Iterative Precision Measurement of Branching Ratios Applied to 5P states in 88Sr+

We report on a method for measuring the branching ratios of dipole transitions of trapped atomic ions by performing nested sequences of population inversions. This scheme is broadly applicable and does not use ultrafast pulsed or narrow linewidth lasers. It is simple to perform and insensitive to experimental variables such as laser and magnetic field noise as well as ion heating. To demonstrate its effectiveness, we make the most accurate measurements thus far of the branching ratios of both 5P1/2 and 5P3/2 states in 88Sr+ with sub-1% uncertainties. We measure 17.175(27) for the branching ratio of 5P1/2-5S1/2, 15.845(71) for 5P3/2-5S1/2, and 0.05609(21) for 5P3/2-4D5/2, ten- fold and thirty-fold improvements in precision for 5P1/2 and 5P3/2 branching ratios respectively over the best previous experimental values.

Empirical measurements of elemental constants are fundamental to the verification and advancement of our knowledge of atoms. One important atomic property is the branching ratio of an electron transition: the ratio of its transition rate to the sum of rates of other decay channels with the same excited state. Measuring these constants accurately is vital for the refinement of relativistic many-body theories and provides a crucial probe in the study of fundamental physics such as parity nonconservation [1][2][3][4][5][6].
Branching ratios for different atomic species are also of great use in a wide range of fields including astrophysics, where analyzing the composition of stars contributes greatly to understanding stellar formation and evolution. Abundances of heavy elements such as strontium are essential for determining the efficiency of neutron capture processes in metal-poor stars, yet can be difficult to determine from emission spectra due to nearby transitions of other elements [7][8][9][10][11][12][13]. Branching ratios of these transitions are therefore vital for quantitative modeling of nucleosynthesis processes [11][12][13].
In addition, precise branching ratios enable the improvement of clock standards, paving the way for better global positioning systems and tests of the timeinvariance of fundamental constants [14]. Atomic clocks using the optical quadrupole transition 5S 1/2 -4D 5/2 in 88 Sr + , one of the secondary clock standards recommended by the International Committee for Weights and Measures [14], have achieved uncertainties at the 10 −17 level [15], more accurate than the current 133 Cs clock standard [16]. To further improve the precision of these systems, it is necessary to reduce uncertainty from the blackbody radiation Stark shift, the dominant source of error in room temperature clocks [2]. Branching ratios measured at the 1% level, combined with high-precision lifetime measurements, would allow for a significant reduction in blackbody radiation shift error by improving the accuracy of static polarizabilities of clock states [2,15].
Despite their relevance, branching ratios of heavy atoms have not been precisely measured for many decades due to the large uncertainties inherent in tradi-tional discharge chamber methods using the Hanle effect [17]. Recent astrophysical studies still use these older experimental results for fitting emission spectra [12,13]. Only in the last decade have there been precision measurements of branching ratios at the 1% level [18][19][20][21] using trapped ions, versatile toolkits for spectroscopy [4,15,22] as well as quantum computation [23]. In particular, pioneering work has been done by Ramm et. al. [18], establishing benchmark results for 40 Ca + and methods for three level lambda transition systems.
Here, we present a novel scheme for measuring the branching ratio of the P 3/2 state of a trapped ion with an iterative population transfer sequence, building upon this prior art. As with [18], we do not require ultrafast pulsed lasers or narrow linewidth lasers for addressing quadrupole transitions, which were used by previous precision measurements of P 3/2 branching ratios [20,24]. Our method uses only two lasers that pump the ion from the ground state to the P 1/2 and P 3/2 excited states and two lasers to unshelve the ion from the metastable states below P 3/2 . For 88 Sr + and analogous species, these lasers are already used for Doppler cooling, making this scheme broadly applicable for many trapped ion systems without the need for additional equipment. Like [18], our method is insensitive to experimental variables such as magnetic field and laser fluctuations, but what we present extends beyond lambda systems to allow branching ratios of more complex systems to be obtained. We demonstrate the effectiveness of our method by making the first precision measurement of the 5P 3/2 branching ratios in 88 Sr + in addition to the most accurate measurement of the 5P 1/2 branching ratios to date.
We begin by briefly describing the procedure for measuring branching ratios of J = 1/2 states using the method by Ramm et. al, which will be a building block for the J = 3/2 system. We use the 5P 1/2 excited state in 88 Sr + as the model system (Fig. 1). We denote the probability of decaying to the ground 5S 1/2 state as p and the long-lived 4D 3/2 state as 1 − p.
At the start of the experiment, the ion is intialized to the ground 5S 1/2 state. In the first step, the 422 nm laser is turned on to invert population to the excited 5P 1/2 arXiv:1605.04210v1 [physics.atom-ph] 13 May 2016 state while we record ion fluorescence at 422 nm. As the ion decays to the metastable 4D 3/2 state, we detect a mean number of photons n = 422 · p/(1 − p), where 422 is the detection efficiency of our system at 422 nm, before the ion is fully shelved. In the second step, the 1092 nm laser is turned on to repump the ion to the excited state, during which we detect 422 photons as it decays to the 5S 1/2 state. The branching ratio p/(1 − p) is therefore equal to the ratio of the number of counts observed during the two time intervals, independent of the collection efficiency.
