Observation of electric quadrupole transitions to Rydberg nd states of ultracold rubidium atoms

Observation of electric quadrupole transitions to Rydberg nd

states of ultracold rubidium atoms

Citation for published version (APA):

Tong, D., Farooqi, S. M., Kempen, van, E. G. M., Pavlovic, Z., Stanojevic, J., Coté, R., Eyler, E. E., & Gould, P. L. (2009). Observation of electric quadrupole transitions to Rydberg nd states of ultracold rubidium atoms. Physical Review A : Atomic, Molecular and Optical Physics, 79(5), 052509-1/7. [052509].

https://doi.org/10.1103/PhysRevA.79.052509

DOI:

10.1103/PhysRevA.79.052509 Document status and date: Published: 01/01/2009

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Observation of electric quadrupole transitions to Rydberg nd states of ultracold rubidium atoms

D. Tong, S. M. Farooqi, E. G. M. van Kempen, Z. Pavlovic, J. Stanojevic, R. Côté, E. E. Eyler, and P. L. Gould

Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA 共Received 13 September 2008; published 15 May 2009兲

We report the observation of dipole-forbidden, but quadrupole-allowed, one-photon transitions to high-Rydberg states in Rb. Using pulsed uv excitation of ultracold atoms in a magneto-optical trap, we excite 5s →nd transitions over a range of principal quantum numbers n=27–59. Compared to dipole-allowed 共E1兲 transitions from 5s→np, these E2 transitions are weaker by a factor of approximately 2000. We also report measurements of the anomalous np3/2: np1/2fine-structure transition strength ratio for n = 28– 75. Both results

are in agreement with theoretical predictions.

DOI:10.1103/PhysRevA.79.052509 PACS number共s兲: 32.70.Cs, 32.80.Ee, 31.10.⫹z

I. INTRODUCTION

Samples of ultracold atoms have a number of benefits for spectroscopic measurements. The low velocities result in re-duced Doppler shifts and long interaction times, significantly reducing both Doppler broadening and transit-time broaden-ing. In addition, the atoms can be highly localized and

pre-pared in specific states. Microwave关1兴 and optical transitions

关2–4兴 involving excited atomic states have been investigated

with ultracold samples, as have photoassociative processes

关5兴, which probe bound molecular states. An example closely

related to the results presented here is the observation of the

Na 3p→4p quadrupole transition in an ultracold sample 关6兴.

In the present work, we use pulsed uv excitation of ultra-cold Rb atoms to measure the oscillator strengths of weak single-photon transitions from the 5s ground state to nd

Ry-dberg states, where nⰇ1 is the principal quantum number. In

zero electric field, there is no Stark mixing and these 5s →nd transitions are dipole 共E1兲 forbidden, but quadrupole 共E2兲 allowed. Such transitions must be considered when us-ing excitation spectra to probe external electric fields or the interactions between ultracold Rydberg atoms. Also, these E2 transitions are potentially useful in extending the number of Rydberg states that can be excited. For example, two-frequency uv light could be used to simultaneously excite

Rb 5s atoms to np and共n−1兲d states via one-photon

transi-tions. Pairs of atoms in these states will have strong

dipole-dipole interactions 关7,8兴. In previous work with Rb, E2

os-cillator strengths to nd states for n = 4 – 9 关9兴 and for n=4

关10兴 were measured. By contrast, we probe states of much

higher n : n = 27– 59. We determine absolute E2 oscillator strengths by combining our measured ratios of signals for the

5s→nd 共E2兲 and 5s→共n+1兲p 共E1兲 transitions with

previ-ously determined absolute E1 oscillator strengths 关11兴. To

compare to our measurements, we calculate the E2 oscillator strengths for high n using phase-shifted Coulomb wave func-tions. This comparison yields reasonably good agreement.

We also present measurements of the anomalous ratio of E1 transition strengths to the two np fine-structure states, J

= 3/2 and J=1/2. This ratio is a sensitive probe of electronic

wave functions in many-electron atoms. Our results, cover-ing the range n = 28– 75, are found to be consistent with a number of previous calculations and with previous measure-ments, most of which involve lower n.

