Measurement of Rb g-series quantum defect using two-photon microwave spectroscopy
MMeasurement of Rb g -series quantum defect using two-photon ng → ( n + 2) g microwavespectroscopy K. Moore ∗ Applied Physics Program, University of Michigan, Ann Arbor, MI 48109, USA
A. Duspayev † , R. Cardman and G. Raithel Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA (Dated: April 27, 2020)We utilize two-photon high-precision microwave spectroscopy of ng → ( n + 2) g transitions toprecisely measure the high-angular-momentum g -series quantum defect of Rb. The samples ofcold Rydberg atoms in the ng state are prepared via a three-photon optical excitation combinedwith controlled electric-field mixing and probed with 40- µ s-long microwave interaction pulses. Wecompare our results with a recent measurement and find our measurement to be consistent andone order of magnitude more precise. The experimental procedure includes a careful cancellationof stray electric fields in all three dimensions. Future extensions towards precision measurements ofatomic polarizabilities are discussed. I. INTRODUCTION
Measurements of atomic transition frequencies are thecornerstone of precision spectroscopy, used in applica-tions ranging from atomic clocks [1] to measuring grav-itational redshifts [2] and the radius of the proton [3].Often, cold atoms are used in these measurements. Al-kali atoms, which have a single valence electron similar tohydrogen, are easier to laser-cool than hydrogen due toa lower recoil energy and near-infrared cooling-transitionwavelengths. However, in an alkali atom such as rubid-ium (Rb), the interaction between the ionic core of theatom and the valence electron depresses the energy lev-els of the valence electron below the expected hydrogeniclevels (the“quantum defect”). In precision spectroscopy,it is imperative to determine this quantum defect for eachcommonly-used alkali species. Moreover, precision mea-surements of quantum defects can serve as a check for ad-vanced theoretical calculations and contribute to a betterunderstanding of the electronic structure in atoms.Here, we measure the high-angular-momentum ng -series quantum defect of Rb ( n is the principal quan-tum number). For electrons in high-angular-momentumstates, the quantum defect is dominated by the po-larizability of the ionic core, which may be extractedfrom high-angular-momentum defect measurements. Inthe most recent experimental measurement of the ng -series quantum defect, microwave spectroscopy of nd → ( n + 1) g transitions [4] was performed, whereas we usesub-THz spectroscopy to measure ng → ( n + 2) g transi-tions in a field-free environment. Our two-photon tran-sition depends only on one set of quantum defects andtakes advantage of equal Land`e- g factors in the lowerand upper states. We measure the ng -series δ and δ ∗ Present address: SRI International, Princeton, NJ 08540, USA † Corresponding author: [email protected] quantum defects (where δ and δ are the Ritz expan-sion coefficients [5]) of Rb with a precision improved byone order of magnitude each, better characterizing thishydrogen-like species for future precision measurements.Our results may pave the way towards an improved Rb + polarizability measurement, which is necessary for theprecision measurement of the Rydberg constant usingcircular Rydberg states [6, 7] and can help to solve theproton radius puzzle, for which an inaccurate Rydbergconstant has been named as a possible answer [3, 8]. II. METHODS
The experimental setup is shown in Fig. 1(a), the en-ergy level diagram is shown in Fig. 1(b), and the tim-ing sequence of the experiment is shown in Fig. 1(d).Atoms are laser-cooled and trapped in a magneto-opticaltrap (MOT). During each experimental cycle, we pre-pare atoms in an initial Rydberg ng state via an on-resonant three-stage optical excitation under simultane-ous application of a perturbative DC electric field. Theweak DC field admixes a small nf character into the ng state, allowing the single-photon 5 d → ng optical tran-sition (see Fig. 1(b)). As it can be seen from the ex-perimental Stark map shown in Fig. 1(c), we observesignificant population in the initial Rydberg ng state,well-isolated from the neighboring nh state and the hy-drogenic manifold. After state preparation, the pertur-bative DC electric field is adiabatically lowered (see Fig.1(d)) to the