Photoionization of nS and nD Rydberg atoms of Rb and Cs from the near-infrared to the ultraviolet spectral region
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Photoionization of nS and nD Rydberg atoms of Rb and Cs from the near-infrared tothe ultraviolet spectral region
Michael A. Viray, ∗ Eric Paradis, and Georg Raithel Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109, USA Department of Physics and Astronomy, Eastern Michigan University, Ypsilanti, Michigan, 48197, USA (Dated: January 8, 2021)We present calculations of the photoionization (PI) cross sections of rubidium and cesium Rydbergatoms for light with wavelengths ranging from the infrared to the ultraviolet, using model potentialsfrom [M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A
49, 982 (1994)]. Theorigins of pronounced PI minima are identified by investigating the free-electron wavefunctions.These include broad PI minima in the nS to ǫP PI channels of both Rb and Cs, with free-electronenergy ǫ , which are identified as Cooper minima. Much narrower PI minima in the nD to ǫF channels are due to shape resonances of the free-electron states. We describe possible experimentalprocedures for measuring the PI minima, and we discuss their implications in fundamental atomicphysics as well as in practical applications. I. INTRODUCTION
Photoionization (PI), or the photoelectric effect, ofatoms is one of the longest-studied processes in atomicphysics, dating back to at least the Bohr model [1]. Be-cause the outermost electron(s) of Rydberg atoms lie athigh energy levels, the minimum photon energy requiredfor Rydberg-atom PI typically is in the milli-eV regime.Hence, the spectral range of the ionizing radiation rangesfrom THz fields through the infrared (IR), visible andultraviolet (UV) spectral regions and higher. Rydbergatoms are known to be extremely sensitive to PI by black-body radiation, a consequence of their extended elec-tronic wavefunctions and large transition electric-dipolemoments [2, 3]. Generally, as the photon energy in-creases, the Rydberg atoms becomes less sensitive toPI due to increasing mismatch between bound-state andfree-electron wavefunctions, and the ionization cross sec-tions drop rapidly. However, in certain PI channels thisgeneral trend is interrupted by pronounced minima inthe PI cross sections as a function of wavelength λ of theionizing field. The PI minima are quite sensitive to theassumed Rydberg-electron model potentials. Hence, ex-perimental PI studies can serve as a method to test andto possibly fine-tune model potentials. Data on Rydberg-atom PI cross sections are also relevant to applicationsof Rydberg atoms in which the rate of laser-induced PImust be minimized.PI minima in atoms can be attributed to Cooperminima or shape resonances, which are, in the follow-ing, briefly explained. Cooper minima, first reportedby John Cooper in 1962 [4], occur as a result of van-ishing PI matrix-element integrals. The PI matrix ele-ments are integrals over expressions that involve a prod-uct of an initial- and a final-state electron wavefunc-tion. The wavefunctions are quasi-periodic and have dif-ferent spatial periods and phases. At certain values of ∗ [email protected] λ , the matrix-element integral can vanish, in remote re-semblance to destructive interference between two out-of-phase periodic functions. The free-state wavefunctionon its own does not exhibit any special behavior at theCooper minima. The associated dips in the PI cross sec-tions as a function of λ can be hundreds of nm wide.Shape resonances, on the other hand, are due to quasi-bound scattering states within an inner well of the rele-vant free-electron potential. As in our case, the potentialbarrier that separates the outer region from the inner wellcan arise from the sum of a short-range attractive poten-tial with a repulsive core and the long-range centrifugalpotential, ~ ℓ ( ℓ + 1) / (2 m e r ), with electron angular mo-mentum ℓ , electron mass m e , and electron radial coor-dinate r . As λ is varied, the free-electron wavefunctioncan pass through a narrow resonance in the inner well,characterized by a large-amplitude quasi-bound scatter-ing state inside the barrier and a π phase shift outside thebarrier. In certain Rydberg-atom PI channels, shape res-onances cause minima in the PI cross sections; these tendto be narrower as a function of λ than Cooper minima.Finding the wavelengths at which PI cross sectionshave minima is interesting from a basic atomic-physicsperspective, because