Quasiclassical theory of quantum defect and spectrum of highly excited rubidium atoms

0.39 MB·English

by  Ali Sanayei

Quasiclassical theory of quantum defect
and spectrum of highly excited rubidium atoms
Ali Sanayei and Nils Schopohl∗
Institut fu¨r Theoretische Physik and
CQ Center for Collective Quantum Phenomena and their Applications in LISA+,
Eberhard-Karls-Universita¨t Tu¨bingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
Jens Grimmel, Markus Mack, Florian Karlewski, and J´ozsef Fort´agh†
Physikalisches Institut and
CQ Center for Collective Quantum Phenomena and their Applications in LISA+,
Eberhard-Karls-Universita¨t Tu¨bingen, Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
(Dated: August 8, 2016)
5 We report on a significant discrepancy between recently published highly accurate variational
1 calculations and precise measurements of the spectrum of Rydberg states in 87Rb on the energy
0 scale of fine splitting. Introducing a modified effective single-electron potential we determine the
2 spectrum of the outermost bound electron from a standard WKB approach. Overall very good
agreement with precise spectroscopic data is obtained.
n
a
J PACSnumbers: 31.10.+z,32.80.Ee

8
2
I. INTRODUCTION reported in [5] choosing

]
h The spectrum of the outermost bound electron of an Zeff(r;l)=1+(Z−1)e−ra1(l)−re−ra2(l)[a3(l)+ra4(l)]
p alkali atom like 87Rb is hydrogen like, but lacks the n2- (3)
- and
t degeneracy of the eigenstates labeled by the principal
n
a quantum number n of the pure Coulomb potential [1],[2] (cid:20) (cid:16) (cid:17)6(cid:21)
1−exp − r
qu 1 V (r;l)= αc rc(l) . (4)
[ En,l =−(n−δ )2. (1) pol 2 r4
l
1 A table of the parameters a1(l), a2(l), a3(l), a4(l), αc,
v This effect is the well-known quantum defect δl, re- and rc(l) can be found in [5].
8 sulting from the interaction of the outermost electron In an attempt to also describe the fine splitting of the
6 with the ionic core of the atom and the nucleus. In excitation spectrum of the outermost electron of 87Rb,
0 a refined version of the statistical Thomas-Fermi the- it has been suggested [4] to superimpose a posteriori a
7 ory [3], an effective potential determining the interac- spin-orbit term
0
tion between the outermost electron and the nucleus can
.
501 atecnctuiraalteVleyffb(re;lm) oddeepleenddbinygaonspthheericdaislltyanscyemrmfertormic pthoe- V(cid:101)SO(r;j,l)= [1−VSαO2V(reff;j(,rl;)l)]2, (5)
center and depending on the orbital angular momentum
1
l∈{0,1,2,...,n−1} [4, 5],[2]: onthepotentialV (r;l),whichtheninfluencesthespec-
: eff
v trum E on the scale of fine splitting and the orbitals
n,j,l
Xi V (r;l)=−2(cid:20)Zeff(r;l) +V (r;l)(cid:21) (2) ψn,j,l(r) accessible to the outermost electron. Here
eff r pol
r 1∂V (r;l)
a V (r;j,l)=α2 eff g(j,l), (6)
Here the function Z (r;l) represents a position- SO r ∂r
eff
dependent weight function that interpolates the value of
the charge between unity for large r and charge number and α = λaBC (cid:39) 1371.036 denotes the fine-structure con-
stant, and
Z neartothenucleusforr →0,andV (r;l)represents
pol
a short-ranged interaction taking into account the static 
0 if l=0,
electric polarizability of the ionic core [1, 6]. 
Overall good agreement with spectroscopic data of al- g(j,l)= (7)
kali atoms (but discarding the fine splitting) has been j(j+1)−l(l+1)−34 if l≥1,
2
where j ∈ (cid:8)l− 1,l+ 1(cid:9). To determine those orbitals
2 2
(with principal quantum number n = n + l + 1 and
r
∗ [email protected] radial quantum number n ∈ N ), a normalizable solu-