pre-determined zero-field value, thereby pro-ducing a sample of pure ng -state atoms. Next, a rectan-gular microwave pulse is applied for τ =40 µ s, drivingthe ng → ( n + 2) g transition. We scan the microwavefrequency across resonance and detect the population inthe target state ( n +2) g via state-selective field ionization(SSFI) [9].In our microwave system, microwaves are first gen-erated in a synthesizer (Agilent N5183A). Next, they a r X i v : . [ phy s i c s . a t m - c l u s ] A p r n m d e t u n i n g (a) (c)(b) (d) ng(n+2)g mm-wave1260nm780nm776nm DCelectric field 780nm776nm1260nmmm-waveDC mixingvoltageSSFIvoltageMOT light DC electric field zxy microwavehorn1260nm,776nm 780nm
FIG. 1: (Color online) Outline of the experiment. (a) Elec-trode, laser, and microwave-horn configuration in the experi-mental setup. The two z -electrodes present in the experimentare not pictured. (b) Level diagram of excitation scheme. (c)Experimental Stark map demonstrating the preparation of39 g population. Black indicates high ( >
40) Rydberg countsper detection cycle. The preparation consists of optical ex-citation of a DC-field-perturbed ng state (red squiggly line)and subsequent ramping of the DC field to zero (black arrow).(d) Partial timing diagram of experiment. Full experimentalperiod is 5 ms and includes MOT loading time, not picturedhere. are frequency-quadrupled in an active frequency mul-tiplier (Norden Millimeter N14-4680). The frequency-quadrupled output power can be varied over a limitedrange by adjusting the input power supplied by the syn-thesizer. While the input-to-output power relation maybe nonlinear, it nevertheless allows us to continuouslyvary the intensity at the location of the atoms over arange that allows us to observe the progression of thespectral lines from being indiscernible from the noise floorto being severely broadened, or until we have spanned theavailable multiplier input-power range. The sub-THz in-tensity control is important in quantifying the AC-Starkshifts of the transitions when evaluating systematic un-certainties.We measure the sub-THz frequency intervals coveringthe ng → ( n + 2) g transition for four choices of n . Foreach interval, we take six data series. Five of the seriesare to evaluate systematics, as outlined in the followingsections, and the final, longer data series is averaged toproduce a measurement result with low statistical uncer-tainty.We have specifically chosen to probe ∆ l = 0 (∆ j = 0)transitions. This allows us to select transitions with equal TABLE I: Summary of corrections. A bar over a zero digitindicates that the digit is significant. See text for the details.Shift(Hz) 38 g → g g → g g → g g → g DC Stark (z) 100 ± ± ± ± ± ± ±
765 459 ± ± ± ± ± ±
978 0 ±
965 0 ±
161 0 ± ±
11 0 ± ±
191 0 ± Land`e-g factors in the lower and upper states, eliminatingbroadening due to the MOT or other external magneticfields. Thus, the selected transitions have no Zeemanshifts.A summary of frequency corrections of our measure-ments is shown in Table I. One can see that the resid-ual DC-Stark shifts represent the largest source of un-certainty. In the following sections, the procedure of theuncertainty evaluation is discussed in more detail.
A. Statistical uncertainty evaluation andmicrowave-frequency calibration
By averaging many data sets in the absence of sys-tematic drift, we observe a Fourier-limited sinc -shapedspectral peak centered on resonance at ν c (Fig. 2). Forour microwave interaction time τ , the expected Fourier-sideband zeros at m ×
25 kHz ( m is a nonzero inte- Transition interval detuning (kHz) -150 0-50-100 50 100 1500.00.30.10.2 F r a c t i o n i n G ν c =191.21592511(22) GHz SystematicStatistical
FIG. 2: (Color online) Two-photon microwave spectrum ofthe 40 g → g transition. Black squares, fraction of atomsdetected in 42 g , detuned from ν c (black line). The statisti-cal uncertainty on ν c is less than the width of the black line.Uncertainty on the final corrected interval frequency (gray re-gion), listed in Table II, reflects systematic uncertainty. Errorbars on data points are SEMs. Red curve is a Lorentzian fit.In this Figure, the systematic corrections exhibited in TableI have not yet been applied to ν c . ger) coincide with local minima observed in the spec-trum. However, these sidebands are not well-resolved.Since a sideband-averaged sinc function approaches aLorentzian, we perform each peak fit using a Lorentzianfunction. We achieve a statistical uncertainty of the linecenters on the 100-Hz level. Since the frequencies ν n ,n of the transition intervals (as defined in Eqn. (2)) exceed0.1 THz, this amounts to a relative frequency uncertaintyof 10 − .In order to achieve this precision, it was critical tolock the internal crystal oscillator of our microwave syn-thesizer to a factory-calibrated, external atomic clock(SRS 725) with a relative uncertainty of ± × − .The absolute instrument uncertainty for the measured ng → ( n +2) g frequency interval is ≈