In Sec. II, the E2 oscillator strength calculations are

de-scribed. In Sec.III, we describe the experimental setup.

Mea-surements of E2 transition strengths and E1 fine-structure transition strength ratios are described and compared to

theory in Secs.IVandV, respectively. SectionVIcomprises

concluding remarks.

II. THEORY

We first compute the E2 excitation probability for a 5s →nd transition by a linearly polarized laser pulse of finite duration. We start with the Hamiltonian for the electric

quad-rupole interaction 关12兴

HQ=

e

2共Eជd· rជ兲共kជ· rជ兲, 共1兲

where Eជdis the electric field, rជthe position of the electron,

and kជ the wave number of the electromagnetic field.

In the present experiment the ground-state atoms have no alignment or orientation, so the excitation probability is not polarization dependent. This allows us to neglect the

hyper-fine structure and to start with a 5s1/2 initial state, which is

not capable of alignment. We nevertheless treat in explicit form the case of linear polarization so that our results can readily be generalized to more complicated cases.

For simplicity, we choose the electric field along z and the wave travelling along x, i.e., Eជd= Ed共t兲eˆz and kជ= keˆx, where

Ed共t兲=Ed0cos␻t. Within the rotating-wave approximation,

Eq. 共1兲 becomes HQ= e 2Ed共t兲kxz = eEd0k 4 xz, 共2兲

and using xz = −r2

冑

2␲₁₅共Y₂1− Y₂−1兲, it can be written in term of

spherical harmonics Y_ᐉmas HQ= eE_d0k 4 r 2

冑

2␲ 15_m=−1

兺

_/2,3/2am−1/2Y2 m−1/2_{, 共3兲} with a−1= + 1 and a1= −1.

For Rb in a magneto-optical trap共MOT兲, the ground state

兩5s1/2 mj=⫾

1

2典 can only be excited to d states via E2

tran-sitions. We consider here the case mj=

1

2; identical results

follow for mj= −

1

2. From the selection rules, mj changes by

⫾1 and the allowed d states 共with j=5

2 or

3

2兲 are 兩n dj mj

= −₂1典 and 兩n dj mj=

3

2典. If we label the amplitude in the

ground 5s state by csand the amplitude in the excited states

兩n d m典 with m=−1

2 or

3

2 by cm, we get the following Bloch

equations: idcs dt =

冑

2␲ 15 eEd0k 4ប 共5s兩r2兩nd兲 ⫻

兺

m=−1/2,3/2

具

1 2, 1 2

兩

a1/2−mY21/2−m兩jm典cm, 共4兲 idcm dt =

冑

2␲ 15 eEd0k 4ប 共nd兩r2兩5s兲具jm兩am−1/2Y2 m−1/2

_兩

1 2, 1 2

典

cs, 共5兲

where共nd兩r2_{兩5s兲 stands for the radial matrix element only. If}

we define cd_j⬅

兺

m=−1/2,3/2

具

1 2, 1 2

兩

Y2 1/2−m_兩jm典a 1/2−mcm, 共6兲 and W⬅

冑

2␲ 15 eE_d0k 2ប 共nd兩r 2_{兩5s兲, 共7兲} ␤2j⬅

兺

m=−1/2,3/2 am−1/2a1/2−m具12, 1 2兩Y21/2−m兩jm典具jm兩Y2m−1/2兩 1 2, 1 2典, 共8兲 we can rewrite the Bloch equations as

idcs dt = W 2cdj, 共9兲 idcdj dt =␤j 2W 2cs. 共10兲

The solutions are simply cs共t兲 = cos W␤jt 2 , 共11兲 cd_j共t兲 = − i␤jsin W␤jt 2 , 共12兲

and the quadrupole excitation probability Pj

Q_{= 1 −兩c}

s兩2 has

the form of a Rabi equation,

Pj Q = sin2W␤jt 2 ⯝ W2␤j 2 t2 4 . 共13兲

The result on the right assumes a short interaction time. We

also note that W is assumed to be real here 共i.e., no chirp兲.

