of-the-art experimental and theoretical results for the electric dipole transition moments for each of the ns ^{2}S_{1/2} → mp ^{2}P_{J} transitions in cesium, where n and m are the principal quantum numbers 6 or 7, and J = 1/2 or 3/2 is the electronic angular momentum of the mp ^{2}P_{J} state. The solid horizontal bar at the center of each line shows the uncertainties of the experimental determinations of these moments, while the open circles [1], asterisks [2], and crosses [3] show the deviation between three theoretical works and the experimental result. Note that most of experimental results have uncertainties of ~0.1% or smaller, and that many of the theoretical results are in excellent agreement with these experiments.

The experimental results are derived from a number of different techniques, from many different groups, including ours. We discuss these moments briefly in the following:

__ 6s – 6p__: In the past thirty years, there have been several measurements of these moments, with remarkable precision and agreement. These measurements were based on fast-beam laser [4, 5], time-resolved fluorescence [6], ultra-fast pump-probe laser [7], photoassociation [8, 9, 10], ground-state polarizability [11] and atom interferometry [51] techniques. The weighted average of these results provides an uncertainty of 0.035% for these two moments.

__ 7s – 6p__: We used a time-resolved florescence technique to determine the lifetime of the 7s

^{2}S

_{1/2}state [13], with a precision of 0.14%. This state spontaneously decays to two states (6p

^{2}P

_{3/2}and 6p

^{2}P

_{1/2}), so the lifetime measurement by itself is not sufficient to determine the individual moments for the two decay pathways. Therefore, we also carried out a second measurement [14], based on two-photon excitation of the 6s

^{2}S

_{1/2}→ 7s

^{2}S

_{1/2}transition with a two-color laser field. By comparing the excitation rate when the two laser fields were parallel or perpendicular to each other over a range of laser wavelengths, we determined the ratio of the two transition moments connecting the 7s

^{2}S

_{1/2}state with the 6p

^{2}P

_{3/2}and 6p

^{2}P

_{1/2}states. The precision of the resulting moments is <0.1%.

__ 7s – 7p__: We were able to use the results described in the previous paragraph, combined with a precision measurement of the Stark shift of the 7s state carried out by Wieman’s group in Boulder [15], to derive a new measurement of the transition moments for the 7s →7p states. The precision of these moments is 0.05%.

__ 6s – 7p__: We carried out a series of measurements [16] of the absorption strength in a Doppler-broadened cesium vapor cell from a laser tuned to the 6s

^{2}S

_{1/2}→ 7p

^{2}P

_{3/2}transition at 456 nm or the 6s

^{2}S

_{1/2}→ 7p

^{2}P

_{1/2}transition at 459 nm. We compared the absorption strength on these lines with that of the 6s

^{2}S

_{1/2}→ 6p

^{2}P

_{1/2}transition at 894 nm to eliminate errors related to the cell length and vapor density. The precision of these moments is 0.10 – 0.16%.

__ 7p – 5d__: In a related study, we determined the lifetime of the 7p

^{2}P

_{3/2}and 7p

^{2}P

_{1/2}states in atomic cesium [17]. The 7p states can decay spontaneously to three lower electronic states (7s

^{2}S

_{1/2}, 6s

^{2}S

_{1/2}, or 5d

^{2}D

_{J}), so we use known moments for the transitions to the 7s

^{2}S

_{1/2}and 6s

^{2}S

_{1/2}states to determine the transition moment to the 5d

^{2}D

_{J}.

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[5] R. J. Rafac, C. E. Tanner, A. E. Livingston, and H. G. Berry, Phys. Rev. A **60**, 3648 (1999).

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[7] B. M. Patterson, J. F. Sell, T. Ehrenreich, M. A. Gearba, G. M. Brooke, J. Scoville, and R. J. Knize, Phys. Rev. A **91**, 012506 (2015).

[8] A. Derevianko and S. G. Porsev, Phys. Rev. A **65**, 053403 (2002).

[9] N. Bouloufa, A. Crubellier, and O. Dulieu, Phys. Rev. A **75**, 052501 (2007).

[10] Y. Zhang, J. Ma, J. Wu, L. Wang, L. Xiao, and S. Jia, Phys. Rev. A **87**, 030503(R) (2013).