it represents a test of the assumedmodel potentials for the Rydberg and the free-electronstates. Knowledge of PI minima can also be benefi-cial from an applications standpoint. Rydberg atomsin optical dipole traps and optical lattices can becomephoto-ionized by the trapping beams themselves. If thetrapping- or lattice-beam wavelength is set at a PI min-imum, the trapped atoms will be less prone to PI bythe trapping beams. The absence of Rydberg-atom PIcould be important, for instance, in experiments on quan-tum simulation and quantum information processing us-ing Rydberg atoms [5–8], quantum control [9, 10], high-precision spectroscopy [11–14], and in large-scale arraysof Rydberg atoms that could potentially be trapped [15].Various experiments and theoretical investigationshave been performed over the years to find PI minima,both from Cooper minima and from shape resonances.Regarding Cooper minima, Zatsarinny and Tayal compu-tationally calculated Cooper minima in potassium [16],which were later verified experimentally by Yar, Ali, andBaig [17]. Beterov et. al. calculated transition probabili-ties and PI cross sections of alkali metals using a Coulombapproximation and a quasiclassical model [18]. Addition-ally, there have been measurements of the Cooper min-imum of atoms in a plasma background [19–21]. Mean-while, shape resonances have been observed in positro-nium [22] and in atom collisions [23–25], and they havebeen used to form Rydberg molecules [26, 27].In this work, we report computational findings onCooper minima and shape resonances in Rydberg-statePI in Rb and Cs. We use model potentials for rubid-ium and cesium from Refs. [28] to determine the ini-tial (bound) and and final (free-state) wave functions ofthe photoionized electrons. We compute PI cross sec-tions across a wide range of λ , and find several Cooperminima and shape resonances. Results are evaluated incontext with free-state wavefunction plots and the un-derlying model potentials, and comparisons between Csand Rb are made. We discuss the viability of measuringthese PI minima experimentally, as well as the relevanceto atomic physics theory and to applications. II. THEORY BACKGROUNDA. Atomic model potentials
We denote the initial Rydberg states | i i = | n, ℓ, m ℓ i with principal, angular-momentum and magnetic quan-tum numbers n, ℓ and m ℓ , respectively, and the photo-ionized free-electron states | f i = | ǫ ′ , ℓ ′ , m ′ ℓ i with free-electron energy ǫ ′ , and angular-momentum and magneticquantum numbers ℓ ′ and m ′ ℓ , respectively.The calculation of PI cross sections requires a proce-dure to calculate the initial-state and free-state wave-functions, ψ i ( r ) = h r | i i and ψ f ( r ) = h r | f i , with relativeelectron position r . The fine structure is neglected inthe present work because it is much smaller than theRydberg-atom binding energy and the energy of the freeelectron. The wavefunction calculation requires a set ofatomic model potentials. Here, we use model potentialsfor Rb and Cs developed and employed in Refs. [28–30].The model potentials V ,ℓ ( r ) include correction termsto the Coulomb potential that yield the correct core-penetration and ion-core polarization quantum defectsof the atomic energy levels for various angular momenta ℓ . The model potentials depend on atomic species and on ℓ , with the potentials of any one species being the samefor all ℓ ≥ V ℓ ( r ) = V ,ℓ ( r ) + ~ ℓ ( ℓ + 1)2 m e r . (1)The wavefunctions are calculated using these potentialswith a numerical method outlined by Reinhard et. al. [31]. B. Calculating PI Cross Sections
PI is an effect of the ˆ A · ˆ p -interaction of the minimal-coupling Hamiltonian [3] in first order. Given a linearlypolarized plane wave with polarization unit vector ˆ n ,wave vector k and angular frequency ω , the partial PIcross section is σ = πe ~ ε m e ωc (cid:12)(cid:12)(cid:12)(cid:12) ˆ n · Z ψ ∗ f e i k · r ∇ ψ i d r (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) E H a (cid:19) , (2)with the atomic energy unit E H ≈ . . The matrix element is computed inatomic units, with free states normalized in units of en-ergy, i.e. h ǫ ′ , ℓ ′ , m ′ ℓ | ǫ ′′ , ℓ ′′ , m ′′ ℓ i = δ ( ǫ ′ − ǫ ′′ ) δ ℓ ′′ ,ℓ ′ δ m ′′ ℓ ,m ′ ℓ ,and the term in parentheses within Equation 2 convert-ing the matrix-element square from atomic into SI units.It is shown elsewhere that for light-induced PI ofRydberg atoms the electric-dipole approximation (EDA), e i k · r = 1, is valid at a level better than 10 − , i. e. electric-dipole-forbidden