After some algebra and using am−1/2a1/2−m= −1, the

ex-pression for ␤2j for j =

5 2 and 3 2 is found to be ␤j 2 =2j + 1 20␲ . 共14兲

Therefore, ignoring the j dependence of 共nd兩r2_{兩5s兲, we}

re-cover the statistical ratio of the 5/2 and 3/2 components for E2 excitations P_j=5Q _/2 P_j=3Q _/2⯝ W2␤_5/22 t2/4 W2␤_3/22 t2/4= 3 2. 共15兲

To compare the measured E2 signal for a 5s→ndd

quad-rupole transition with a nearby E1 5s→npp dipole transition,

we compute the ratio ␩ between the E2 and E1 excitation

probabilities, assuming Pjdip_p = sin2␻j_pt/2 and the same pulse

duration t for both cases

␩j_p,j_d= Pj_d Q Pj_p dip⯝ W2␤j_d 2 ␻j_p 2 . 共16兲

For 5s→nppjtransitions with polarized light共Eជp= Epeˆz兲, we

have for the E1 Rabi frequency

␻j_p= eDj_pEp0 ប , 共17兲 with eDj_p=

冑

fj_p 6បe2 2me2␲␯

冉

1 2 1 jp 1 2 0 − 1 2

冊

2 , 共18兲

where fj_pis the oscillator strength to the jpcomponent and␯

is the transition frequency in Hz. From the definition of W

and the result for␤2_{, we have}

␩j_p,j_d= 共2jd+ 1兲 150 E_d02 E_p02 k2 4Dj_p 2兩共nd兩r 2_兩5s兲兩2 _共19兲 =6.421⫻ 10−10共2jd+ 1兲 fj_p Id Ip 兩共nd兩r2_兩5s兲兩2_, 共20兲

where共nd兩r2_{兩5s兲 is in atomic units and I}

dand Ipare the laser

intensities used to excite the npp and ndd states, respectively.

We also use the fact that k = 2␲␯/c, where for these

high-Rydberg states we take h␯⯝h共1.010⫻1015 _{Hz兲 as equal to}

the 5s ionization energy. We note that Eqs.共16兲 and 共19兲 are

valid even in the presence of a frequency chirp.

It is easy to modify Eq.共20兲 to find the ratio␩of the total

signal sizes summed over fine-structure components. This

requires only summation over jd=

5

2 and

3

2 in the numerator

and replacement of fj_p in the denominator by the total E1

oscillator strength to the p state, fp,

␩=6.421⫻ 10 −9 fp Id Ip 兩共nd兩r2_兩5s兲兩2_{. 共21兲}

Finally it is useful to relate the ratio␩, which in general is

polarization dependent, to the ratio of oscillator strengths

fd共nd兲/ fp共np兲, which is not. In the present experiment the

absence of alignment in the initial state nullifies this

distinc-TONG et al. PHYSICAL REVIEW A 79, 052509共2009兲

tion and it is easily shown that ␩= fd共nd兲/ fp共np兲, where the

E2 oscillator strength is given by fd共n兲=6.231

⫻10−54_␯3_兩共nd兩r2_兩5s兲兩2_.

Although the E1 oscillator strengths to the high-n p states

are well known 关11兴, the quadrupole oscillator strengths fd

are not. To evaluate them numerically for comparison to our experiment, we have computed the quadrupole matrix

ele-ment 兩共nd兩r2兩5s兲兩2 using the model potential for Rb 5s

pro-duced by Marinescu et al. 关13兴 and phase-shifted Coulomb

wave functions for the Rydberg states. This precise one-electron model potential was developed to represent the mo-tion of the valence electron in the field of the closed

alkali-metal positive-ion core. In Table I we show the calculated

radial matrix elements and the corresponding values of the

total oscillator strength fd. To evaluate their accuracy, we

compare them to measured values of E2 transitions by

Ni-emax 关9兴 and calculated values by Warner 关14兴 for low nd

states. For nd⬍10, our results differ by 20–35 % from those

of Warner and by 40– 70 % with those of Niemax, although the agreement with Niemax’s values seems to get better

rap-idly with growing nd, falling to 5% for nd= 9. Overall, we

estimate the error of our calculated oscillator strengths to be roughly 40– 60 %.