10 Hz. This instru-ment uncertainty is well below systematic uncertainties.For the 40 g → g frequency-interval measurement,we used a different atomic clock (DATUM LPRO) be-cause the SRS 725 was not available. The LPRO clockhad an unknown calibration and a maximum relative un-certainty of 10 − due to aging. We have determined theLPRO clock shift by beating the LPRO clock with thecalibrated SRS 725 used in the data sets for n = 38, 39,and 41. We determined that the LPRO runs faster bya relative amount of 2 . × − , showing that a correc-tion accounting for the LPRO’s clock shift was impor-tant. The frequency correction applied to the 40 g → g measurement that results from the LPRO clock shift isexplicitly listed in Table I. B. Systematic uncertainty evaluation
In this experiment, systematic uncertainties includeshifts in the resonance frequency due to electromagneticexternal fields or Rydberg-atom interactions.We have timed the excitation sequence such that opti-cal light is not present during the measurement interval,eliminating optical AC Stark shifts (see Fig. 1(d)).In order to minimize the shifts due to static electricfields, which are controlled in all three direction, we fol-low the standard procedure [4]. These field zeros aredetermined by varying a field direction E i ( i = x, y, z )while holding the other directions fixed until a minimumshift of the transition frequency ν is observed. This is de-termined by measuring ν as a function of the DC tuningvoltage that corresponds to field direction i and fittingthe result to a parabola. The uncertainty in the parabolicfit determines the uncertainty in the residual DC Starkshift contributed by the field direction i . The procedureis iteratively performed for all directions. The correctionsand uncertainties derived from this procedure are listedin Table I (the listed uncertainties include the noise of thetuning voltage sources). The uncertainty correspondingto the DC Stark shift in the z -direction is the dominantsystematic. We attribute this to the fact that the z -direction field is applied via a high-voltage amplifier thatis needed for SSFI. P e a k w i d t h ω L ( k H z ) Synthesizer output power (mW)1.9 2.3222.199200222.199280222.199360 0.150.450.30 F r a c t i o n i n G F r e q u e n c y ( G H z ) P e a k s h i f t Δ ( k H z ) ω L - γ (kHz ) Δ ( k H z ) (a)(b) (c) FIG. 3: (Color online) AC Stark shift analysis for the 38 g → g transition. This analysis is representative of the proce-dure used for all measured transitions. (a) Power map dis-playing synthesizer power as a function of interval frequency.Power broadening is evident. (b) Plot of measured peakwidths ω L (black squares, left axis) and peak center-frequencyshift γ (blue circles, right axis) versus synthesizer power.Expected Fourier limit γ F (red horizontal dashed line), ± γ (thick cyan line) are also displayed.Data points to the left of the vertical dashed line contributeto γ . (c) ∆ as a linear function of ω L − γ . Black squaresare the data points. Red line is a linear fit. In (a)-(b), notethat synthesizer output power is not linearly proportional tomicrowave intensity at the location of the atoms. In (b)-(c),∆ is detuned from the final measured value in Table II. Allerror bars are propagated from the standard errors of peakfits (Lorentzian model). The microwave intensities at the atom location are un-known due to nonlinearities in the frequency multiplierand other unknowns in the system. For that reason, wehave determined the AC Stark shift (and correspond-ing uncertainty) for each measured transition by analyz-ing the relationship between the power-broadened peakwidth and the center frequency location as the microwaveintensity is continuously varied (Fig. 3(a)). As the