transitions have cross sectionsthat are smaller than those of the dipole-allowed ones bya factor of at least ten thousand. We make the EDAand average the cross sections over the initial-state mag-netic quantum number, m ℓ , to obtain the shell-averagedpartial cross section,¯ σ ǫ ′ ,ℓ ′ n,ℓ = πe ~ ǫ m ωc ℓ > (2 ℓ + 1) | M | (cid:18) E H a (cid:19) , (3)where M is a radial matrix element in atomic units, M = Z ∞ u ǫ ′ ,ℓ ′ ( r ) (cid:20) u ′ n,ℓ ( r ) ∓ u n,ℓ ( r ) r ℓ > (cid:21) dr. (4)There, the upper sign is for ℓ > = ℓ ′ = ℓ + 1 and thebottom sign for ℓ > = ℓ = ℓ ′ + 1. The functions u ∗ ,ℓ ( r )are given by u ∗ ,ℓ ( r ) = rR ∗ ,ℓ ( r ), with the usual radialwavefunction R ∗ ,ℓ ( r ), and ∗ = n and ∗ = ǫ ′ denotingthe principal quantum number of the bound- and theenergy of the free-electron state, respectively. The free-state energy follows from the wavelength of the PI light, λ , and the binding energy of the Rydberg atom. The free-state energy in atomic units is ǫ ′ = 2 πa / ( αλ ) − / (2 n ∗ ),where λ is entered in meters, the fine structure constant α , the effective quantum number n ∗ = n − δ n,ℓ , and thequantum defect δ n,ℓ .For z -polarized light, m ℓ is conserved, and the m ℓ -dependent PI cross sections follow from the shell-averaged ones via σ ǫ ′ ,ℓ ′ z,n,ℓ,m ℓ = 3( ℓ > − m ℓ )(2 ℓ > + 1)(2 ℓ > −
1) (2 ℓ + 1) ℓ > ¯ σ ǫ ′ ,ℓ ′ n,ℓ , (5)with similar expressions applicable to other light polar-izations. Therefore, it is sufficient to discuss the PI be-havior in terms of the shell-averaged ¯ σ ǫ ′ ,ℓ ′ n,ℓ .It is noted that the matrix element in Eq. 4 fol-lows directly from the ˆ A · ˆ p -interaction and the EDA.The matrix-element form in Eq. 4 is known as veloc-ity form [3, 32]. In the case that the atomic poten-tial is velocity-independent (not including the centrifu-gal term), the matrix elements can be converted intolength form [3, 32], allowing an alternate, commonly usedmethod to compute electric-dipole matrix elements. Ifthe potentials are ℓ -dependent, the velocity form mustbe used. We did confirm in our work that the PI crosssections that follow from matrix elements calculated inthe velocity and in the length forms are identical for ℓ ≥
4. This is expected, because for ℓ ≥ ℓ -independent model potential applies (namely, V ,ℓ ≥ = V ,ℓ =3 ) to compute both the bound- and free-state wavefunctions. As ℓ approaches zero, deviationsbetween velocity- and length-form cross sections increaseand reach about 15% relative difference for ℓ = 0 (awayfrom Cooper minima). Also, the exact λ -values atwhich the Cooper minima and shape resonances occurare slightly different. These deviations are due to thefact that the length form becomes increasingly inaccu-rate when ℓ approaches 0, because the model potentialsbecome increasingly ℓ -dependent, whereas the velocityform remains accurate. For the PI cross-section calcula-tions presented in this paper, we have chosen ℓ = 0 and ℓ = 2, because these are the experimentally most rele-vant cases. For these ℓ -values it is important to use theexpressions given in Eqs. 3 and 4, which are in velocityform.We note that the difference between electric-dipole ma-trix elements calculated in length and velocity forms isorders of magnitude lower for bound-bound transitionsbetween Rydberg states than it is for Rydberg-atom PI.For bound-bound Rydberg transitions, which are in themicrowave to THz range, the length-form matrix ele-ments are accurate enough for most purposes. However,the length form is not generally applicable to bound-free transitions, particularly in the optical regime. Thisfinding can be explained, qualitatively, through the factthat the lower the transition energy is the more thematrix-element integral becomes dominated by contri-butions from the outer reaches of the electron configu-ration space, where all model potentials converge into an ℓ -independent Coulomb potential. III. RUBIDIUM PI CROSS-SECTIONS
Figure 1(a) shows shell-averaged cross sections of threePI channels of Rb Rydberg atoms, namely ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =0 ( λ ),¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =2 ( λ ), and ¯ σ ǫ ′ ,ℓ ′ =3 n =35 ,ℓ =2 ( λ ), for wavelengths rangingfrom the deep UV to the near-IR regime. The free-electron energy in atomic units is ǫ ′ = 2 πa / ( αλ ) − / (2 n ∗ ). Aside from the resonant features discussedlater, it is seen that the S → P cross sections are fairlysmall over the entire range, topping out at only about 20 times the Thomson cross section, σ T = 0 .