III. EXPERIMENT

In our experiments, we use pulsed uv excitation to probe

high Rydberg states of ultracold 85Rb atoms. These pulses

are generated by pulsed amplification of cw light at

⬃594 nm followed by frequency doubling in a ␤ barium

borate共BBO兲 crystal. The cw light is generated by a

single-frequency tunable ring dye laser system 共Coherent 699-29

with Rhodamine 6G dye兲 pumped by an argon-ion laser. This

cw light seeds a three-cell amplifier chain in which the

cap-illary dye cells are pumped by the second harmonic共532 nm兲

of an injection-seeded pulsed Nd:yttrium aluminum garnet 共YAG兲 laser. The frequency-doubled pulses at ⬃297 nm are

typically 5 ns in duration 共full width at half maximum,

FWHM兲 and have energies up to 500 ␮J/pulse. The uv

bandwidth of ⬃140 MHz, measured by scanning over the

30p resonance at low uv intensity 共18 kW/cm2_{兲, is roughly}

twice the Fourier transform limit and is determined by

opti-cal phase perturbations in the pulsed amplifier 关15兴.

The ultracold 共T⬃100 ␮K兲 sample of 5–10⫻106 Rb

atoms is generated in a diode-laser-based vapor-cell MOT in

which densities up to 1011 cm−3 are achieved. Rydberg

ex-citation is performed by focusing the uv light into the MOT

cloud, yielding a cylindrical excitation volume ⬃500 ␮m

long and⬃220 ␮m in diameter共FWHM兲. Although the use

of a MOT is not absolutely essential to this experiment, it provides a collision-free environment and eliminates Dop-pler broadening, which would produce a 1.36 GHz linewidth at 297 nm and 300 K. To prevent direct photoionization from

the 5p_3/2 state by the uv light, the trapping and repumping

beams are turned off with acousto-optical modulators about

2 ␮s before the uv pulse arrives. Usually, the trapping light

is switched off slightly 共⬃500 ns兲 before the repumping

light in order to ensure that all atoms are in the 5S_1/2共F=3兲

level when the uv pulse arrives. However, we can vary the effective atomic density without affecting other properties of

the MOT by delaying the turn-off of the repump laser 关16兴.

This allows the trapping light to optically pump atoms from F = 3, the hyperfine level probed by the uv laser, into F = 2.

The time scale for this population transfer is ⬃100 ␮s.

The MOT is located between a parallel pair of 95% trans-parent grids separated by 2.09 cm. These grids allow control of the static electric field during Rydberg excitation as well as pulsed-field ionization of the Rydberg atoms and extrac-tion of the resulting ions. The applied field is perpendicular to the linear polarization of the uv light. There is also a stray electric field of up to 100 mV/cm, probably due to gradual chemical degradation of the copper field grids, that must be taken into account when analyzing the results. Within 100 ns after the Rydberg atoms are created, a pulsed field of ⬃1500 V/cm is applied, ionizing states with principal quan-tum numbers as low as n = 25. Photoionization by the exci-tation laser can be neglected because high-n Rydberg states have very low photoionization cross sections for uv radiation

关17兴. We estimate this probability at well under 1% even for

the worst case 共low n and high laser power兲. The ions are

detected with a discrete dynode electron multiplier 共ETP

model 14150兲. A boxcar averager is used to select the desired time-of-flight window.

A typical excitation spectrum for n = 30 is shown in Fig.1.

The 5s_1/2→npJ E1 transitions are visible, as well as the

nearby 5s_1/2→共n−1兲dJ E2 transitions. Transitions to

共n+1兲s1/2 states are not observed because they are both

di-pole and quadrudi-pole forbidden. For this scan, the stray elec-tric fields were minimized as described below, resulting in

negligible Stark mixing. Thus the entire 5s_1/2→ndJ signal

can be attributed to the E2 transition.