For the more complex 5P 3/2 state, which decays to three instead of two states, we denote the probability of decaying to the 5S 1/2 , 4D 3/2 , and 4D 5/2 states as q, r, and s = 1 − q − r respectively ( Fig. 1). To measure the 5P 3/2 branching ratios q/(1 − q), r/(1 − r), and s/(1 − s), we begin with a sequence analogous to the 5P 1/2 sequence, this time detecting photons at both 408 nm and 422 nm. Starting again with the ion in the 5S 1/2 ground state, we first pump the ion into the excited 5P 3/2 state with the 408 nm laser (Step A). We detect a mean number of photons from the ion where 408 is the detection efficiency at 408 nm. We now turn on the 1033 nm laser, which drives the ion to the 5P 3/2 state if it was in the 4D 5/2 state and does nothing otherwise (Step B). We detect a mean number of 408 nm photons in this step. We can obtain the 5P 3/2 -4D 5/2 branching ratio s/(1−s) from the photon count ratio of the previous two steps.
To measure the other two branching ratios, we note that their values are contained in the state of the ion after Step B-the population split between the 5S 1/2 and 4D 3/2 states. To obtain this information, we now turn on the 422 nm laser to pump all 5S 1/2 population into the 4D 3/2 state (Step C). We detect photons at 422 nm. Finally, turning on the 1092 nm laser repumps all of the population to the 5S 1/2 state and we detect 422 photons (Step D), which is necessary for canceling the detection efficiency 422 .
Since we can determine p experimentally with the 5P 1/2 branching ratio sequence, we can solve for the 5P 3/2 branching ratios without knowing 422 or 408 : As with the 5P 1/2 measurement scheme by Ramm et. al., our sequence of population transfers is insensitive to detection efficiencies and many experimental variables. The long-lived shelving states 4D 3/2 and 4D 5/2 allow for the length of the measurement to far exceed the timescale needed for population transfer, rendering the measurement independent of laser power and frequency fluctuations as well as ion heating. There are no coherence effects or dark resonances since only one laser is on at a time, so our method is also insensitive to micromotion and magnetic field fluctuations. This distinguishes our method from a proposed P 3/2 branching ratio measurement scheme [26], which not only requires an extra laser for the 5P 3/2 -4D 3/2 transition but also that two lasers be alternately pulsed for each step to avoid dark resonances, making the measurement sequence significantly longer.
To demonstrate this method, we experimentally measure the 5P 1/2 and 5P 3/2 branching ratios in 88 Sr + . We trap single 88 Sr + ions using a surface electrode Paul trap fabricated by Sandia National Laboratories [27]. RF and DC confining fields are set such that the axial secular frequency of the ion is 600 kHz, with radial frequencies in the 3-4 MHz range and a 15 degree tilt in the radial plane. A magnetic field of 5.4 Gauss is applied normal to the trap to lift the degeneracy of the Zeeman states. Fluorescence from the ion is collected along the same axis by an in-vacuum 0.42 NA aspheric lens (Edmunds 49-696) into a single photon resolution photomultiplier tube (PMT, Hamamatsu H10682-210) with a filter that only passes light between 408 and 422 nm (Semrock FF01-415/10-25). The PMT signal is counted by an FPGA with arrival time binned into 2 ns intervals. The overall detection efficiency of the setup is approximately 4 × 10 −3 at both 422 nm and 408 nm. Dipole transitions of the ion are addressed using frequency-stabilized diode lasers. To execute the experimental sequence, we switch laser beams on and off using acousto-optic modulators (AOMs) driven by FPGA-controlled direct digital synthesizers.
Each branching ratio measurement cycle begins with 100 µs of Doppler cooling using 422 nm and 1092 nm lasers. Subsequently, we turn on only IR lasers for 20 µs to ensure the ion is in the 5S 1/2 state, then perform the experimental sequence. We ran the 5P 1/2 and 5P 3/2 branching ratio measurement sequences for 1.9 × 10 8 and 6.4 × 10 7 cycles respectively for a run time of 13 and 6 hours each. For each step within the experimental sequence, the laser is turned on twice: first the data interval where population transfer occurs, then the background interval that is subtracted from the data interval to obtain only fluorescence from the ion. For the 5P 1/2 experiment, the 422 nm and 1092 nm intervals are 35 µs and 25 µs in length respectively for both data and background, with 1 µs between each interval, and the 5P 3/2 experimental sequence is depicted in Fig. 2. To arrive at final values for the branching ratios, we carefully calibrated the systematic sources of error in our experiment, which are summarized in Table I.