665 barn. Inthe near-IR spectral range, nS -type Rydberg atoms havePI cross sections that are lower than those of other low- ℓ PI channels by up to about three orders of magnitude.In contrast, the D → P and D → F channels in Fig. 1(a)follow the generic trend that PI rates rapidly increase atlonger wavelengths. Between λ = 200 nm and 700 nm,the PI cross sections of those channels increase by one totwo orders of magnitude. Typically, the PI threshold willpeak at the PI threshold, with depends on n and ℓ andis in the far-IR regime.The plot ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =0 ( λ ) in Fig. 1(a) shows a minimumcentered at λ = 536 nm. As the S → P channel is theonly PI channel of S -type Rydberg atoms, the total PIcross section equals the partial cross section ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =0 ( λ ),and is below σ T over a range 490 nm . λ .
570 nm.Taking into account the asymmetry of the PI minimumin ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =0 ( λ ), we define the full width at half depth(FWHD) of the minimum as the range over which thecross section dips below half of the PI maximum seenin the UV range. For the Rb Cooper minimum, the PImaximum in the UV range is 10 barns, so the FWHD isthe range over which the cross section dips below 5 barns.The FWHD is ≈
240 nm, with the FWHD-range cover-ing the spectral region 400 . λ .
640 nm. The largewidth of the PI minimum in the S → P channel servesas an indicator that this is a Cooper minimum, as willbe proven below.The D → F channel shows a minimum centered at λ = 366 nm with a FWHD of only about 10 nm. Thenarrow width of this minimum is a first indicator that thisminimum is different in nature from the minimum in the S → P channel; below we will show that the minimumin the D → F channel is due to a shape resonance. The D → P channel has no PI minimum within the rangedisplayed in Fig. 1(a). We also did not find any minimumover an extended search across a wider range from 100 nmto 2 µ m (not shown). The total shell-averaged PI crosssection of Rb 35 D is given by the sum of the partialcross sections, ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =2 ( λ ) + ¯ σ ǫ ′ ,ℓ ′ =3 n =35 ,ℓ =2 ( λ ). Hence, theobservable cross section of the 35 D -state is the sum oftwo of the curves in Fig. 1(a). The total 35 D PI crosssection therefore has a minimum of about 200 barns at366 nm, and rises to about 3000 barns several tens of nmaway.The physical differences between the PI minima in theRb S → P and D → F channels become apparent whenlooking at the inner regions of the relevant free-electronmodel potentials, V ℓ ( r ), and the free-electron wavefunc-tions. In Fig. 1(b) we show V ℓ ( r ) for ℓ = 0 to 3 and overthe range r ≤ a , and in Figs. 1(c) and (d) free-electronwave-function moduli for the S → P and D → F ioniza-tion channels, respectively, over the range r ≤ a .Figs. 1(b)-(d) are focused on the central atomic region,where phase shifts and shape resonances determine thePI behavior. In Fig. 1(c), the free-electron wavefunc-tion does not exhibit any noteworthy feature, as λ passes V ℓ ( a t o m i c u n i t s ) (a) (b)(c) (d) | , ⟩ | , ⟩ -144 eV +16.2 eV FWHDFWHD
FIG. 1. Rubidium: (a) Partial PI cross sections of 35 S and 35 D Rb Rydberg atoms vs PI wavelength λ : S → P (blacksquares), D → P (red circles), and D → F (green triangles). The dashed line shows the Thomson scattering cross section( σ T = 0 .