At a given value of n, the E1 and E2 transitions we mea-sure have oscillator strengths that differ by 3 orders of mag-nitude. In addition, their strengths vary significantly with n. To avoid possible problems with either the limited dynamic range of our detector or saturation of the transitions them-selves, we adjust the uv intensity to maintain similar signal

sizes for all scans. Low intensities 共e.g., 140 kW/cm2 _for

TABLE I. Electric quadrupole radial matrix elements and oscil-lator strengths f_d for 5s→nd transitions. Matrix elements are in units of a₀2and oscillator strengths are in units of 10−10_{. Columns 2}

and 3 show calculated values. Experimental values in column 4 are obtained from measured signal size ratios␩as described in Sec.IV.

n 兩共nd兩r2_兩5s兲兩2 ₁₀10_f d,calc 1010fd,expt 24 1.14⫻10−1 7.15 27 7.84⫻10−2 _{4.96 7.45⫾1.2} 29 6.27⫻10−2 3.97 5.78⫾0.65 34 3.82⫻10−2 _{2.43 3.68⫾0.73} 39 2.49⫻10−2 1.59 2.65⫾0.45 44 1.72⫻10−2 _{1.10 1.83⫾0.29} 47 1.66⫾0.24 49 1.23⫻10−2 _{0.79 1.27⫾0.17} 54 9.13⫻10−3 _{0.58 1.30⫾0.21} 59 6.95⫻10−3 _{0.45 1.00⫾0.24}

n = 28兲 are used to observe the relatively strong E1 transitions

共5s1/2→npJ兲, while higher intensities 共e.g., ⫻1000兲 are used

to drive the weaker E2 transitions to共n−1兲dJ. At each n, the

ratio of oscillator strengths is obtained by dividing the mea-sured E2:E1 signal ratio by the ratio of intensities used. A polarizer and half-wave plate combination and neutral den-sity filters are used to vary the intenden-sity. The relative inten-sities used for each scan are directly measured to within 5% with a uv photodiode. Using this scheme, the residual errors due to detector nonlinearity for comparing the signal strengths of the various fine-structure components are no more than 2 – 3 %.

Since E2 transitions are one-photon processes occurring in individual atoms, the corresponding signals should be lin-ear with respect to both atomic density and uv intensity.

These dependencies are verified in Figs. 2共a兲 and 2共b兲,

re-spectively. This verification is particularly important because

it rules out the possibility that these 5s1/2→ndJfeatures are

variations of previously observed two-photon molecular

resonances 关18–20兴. The molecular resonances are due to

avoided crossings between long-range molecular potentials of two Rydberg atoms and occur at the average energy of two states that are strongly coupled to np states. For

ex-ample, the 共n−1兲d and ns states are dipole coupled to np,

leading to a molecular resonance at the average energy of these states. Since two excited atoms are involved, the mo-lecular resonance signal is quadratic in both atomic density

and uv intensity关18兴. In principle, a molecular resonance of

this type could occur at the frequency of the 5s_1/2→ndJ

tran-sition via two-photon excitation of a pair of atoms to an

ndJ+ ndJconfiguration. The measured density and intensity

dependencies indicate that our ndJsignals are instead due to

single-atom E2 transitions.

Because the Stark effect can mix closely spaced p and d states, a stray electric field can induce an E1 transition

am-plitude to an ndJstate. We have carefully studied the effect

of an applied electric field on our nd signals and conclude

that for n⬍49, Stark mixing is negligible, and for 49ⱕn

ⱕ59, corrections for the stray electric field can be made. Because the Stark mixing increases rapidly with n, our nd

signals are dominated by the stray field for n艌69. We use

the measurements at high n to correct the signals at lower n. Using the parallel grids in the MOT chamber, we are able

to apply a uniform field Fzand thus cancel any z component

of the stray field. However, we cannot eliminate other

com-ponents 共or gradients兲 of the stray field. To determine the

effects of the residual stray field, we use high-n states共e.g.,

n = 74兲 to measure the nd signal as a function of Fz, as shown

in Fig.3. For small fields, the Stark-mixed p-state amplitude

is linear in Fzand the measured nd signal Sd should be

qua-dratic in Fz,

Sd= S0+␣共Fz+ Fz0兲2. 共22兲

Fitting the data to the Stark parabola of Eq.共22兲 yields the z

component of the stray field Fz0, the Stark coefficient␣, and

the minimum signal S0. After having determined Fz0

= 31 mV/cm at several values of n共69,74,84兲, we set Fzto

cancel this component of the stray field for subsequent scans at lower n.