The polarization alignment error arises from the Hanle effect and is a function of the magnetic field, detector position, and incident laser direction and polarization. In the 5P 1/2 system, the m = ±1/2 sublevels both emit radiation isotropically with 1:2 ratios of π-to σ-polarized light regardless of magnetic and electric fields, so this does not affect the measurement [28]. However, the Hanle effect is a major source of error for the 5P 3/2 system as the ratio of emitted π-to σ-polarized photons is 0:1 for m = ±3/2 sublevels and 2:1 for m = ±1/2 sublevels. The ratio of π to σ light emitted from Step A and Step B will therefore not be equal in general, biasing the fluorescence ratio.
To resolve this problem, we linearly polarize 408 and 1033 nm light to an axis 54.7 • (the magic angle [29]) with respect to the magnetic field, which is set orthogonally to the laser beam. At the magic angle, the ratio of π-to σ-polarized light emitted during Steps A and B are both equal to 1:2. The difference between radiation patterns of π and σ photons and any birefringence effects in the detection system cancel out. We use a Glan-Taylor polarizer with >50 dB attenuation of the orthogonal polarization to align the 408 and 1033 nm laser polarizations to within 0.2 degrees of the magic angle. The error in aligning the laser polarization with respect to the magnetic field and setting the magnetic field to be orthogonal to the laser beam accounts for the polarization alignment error in Table I   curves using optical Bloch equations to determine the shift and uncertainty contributed by each error source. PMT dead time, calibrated to be 20±1 ns for our system using the method by Meeks and Siegel [30], leads to more undercounting at higher count rates. Finite laser durations reduces the fluorescence from the ion in each step in addition to preparing states imperfectly. The small amount of laser light still present when the AOMs are switched off (extinction ratios >60 dB) leads to slight coupling between undesirable states. The finite lifetime of the 4D 3/2 and 4D 5/2 states lead to extra counts in the blue intervals and reduced counts in the IR intervals. Off-resonant excitations, where the ion is excited to the wrong state by a collision or far-detuned laser, are found to contribute negligible errors to our system based on measuring the frequency of dark events while Doppler cooling the ion. We find that these sources of systematics do not limit our current level of accuracy. We also verify that the fluorescence from the ion is normally distributed when binned into 500,000 measurement cycles.
The largest source of error for both 5P 1/2 and 5P 3/2 branching ratios is from counting statistics. This can be improved via either more measurement cycles, more ions, or greater collection efficiency, though for the latter two methods it is important to take into account the increased error from PMT dead time. Other errors can also be reduced via improvement of the experimental apparatus, such as more careful alignment of the laser polarization and using a PMT with less dead time. The only fundamental limitation to the accuracy of the technique is the uncertainty on the finite lifetimes of the 4D 3/2 and 4D 5/2 states, which restricts the length of the population inversion sequence, but the limit is many orders of magnitude below the current level of accuracy.
After applying systematic shifts and propagating uncertainties, we obtain for the 5P 1/2 branching ratio p/(1 − p) = 17.175 (27)  The uncertainty of our results is at a level smaller than the discrepancy between previous experimental and theoretical results, as shown in Fig. 3. Our value for the 5P 1/2 -5S 1/2 branching ratio agrees with the recent trapped ion experiment done by Likforman et. al. [19] as well as theoretical values of Safronova [1] and Jiang et. al. [2], while disagreeing with the gas discharge chamber experiment by Gallagher [17]. For 5P 3/2 , only the 5P 3/2 -5S 1/2 branching ratio has been previously reported, also by Gallagher, which our value is in agreement with. We are also in agreement with theory values of Safronova and Jiang et. al. for all 5P 3/2 branching ratios. We note that Safronova's theoretical values have been found to be in good agreement with precision measurements of branching ratios and dipole matrix elements in other elements [18,22,[31][32][33]. We obtain a ten-fold improvement in accuracy for the 5P 1/2 branching ratios over Likforman et. al. and a forty-fold improvement for the 5P 3/2 -4D 5/2 branching ratio over Gallagher.
In summary, we have introduced a novel method for measuring the branching ratio of the J = 3/2 state in trapped ion systems and demonstrated its effectiveness by measurements in 88 Sr + at the sub-1% level. Our scheme, as with the Ramm et. al. method for J = 1/2 states, uses only dipole transition addressing lasers and is insensitive to detector efficiencies, laser and magnetic field fluctuations, as well as ion heating and micromotion. The branching ratios of many higher-up excited states, such as the 6S 1/2 and 5D 3/2 states in 88 Sr + , can also be measured by applying this method of chaining measurement sequences of successive states with selective detection of dipole transitions. This scheme is also broadly applicable to excited states in other elements with a similar structure of decaying into a ground state and longlived states, such as secondary clock standards 199 Hg + and 171 Yb + , for which greater branching ratio and lifetime precision can reduce uncertainty from blackbody radiation as well [35,36].
The authors acknowledge helpful discussions with Hartmut Häffner, Christian Roos, and Luca Guidoni. This work was supported in part by the IARPA MQCO program and by the NSF Center for Ultracold Atoms.