665 barn), which is the elastic photon scattering cross section of the Rydberg atoms. In the gray region, elasticscattering exceeds PI. (b) Core region of the potentials V ℓ ( r ) for ℓ = 0 , , , P -state free-electron wavefunction vs λ , with the Cooper minimum of the PI cross section indicated as a dashed line. The free-electron wavefunctionexhibits an oscillatory pattern with a smooth, gradual phase shift as a function of λ , with no marked behavior at the Cooperminimum. (d) D -state free-electron wave function vs λ , with the shape resonance in the D → F PI cross section indicated asa dashed line. The plot shows a quasi-bound state centered at the shape resonance. through the PI minimum. There merely is a gradualphase shift of the wavefunction due the slightly changingde-Broglie wavelength of the free electron. Hence, thePI minimum is due to incidental near-perfect destructiveinterference between bound-state and free-electron wave-functions (where the bound-state function shows up in amodified form; see Eq. 4). This fact makes the PI mini-mum in the Rb S → P channel a Cooper minimum. Dueto the rather gradual change of the free-state wavefunc-tion, the Cooper minimum is comparatively wide in PIwavelength and free-electron energy. The Cooper mini-mum relates to the fact that the bound-state and free-state quantum defects differ by about 1/2 in Rb (thequantum defects are near 3.13 for S and 2.65 for P ).The free-state wavefunction of the D → F channel ex-hibits a resonant structure at the PI minimum, whichmanifests in the region of high wavefunction amplitudenear r ≈ a , as well as a phase shift by ≈ π in the outer,oscillatory region of the wavefunction. The resonance oc-curs at the λ -value of the PI minimum, and it has a widthin λ of only a few tens of nm. Another piece of insight fol-lows from the V ℓ ( r ), plotted in Fig. 1(b). There, it is seen that V ℓ =3 ( r ) exhibits an inner well formed by the cen-trifugal potential and the core region of the model poten-tial V ,ℓ =3 ( r ) for r . . a . (Note Eq.1 regarding the def-inition of V ,ℓ =3 and V ℓ =3 ). Of the four potentials shown,the ℓ = 3 potential is the only one with an inner poten-tial well and a centrifugal barrier. The potential barrierpeaks at an energy of about +16 . −
144 eV. It is apparentthat the resonance, which is at about ǫ ′ = 3 .
37 eV, is thelowest and only quasi-bound state in the inner potentialwell of V ℓ =3 ( r ). These findings sum up to the statementthat the PI minimum in the D → F channel is due to ashape resonance. The phase shift of π of the scatteringwavefunction across the resonance, seen in Fig. 1(c), isanother telltale sign of a shape resonance [33].After identifying the physical origins of the PI min-ima, we wish to comment on a length-form calculationof the PI cross sections, which has yielded qualitativelysimilar behaviors with quantitatively notable differences(not shown). For instance, the Cooper minimum in thelength-form result for Rb ¯ σ ǫ ′ ,ℓ ′ =1 n =35 ,ℓ =0 ( λ ) is shifted up inwavelength by about 50 nm relative to the velocity-formresult. While the (unphysical) shift in the length-formresult is less than the width of the minimum, it is largeenough to be of significance. The shift represents a casein which the length-form calculation exhibits a signifi-cant error relative to the velocity-form calculation. Wereiterate here that the velocity-form calculation is cor-rect, while the length-form is expected to show an errorbecause of the ℓ -dependence of the model potentials V ,ℓ .For ℓ ≥ ℓ ) and free-electron ( ℓ ′ = ℓ ±
1) states areidentical (namely, V ,ℓ =3 ).It is also worth commenting on free-electron photonscattering due to the A -term in the atom-field interac-tion [3]. Since the λ -range discussed in our work exceedsthe atomic energy scale by orders of magnitude, the pho-ton scattering has a cross section given by the Thomsonscattering cross section, σ T = 0 .