The resulting minimum nd signal, S0 in Eq.共22兲, can be

written as S0= S0F+ SE2, where S0Fis the contribution from

Stark mixing due to components of the stray field that are not

canceled and S_E2is the E2 signal that we are trying to

ex-tract. These two contributions are dominant at high and low n, respectively. At a fixed uv intensity, the Stark-mixed nd

signal S0Fscales as 共nⴱ兲7关21兴, where the effective principal

quantum number nⴱ= n −␦ and the quantum defect ␦ is

1.3472 for high-n d states关22兴. Meanwhile, the 共n+1兲p

sig-nal Sp scales as共nⴱ兲−3 at high n关11兴. The normalized

Stark-mixed nd signal S0F/Sp should therefore scale as共nⴱ兲10.

Be-3 Be-3 5 4 9 . 7 2 9 3 3 5 4 9 . 5 9 5 3 3 5 4 7 . 5 3 1 3 3 5 4 7 . 3 9 7 3 3 5 4 4 . 3 2 4 3 3 5 4 4 . 1 9 0 S ig na lS iz e (a rb .u ni ts ) U V F r e q u e n c y (c m - 1 ) I0 1 0 - 3 _I 0 3 1 s1 / 2 3 0 p 1 / 2 3 0 p 3 / 2 2 9 d3 / 2 2 9 d 5 / 2 E ne rg y E 1 X E 2 2 9 d 3 0 p 3 1 s 5 s

FIG. 1. Spectrum showing 5s_1/2→30p_J 共E1兲 and 5s_1/2→29d_J 共E2兲 transitions with no external electric field. The location of the 5s_1/2→31s_1/2 transition, which is both E1 and E2 forbidden, is indicated by the arrow. The laser intensity for the 29d and 31s portions of the scan is I₀= 18.8 MW/cm2_{, while for the 30p region,}

the intensity is reduced by a factor of 1000 to avoid saturation effects. Inset shows the energy levels共not to scale兲 and associated E1 and E2 transitions. 0 2 4 6 8 1 0 ( b ) S ig na lS iz e (a rb .u ni ts ) D e n s i t y ( 1 0 1 0 c m - 3) I = 5 4 . 8 M W / c m 2 ( a ) 0 2 0 4 0 6 0 8 0 1 0 0  = 8 . 0 6 x 1 01 0/ c m 3 U V I n t e n s i t y ( M W / c m 2)

FIG. 2. 共a兲 59d signal as a function of atomic density at a fixed intensity of 55 MW/cm2_{.共b兲 59d signal as a function of intensity}

at a fixed density of 8.1⫻1010 _cm−3_{. Linear fits through the origin}

are shown for both cases.

TONG et al. PHYSICAL REVIEW A 79, 052509共2009兲

cause of this rapid n scaling, S0F/Sp is the dominant

contribution to S₀/Spfor nⱖ69. We isolate this Stark

contri-bution for these high n’s by subtracting the normalized E2

contribution SE2/Spfrom the measured values of S0/Sp. Here

we assume that SE2/Spis given by its average low-n value of

6⫻10−4 _{共see Fig.4兲, which amounts to 17% of S}

0/Sp at n

= 69. We now correct the low-n共nⱕ59兲 data for Stark mixing

by fitting the S0F/Spvalues for high n共n=69–89兲 to the

ex-pected 共nⴱ兲10 _{scaling and then extrapolating this scaling to}

lower n. Typically at n = 59 the Stark correction is 51%, but it

drops rapidly to 6% at n = 49. The E2 subtraction from the high-n data has only a small effect on the Stark corrections to the lower-n data, e.g., it changes the n = 59 Stark correction from 60% to 51%. We note that the noncancelable stray field

determined from the high-n data is given by共S_0F/␣兲1/2and is

typically 80–100 mV/cm.