full-width-half-maxima ω L of the Lorentzian spectral profilesbroaden as a function of microwave power, microwave-induced AC Stark shifts ∆ of the two-photon transi-tion frequency between lower state | ng (cid:105) = | (cid:105) and upperstate | ( n + 2) g (cid:105) = | (cid:105) become evident. We observe thisrelation experimentally (Fig. 3(c)). Explicitly [10],∆ = β ( ω L − γ ) . (1)Here, β is a proportionality constant that includes theintermediate AC Stark shifts of the transition, the two-photon state-coupling, the natural decay rate of state | (cid:105) ,and the decay rate γ / | (cid:105) − | (cid:105) coherence ρ .We expect the spectral linewidth ω L to approach theFourier-limited linewidth γ F = 0 . /τ = 22 . γ to be theweighted average of the peak widths that fall within ±
50% of γ F (the gray region in Fig. 3(b)). Results are γ g − g = 29 . ± . γ g − g = 29 . ± . γ g − g = 27 . ± . γ g − g = 25 . ± . γ F , indicating that addi-tional sources of dephasing in the system have almost noeffect.For each transition, we use these values of γ to plot∆ versus ω L − γ (Fig. 3(c)) as we increase the mi-crowave intensity at the location of the atoms. The linearrelationship is consistent with the prediction of Eqn. (1).Moreover, the cluster of data points near the origin in-dicates that there is no measurable AC-Stark shift ∆ for the peaks of width ω L ≈ γ . The peak width of Average Rydberg counts per detection cycle P e a k s h i f t ( k H z ) F r a c . G Detuning (kHz)0-100 100Detuning (kHz)0.000.30 F r a c . G FIG. 4: (Color online) Rydberg-Rydberg shift analysis forthe 40 g → g transition. This analysis is representative ofthe procedure used for all measured transitions. Data points(black squares) represent measured peak locations versus av-erage Rydberg counts per detection cycle, detuned from thefinal measured value in Table II. Vertical error bars are thestandard errors of fit (Lorentzian model). Horizontal errorbars are the standard deviations of the counts. Insets displaymeasured resonance peak (black squares) and Lorentzian fit(red curve), detuned from final measured value in Table II.The dashed red curve indicates that a Lorentzian is not anideal fit function for the asymmetric peak. the spectrum obtained in each final data set is equal toor slightly less than γ to within the Lorentzian-fit un-certainty. Therefore, we take the AC Stark shift to bezero for all transition frequencies reported in Table II.For each transition, the uncertainties of the zero-shifts(Table I) are given by the uncertainty in the weightedaverage of the center frequencies of all data points thatcontribute to the calculation of γ (for n = 38, the pointswithin the gray region in Fig. 3(b)).Finally, we have identified a maximum number of Ryd-berg atoms that may be excited per detection cycle with-out inducing collisional shifts affecting the result, therebyoptimizing the signal/noise ratio while minimizing thatsource of uncertainty. In Fig. 4, we show the mea- (b)(a) n *-2 δ * ( n , n + ) *=0.00399976100-10-2020 R e s i d u a l fi t e rr o r ( k H z )
38 39 40 41 n δ = 0.0039997(26)δ = -0.0209(26) A FIG. 5: (Color online) Quantum defect determination. (a)Determination of δ ∗ . Black circles are data. Error bars are er-rors propagated from the final frequency-interval results (seeTable II). Red line is a weighted linear fit to data. (b) De-termination of δ and δ . Residual errors in nonlinear least-squares fit to data using model in Eqn. (2) are initializedusing δ ∗ and δ ∗ . Error bars are errors propagated from thefinal frequency-interval results (see Table II). See text for de-tails. sured peak position versus average total detected Ry-dberg counts per cycle for the 40 g → g transition.Although an increase in detected Rydberg counts canimprove the signal/noise ratio, Rydberg-Rydberg inter-actions cause a red shift of the detected transition fre-quency, as well as asymmetric broadening (compare in-sets of Fig. 4). The asymmetric lineshape observedat high Rydberg counts (in the inset of Fig. 4) is at-tributed to an attractive long-range electrostatic van derWaals interaction between Rydberg atoms. Although aLorentzian is not an ideal choice as a fit function forthese asymmetric shapes, it still provides an approximatevalue for the peak shift and broadening as a function ofdetected Rydberg counts. Importantly, for each tran-sition, we observe that below about ten detected Ryd-berg counts, the peak position and width converge to theinteraction-free values. Therefore, we limit our final datasets to this Rydberg-count region. III. RESULTS