665 barn. For a trulyfree electron, the Compton (recoil) energy would be inthe range of h ×
10 GHz. This recoil energy would be toosmall to cause atomic bound-bound transitions, nor doesit cause photo-ionization of the atom (as in typical in-stances of the Compton effect). Therefore, the Thomsonscattering of a Rydberg electron is perfectly recoil-freeand elastic (except for a very small recoil of the entireatom). In the gray regions in Figs. 1 and 2, the elastic(Thomson) scattering rate exceeds the PI rate. Compar-ing the two effects, we further note that the Thomsonscattering is due to the A -term in the atom-field inter-action, and it occurs in the outer reaches of the Rydbergatom, r & a , where the Rydberg electron resides withnear-unity probability, whereas PI (photo-electric effect)is due to the A · p -term and occurs in the atomic core, r . a .The dependence of elastic photon scattering and PIon principal quantum number n also is of interest. Theelastic cross section, σ T , is independent of n . In contrast,the PI cross sections, away from the resonances, have ageneric scaling close to ∝ n ∗− , with n ∗ = n − δ ℓ de-noting the effective quantum number, and δ ℓ the leadingterm of the quantum defect. We have verified this scal-ing in additional calculations (which are not presented indetail). IV. CESIUM PI CROSS-SECTIONS
Figure 2 (a) shows partial PI cross sections for Cs 35 S and 35 D , for the S → P , D → P , and D → F ionizationchannels. In this figure, which is organized analogous toFig. 1, we see that all three partial PI cross section chan-nels have minima, namely Cooper minima for S → P and D → P , and a shape resonance for D → F . ThePI minima are 100 to 200 nm deeper in the UV than inRb. The PI cross section of the 35 S state barely risesabove the elastic scattering cross section, σ T , across thedisplayed range, making S -type Rydberg atoms of Cs es-sentially PI-free at all wavelengths shorter than about 500 nm. Otherwise the trends observed in Fig. 2 (a) fol-low those of Rb. As in Rb, in Cs the shape resonanceis considerably narrower than the Cooper minima. TheCooper minimum in the D → P channel is of little rel-evance, because in the total PI cross section it will benear-invisible against PI on the D → F channel.The potential curves and free-state wavefunction mapsfor Cs, shown in Fig 2 (b) and Fig 2 (c-e), respectively,present a situation that is similar to that in Rb. TheCooper minima in the S → P and D → P channelshave a FWHD of about 100 nm and are characterized byfree-state wavefunctions with smooth, λ -dependent phasechanges and without any resonant behavior. The ℓ = 3potential is the only one that features a relevant barrier,which is located at r ≈ a and peaks at 39.2 eV. Thepotential well inside the barrier bottoms out at −
335 eV.The lowest electron “state” in the well is a quasi-boundpositive-energy resonance associated with the shape res-onance at λ = 150 nm in the partial PI cross section onthe D → F channel. V. DISCUSSION
The PI cross sections warrant an experimental inves-tigation because of the importance of optical traps ofRb and Cs Rydberg atoms in the applications mentionednear the end of Sec. III, where atom loss and decoher-ence must be avoided. In our paper we stress that themodel potentials used in the calculations play a centralrole. It is apparent that the positions of the PI min-ima are very sensitive to the potentials and the resultantphase shifts and quasi-bound states near and inside theRydberg atoms’ ionic cores. Noting the large depth andthe small range of the inner wells of the ℓ = 3 potentials,one may expect that a measurement of the shape reso-nances will present a particularly sensitive test for the ℓ = 3 model potentials.Considering the widths of the PI minima, one fruitfulexperimental approach is to use a tunable pulsed laser tophoto-ionize a sample of N cold Rydberg atoms and tocount the ions using a single-particle counter. The lattermay utilize, for instance, a micro-channel plate or a chan-neltron, which are capable of single-ion counting with ef-ficiencies of & ∼ N/ σ the fluence F of the pulseshould be in the range hc σλ & F & hcN σλ (6)To measure the shape resonance of 35 D in Rb, thisrelation would have to be satisfied for σ ranging between σ min ∼
200 barn and σ max ∼ N = 10 Rydberg atoms, Eq. 6 V ℓ ( a t o m i c u n i t s ) (a) (b)(c) (d) (e) | , ⟩ | , ⟩ | , ⟩ -335 eV +39.2 eV FWHDFWHDFWHD
FIG. 2. Cesium: (a) Partial PI cross sections of 35 S and 35 D Cs Rydberg atoms vs PI wavelength λ : S → P (black squares), D → P (red circles), and D → F (green triangles). The Thomson scattering cross section, σ T , is indicated as in Fig. 1. (b)Potential curves V ℓ ( r ) for ℓ = 0 , , , P -state free-electron wave function vs λ , with the S → P Cooperminimum of the PI cross section indicated as a dashed line. (d) Same as (c), with the D → P Cooper minimum of the PI crosssection indicated as a dashed line. (e) D -state free-electron wave function vs λ , with the shape resonance in the D → F PIcross section indicated as a dashed line. translates into hc σ max λ & F & hcN σ min λ × mJmm & F & . × − mJmm . (7)Noting that the PI laser could have an area of severalmm , it is seen that the pulse fluence F required to mea-sure the shape resonance of Rb 35 D lies within fairlycomfortable limits. A pulse energy of a few tens of µ Jper pulse could be sufficient to map out the shape reso-nances.To measure the Cooper minimum of Rb 35 S , we set σ min = σ T = 0 .