IV. E2 OSCILLATOR STRENGTHS

In Fig.4, we show the n dependence of the ratio␩of the

E2 oscillator strength fd共n兲 共for 5s→nd transitions兲 to the E1

oscillator strength fp共n+1兲 关for the closest np transition, 5s

→共n+1兲p兴. This ratio is obtained from the experimental data by taking the ratio of the nd signal per unit intensity to the

共n+1兲p signal per unit intensity. As discussed in Sec.III, we

use higher 共lower兲 intensities for the E2 共E1兲 transitions in

order to have comparable signal sizes. The nd and 共n+1兲p

signals are determined by summing the areas under each fine-structure peak using Gaussian fits, yielding oscillator strengths that are summed over J. The relative strengths for

the different J’s are discussed in Sec.V. The results in Fig.4

with n艌49 have also been corrected for Stark mixing of the

nd states by the residual field as described in Sec. III.

Un-certainties in the data include statistical contributions as well as contributions from signal area determinations, relative in-tensity measurements, and the Stark corrections. Over the

range n = 27– 59, the ratio fd共n兲/ fp共n+1兲 is seen to be

rela-tively constant, with a possible slight increase with n.

Figure 4 also shows predicted oscillator strength ratios,

obtained by combining the fdvalues in TableIwith

electric-dipole oscillator strengths fpof Ref.关11兴. Using the

extrapo-lation method shown in Fig. 2 of Ref.关11兴, we find

fp共n兲 ⯝

0.0234

共nⴱ兲3 +

1.58

共nⴱ兲5, 共23兲

where nⴱ= n − 2.6415 for the high-n p3/2states关22兴. These E1

oscillator strengths should be accurate to within at worst 15%. Although the calculated values are typically about 40% smaller than the measurements, they are consistent given the

40– 60 % uncertainty of the calculations. Figure 4 also

shows the calculated values scaled by a factor of 1.6 to aid comparison of the n-dependence between theory and experi-ment. In this regard the agreement is good except at high n, where the measured ratio is slightly larger than the calculated trend would indicate. This may be due to a slight underesti-mate of the Stark mixing correction, which is almost 1 order of magnitude larger at n = 59 than at n = 49.

Because the E1 oscillator strengths are accurately known, it is also possible to obtain experimental determinations of

the absolute E2 oscillator strengths fdfrom the data in Fig.4.

The final column of TableIshows the resulting f values. To

the best of our knowledge, there are no previous high-n fd

measurements in Rb with which to directly compare our

re-sults. Niemax 关9兴 reported values for n=4–9, including fd

= 1.3⫻10−8 _{共summed over J兲 at n=9. Although we cannot}

extrapolate meaningfully to our range of high n, his f values do decrease sharply with n. We note that the exceptionally

large fd共n兲/ fp共n+1兲 ratios we measure at high n are due

mainly to the anomalously small size of fp, which results

- 2 0 0 - 1 0 0 0 1 0 0 2 0 0 S ig na lS iz e (a rb .u ni ts ) A p p l i e d F i e l d F ( m V / c m )

FIG. 3. 74d signal as a function of applied electric field Fz. A

least-squares fit to Eq.共22兲 is also shown. From the fit we find that

the minimum signal occurs at Fz= 31 mV/cm and that at this

mini-mum, the residual stray field is 78 mV/cm.

3 0 4 0 5 0 6 0 0 2 x 1 0 - 4 4 x 1 0 - 4 6 x 1 0 - 4 8 x 1 0 - 4 1 x 1 0 - 3 fd ( n ) :fp ( n+ 1 ) n

FIG. 4.共Color online兲 Ratio of E2 to E1 oscillator strengths as a function of n. The results for n艌49 have been corrected for Stark mixing by the residual stray field. The solid line is the predicted ratio and the dotted line is the prediction scaled by 1.6 for easier comparison to data共see text兲.