We list the systematic corrections for each measuredtransition frequency in Table I. The uncertainties havebeen reported to three significant digits, so as to not loseinformation in the calculations of the corrected transitionfrequencies (which are rounded to two significant digitsin Table II) [11].To obtain values for the ng -series δ and δ quantumdefects, we follow the procedure of [12]. First, we obtainan average quantum defect δ ∗ ( n, n + 2) for the pair ofappropriate levels using ν n ,n = R Rb c (cid:18) n − δ ( n )) − n − δ ( n )) (cid:19) , (2)where R Rb is the Rydberg constant using the rubidiumreduced mass. Here, we substitute δ ( n ) = δ ( n ) = δ ∗ ( n, n + 2). In Table II, we list the final reported transi-tion frequencies ν n,n +2 , including corrections from TableI, along with the corresponding values of δ ∗ ( n, n + 2).Next, in Fig. 5(a) we plot δ ∗ ( n, n + 2) versus n ∗− A ,where n ∗ A = n + 1 − δ G . Here, δ G is the previous valueof δ determined by [4]. Extrapolating to n ∗− A = 0, weobtain an initial estimate δ ∗ = 0 . δ ( n ) = δ ∗ + δ / ( n − δ ∗ ) to solve for δ .Averaging the four results yields a preliminary value δ ∗ = − . δ ∗ and δ ∗ as initial values for the two free pa-rameters δ and δ , we perform a nonlinear least-squaresfit of Eqn. (2) to the transition-frequency values listed inTable II, where n is the independent variable. The resid-ual error is plotted in Fig. 5(b). We use a Levenberg-Marquardt algorithm with assigning weights of the datapoints to 1 /σ i ( σ i being a frequency uncertainty of the i -th data point). TABLE II: Summary of the results for transitions n → n + 2.See text for the details. n Transition frequency (GHz) δ ∗ ( n, n + 2)38 222.199268(14) 0.00397661(84)39 205.9325351(58) 0.00397784(38)40 191.2159300(47) 0.00397919(33)41 177.8690737(46) 0.00397980(36) Our final results for the ng -series quantum defects are δ = 0 . δ = − . ng -series measurements[4], but at least one order of magnitude more precise.Finally, we have explored how to use our precise ng -series energy-interval measurements to extract a new es-timate of the Rb + polarizability, using the polarizabilitymodel found in [4]. Unlike [4], however, our measure-ments are constrained to energy intervals between thesame l . Consequently, to extract α d and α q , we must in-stead adapt the method of [4] to measurements of shifts oftransition energies relative to their quantum-defect-freevalues, δW , by writing (in atomic units)2 δW (cid:104) /r (cid:105) D = α d + α q (cid:104) /r (cid:105) D (cid:104) /r (cid:105) D . (3)Here (cid:104) /r i (cid:105) D = (cid:104) /r i (cid:105) n,l − (cid:104) /r i (cid:105) n +2 ,l . Using Eqn. (3) toplot our data, we find that our data is consistent with thepolarizability results in [4]. However, as can be seen byexamining the (cid:104) /r i (cid:105) n,l functions in [13], the variationin (cid:104) /r i (cid:105) n,l has a much stronger l -dependence than n -dependence at the quantum numbers used in this experi-ment. As a result, the strong n -dependence of δ ∗ ( n, n +2)evident in Table II is not sufficient to allow for a precisedetermination of α d and α q . Therefore, to reach a defini-tive improvement on the measurements of α d and α q , inthe future a measurement of the nh -series quantum de-fect will be required on our setup as well. The ∆ l = 0sub-THz method presented in this paper is well-suited touse for such a measurement, and thus to determine im-proved Rb + polarizability values in future experiments.The systematics listed in Table I suggests that the sub-stantial experimental improvement is additionally possi-ble through a re-design of the SSFI apparatus. ACKNOWLEDGMENTS
KRM acknowledges support from the University ofMichigan Rackham Pre-Doctoral Fellowship. This workwas supported by NSF Grant No.PHY1506093 andNASA Grant No. NNH13ZTT002N NRA.
Note added. -
Recently, we became aware of re-lated work that reported precision measurements of theRb-core dipole and quadrupole polarizabilities and sub-sequently extracted quantum defects for g − , h − and i − series using n = 17 , ,
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