67 barn, the elastic photon scatteringrate, and σ max = 20 barn. In this case, the limitingexperimental requirement is F & hcN σ min λ , or , F & . . (8)It is seen that for a beam with several mm in cross sec-tion a pulse energy of a few mJ per pulse could be suffi-cient to map out the Cooper minimum. This pulse energycould be delivered, for instance, by a nanosecond pulseddye laser, pumped with a harmonic of a pulsed YAG laser. Noting that with decreasing n the PI cross sec-tions generally increase as n ∗− , additional experimentalflexibility would be afforded by lowering n .A main issue with measuring the PI minima is thelasers that would be required to run these experiments.The cesium shape resonance, for example, is centered at150 nm, which is not an easily accessible wavelength.This wavelength could be reached by running a 600 nmlaser through frequency-doubling crystals, but the setupwould be expensive and inefficient. Additionally, 150 nmlight is readily scattered in air, so the laser beam pathsmust be short to avoid significant beam attenuation.Of the five PI minima we found, we surmise that theeasiest one to experimentally investigate is the Rb shaperesonance (see Eq. 7). This minimum is centered at366 nm, which can be accessed by running a pulsed dyelaser (PDL) with a dye such as LDS-720, and then send-ing the PDL beam through a doubling crystal. The Rband Cs S → P Cooper minima are the second-best candi-dates for measurement because of their accessible wave-lengths, but the cross sections expected for these channelsare generally low. The resultant condition on the fluence(see Eq. 8) will make this effort more challenging.The small total PI cross sections of Rb and Cs nS Rydberg states makes these states ideal for applicationsof optically excitable, laser-trapped Rydberg atoms. TheCooper minimum in Rb ¯ σ ǫ ′ ,ℓ ′ =1 n,ℓ =0 ( λ ) is near 532 nm, thesecond harmonic of YAG and similar lasers, which can de-liver sufficient power for dipole and optical-lattice trapsfor Rydberg atoms [9, 11, 12, 34]. Laser-trapped, prac-tically PI- and decoherence-free Rydberg atoms can beuseful in applications in which atomic decay and decoher-ence must be minimized, such as in quantum simulation,quantum information processing and high-precision spec-troscopy. VI. CONCLUSION
We have calculated the partial PI cross sections ofRb and Cs 35 S and 35 D Rydberg atoms from the UVinto the near-IR spectral regime. We have identified oneCooper minimum and one shape resonance in Rb, andtwo Cooper minima and one shape resonance in Cs. Infuture work, one may investigate the PI cross sectionsexperimentally. The exact wavelengths of the PI minimawill be useful to know for the design of optical dipoletraps and optical lattices for Rydberg atoms. For in-stance, traps for Rb nS Rydberg atoms could benefitfrom the Cooper minimum of Rb near λ = 536 nm. Fur- ther, the Rb shape UV lattice would be effective for Rb nD Rydberg atoms. UV lattices are uncommon, but theyhave been used in the past to trap mercury [35].Similar calculations can be performed for any element,as long as there is a model potential to use. In this vein,the same study could be conducted for other alkali metalssuch as potassium and sodium. While there have beenarticles in the past that have reported on Cooper minimain these elements, there may be other Cooper minima andshape resonances that are unknown. The studies couldalso be expanded out of the alkali metal group into othercommonly studied species, such as Sr, Yb and Ca, whichmay have interesting PI behavior due to the presence oftwo valence electrons. The elastic photon scattering ofRydberg atoms, which has a cross section equivalent tothe Thomson scattering cross section, may also deservea future study.
ACKNOWLEDGMENTS
This work was supported by NSF Grant No. PHY-1707377. We thank Callum Jones of the University ofCalifornia, Los Angeles, and Shruti Paranjape of the Uni-versity of Michigan for valuable discussions. [1] N. Bohr, On the constitution of atoms and molecules,Philos. Mag. , 1 (1913).[2] T. F. Gallagher, Rydberg Atoms (Cambridge UniversityPress, Cambridge, 1994).[3] H. Friedrich,
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