from the broad Cooper minimum located just above the

photoionization threshold关23兴. This is in sharp contrast with

the situation at low n. For example, using the measured

val-ues of fd共n=4兲 关9兴 and fp共n=5兲 关11兴, the ratio is 2.25

⫻10−6_{, 2 orders of magnitude smaller than at high n.}

V. RATIO OF J-DEPENDENT OSCILLATOR STRENGTHS

Our narrow-band uv excitation of an ultracold sample al-lows us to resolve the Rb np fine-structure splitting up to n = 75. From these spectra, an example of which is shown in

Fig.1, we can obtain the ratio of fine-structure E1 oscillator

strengths, ␳共n兲= f_3/2/ f_1/2, for transitions 5s_1/2→np_3/2 and

5s1/2→np1/2. This ratio is taken to be the ratio of areas under

the respective spectral peaks. The variation of ␳ with n is

shown in Fig. 5. The uncertainties include statistical

contri-butions as well as contricontri-butions from measurements of the laser intensities and signal areas. Over the range of n

ex-plored, 28–75,␳ is rather constant.

A weighted average of these data yields ␳= 4.57⫾0.46.

The overall uncertainty is dominated by the uncertainties in measuring the uv intensities and signal sizes. This is clearly inconsistent with the statistical value of 2.0 based on the 2J + 1 degeneracies alone. This anomaly has been previously noted and is attributed to the interplay of the sporbit in-teraction, core polarization, and cancelation effects in

transi-tion dipole moment matrix elements 关24兴. There are several

previous experimental measurements of the ratio␳ at low n

关11,25,26兴 and one at high n 关27兴. It has the expected

statis-tical value of 2.0 at n = 5 关11兴, then rises with increasing n

before leveling off above n = 20. The highest n for which an

individual value has been previously reported is ␳共n=20兲

= 4.9⫾0.2 关26兴. A value of␳共n=25兲=5.1 is quoted in 关25兴.

The average value over the range n = 29– 50 was determined

to be ␳= 5.9⫾1.4 关27兴. All of these results are consistent

with our present measurements.

A number of calculations of␳have been performed at low

n共ⱕ10兲 关14,28兴, intermediate n共ⱕ25兲 关24,29–31兴, and high

n共ⱕ80兲 关32兴. The predictions of 关31,30兴, when extrapolated

to high n, are significantly higher than our measurements.

The extrapolated results of关29兴 appear to be marginally

con-sistent but slightly higher. The variational and frozen-core

calculations of关32兴 agree well with our results. Our

measure-ments are also consistent with the extrapolated RMP+ CPIB

version of the results of Migdalek and Kim关24兴. This

calcu-lation uses a relativistic model potential and incorporates core-polarization effects in both the potential and the transi-tion dipole moment operator.

We also measure the ratio f_5/2/ f_3/2 of E2 oscillator

strengths for the 5s_1/2→nd_5/2 and 5s_1/2→nd_3/2 transitions.

Because the nd fine-structure splittings are smaller than for

np, we are only able to resolve the ndJlevels up to n = 48. We

find that f_5/2/ f_3/2= 1.60⫾0.16, independent of n over the

range n = 28– 48. This is consistent with the statistical value of 1.5. To the best of our knowledge, there are no calcula-tions of this E2 oscillator strength ratio, but there is also no reason to expect nonstatistical behavior.

VI. CONCLUSIONS

In summary, we have measured oscillator strengths for electric quadrupole transitions to highly-excited nd states of Rb. We find that these E2 transitions are weaker than the E1 transitions to nearby np states by a slightly n-dependent

fac-tor of only⬃2000. We have also determined the relative E1

transition strengths to the two np fine-structure states, J

= 3/2 and J=1/2. For the states we have investigated, the

measured ratio of 4.57⫾0.46 is independent of n and differs

dramatically from the statistical value of 2.0 expected from degeneracies alone. Both anomalies, the unexpectedly large E2 to E1 transition strength ratio and the nonstatistical np fine-structure transition strength ratio, owe their origin in

part to the pronounced Cooper minimum 关23兴 for 5s→np

transitions that lies just above threshold in Rb关24兴.

ACKNOWLEDGMENT

We gratefully acknowledge funding support from NSF Awards No. PHY-0457126 and No. PHY-0653449.

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