[11] J. M. Amini and H. Gould, Phys. Rev. Lett. **91**, 153001 (2003).

[12] M. D. Gregoire, I. Hromada, W. F. Holmgren, R. Trubko, and A. D. Cronin, Phys. Rev. A **92**, 052513 (2015).

[13] G. Toh, J. A. Jaramillo-Villegas, N. Glotzbach, J. Quirk, I. C. Stevenson, J. Choi, A. M. Weiner, and D. S. Elliott, Phys. Rev. A **97**, 052507 (2018).

[14] G. Toh, A. Damitz, N. Glotzbach, J. Quirk, I. C. Stevenson, J. Choi, M. S. Safronova, and D. S. Elliott, Phys. Rev. A **99**, 032904 (2019).

[15] S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A **59**, R16 (1999).

[16] A. Damitz, G. Toh, E. Putney, C. E. Tanner, and D. S. Elliott, Phys. Rev. A **99**, 062510 (2019).

[17] George Toh, Nathan Chalus, Andrew Burgess, Amy Damitz, Poolad Imany, Daniel E. Leaird, Andrew M. Weiner, Carol E. Tanner, and D. S. Elliott**, **Phys. Rev. A 100, 052507 (2019).

##
Scalar polarizability for the 6s ^{2}S_{1/2} → 7s ^{2}S_{1/2} transition

The 6s ^{2}S_{1/2} → 7s ^{2}S_{1/2} transition is forbidden to first order according to electric dipole selection rules. It is weakly allowed, however, through a magnetic dipole interaction, a Stark-induced interaction, or through an electric quadrupole interaction. The scalar polarizability α is a parameter used to characterize the strength of the Stark-induced interaction. That is, the Stark-induced transition amplitude A_{St} = αεE, where ε is the laser electric field amplitude, and E is the static electric field strength. We can think of the Stark-induced transition in terms of the distortion of the electronic structure of the atoms when one applies a static electric field to the region in which the atom resides, essentially mixing the eigenstates of the system, and allowing a weak electric dipole transition. The scalar polarizability α is the relevant parameter when the static electric field and the laser polarization driving the transition are parallel to one another. When these two fields are mutually perpendicular, the transition strength is governed by a related parameter known as the vector polarizability, β. Measurements of the weak amplitude depend critically on the precise determination of these polarizabilities, as the laboratory measurement typically yields the value of a ratio such as E_{PNC}/β. There is currently a serious discrepancy between two competing methods of determining the vector polarizability β for the 6s ^{2}S_{1/2} → 7s ^{2}S_{1/2} transition in cesium. In one method, which has for the past 20 years been the more precise, a theoretical calculation of the hyperfine component of the magnetic dipole moment M1_{hf} for the 6s ^{2}S_{1/2} → 7s ^{2}S_{1/2} transition [1] is used with a laboratory determination of the ratio M1_{hf}/β[2], resulting in β = 26.957 (51) a_{0}^{3} [1], where a_{0} is the Bohr radius. We recently used the precise electric dipole matrix elements discussed above to calculate a new, high precision value for the scalar polarizability α = -268.82 (30) a_{0}^{3} [3]. Then combining this result with a laboratory measurement of α/β from Wieman’s group [4], we determine β = 27.139 (42) a_{0}^{3} [3]. While the uncertainty of our new determination is slightly improved relative to the former value, these two values of the vector polarizability are not in good agreement with one another, differing by 0.182 (66) a_{0}^{3}. Resolution of this discrepancy is critical. To this end, we are currently setting up a new experimental determination of α/β. We also have plans for a new measurement of M1_{hf}/β. Between these two measurements, we hope to find consistency between these two methods of determining the vector polarizability, and to find consensus on the proper value of the vector polarizability.

[1] V. A. Dzuba and V. V. Flambaum, Phys. Rev. A **62**, 052101 (2000).

[2] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. **82**, 2484 (1999).

[3] George Toh, Amy Damitz, Carol E. Tanner, W. R. Johnson, and D. S. Elliott, Phys. Rev. Lett. **123**, 073002 (2019).

[4] D. Cho, C. S. Wood, S. C. Bennett, J. L. Roberts, and C. E. Wieman, Phys. Rev. A **55**, 1007 (1997).