r 0
† [email protected] tiontotheSchr¨odingereigenvalueproblemfortheradial
2
wavefunctionU (r)=rR (r)andassociatedeigen- thusengendersasystematic(small)errorofthematrixel-
n,j,l n,j,l
values E <0 is required: ementscalculatedin[7]onthefine-splittingscale. When
n,j,l
employingsubstantiallymoreSTOsthiserrorwouldcer-
(cid:20) d2 l(l+1) (cid:21) tainly become larger. With N = 500 STOs the discrep-
−dr2 + r2 +V(cid:101)(r;j,l)−En,j,l Un,j,l(r)=0, ancy of these theoretical results with the high precision
(8) spectroscopic data, as shown in Tables I and II, is far
where too large to be corrected by simply replacing VSO(r;j,l)
with V(cid:101)SO(r;j,l). Hence another explanation is required.
V(cid:101)(r;j,l)=Veff(r;l)+V(cid:101)SO(r;j,l) (9)
denotes the effective single-electron potential. II. QUASICLASSICAL APPROACH AND FINE
Ahighlyaccuratevariationalcalculationoftheexcita- SPLITTING OF THE HIGHLY EXCITED 87RB
tionspectrumoftheoutermostelectronof87Rbhasbeen
carried out recently [7], in which the authors expand the In 1941 alkali atoms have already been studied in the
radial wavefunction of the Schr¨odinger eigenvalue prob- context of modern quantum mechanics in the seminal
lem (8) in a basis spanned by 500 Slater-type orbitals workbyGoeppertMayer[3],whoemphasizedtheexcep-
(STOs). Ontheotherhand,modernhighprecisionspec- tional role of the l = 1 and l = 2 orbitals. According
troscopyofRydberglevelsof87Rbhasbeenconductedre- to Goeppert Mayer, the outermost electron of an alkali
cently. Millimeter-wavespectroscopyemployingselective atomisgovernedbyaneffectiver-dependentchargeterm
field ionization allows for precise measurements of the
energy differences between Rydberg levels [8]. An inde- Zeff(r)=1+(Z−1)F(r), (12)
pendent approach is to perform purely optical measure-
wherethefunctionF(r)hasbeendeterminedbyemploy-
ments on absolute Rydberg level energies by observing
ing the semi-classical statistical Thomas-Fermi approach
electromagnetically induced transparency (EIT) [9, 10].
to the many-electron-atom problem, posing the bound-
However,thereisasystematicdiscrepancybetweenvari-
aryconditionsaslim F(r)=1andlim F (r)=0.
ational calculations and the spectroscopic measurements r→0 r→∞
As discussed by Schwinger [12], this approach ceases to
of the fine splitting
be valid in the inner shell region Z−1 < r < Z−31 of
∆E =E −E (10) the atom. Therefore, taking into account the fine split-
n,l n,l−1,l n,l+1,l
2 2 ting in the spectrum of the outermost electron of alkali
as shown in Tables I and II. Given the fact that the er- atoms a posteriori by simply adding the phenomenolog-
rorbarsoftheindependentexperiments[8,10]arebelow ical spin-orbit term (5) to (2), resulting in the effective
1.1MHz down to 20kHz, and on the other hand con- single-electronpotential(9),seemstobequestionableon
sidering the high accuracy of the numerical calculations general grounds in that inner shell region.
presented in [7], such a discrepancy between experiment On a more fundamental level, the treatment of rela-
and theory is indeed significant. tivistic effects in multi-electron-atom spectra requires an
So, what could be the reason for the reported discrep- apriori microscopicdescriptionbasedonthewell-known
ancies? First, it should be pointed out that in the vari- Breit-Pauli Hamiltonian [13, 14]
ational calculations [7] a slightly different potential was
H =H +H +H . (13)
used, that is, nr rs fs
Here H is the ordinary nonrelativistic many-electron
V (r;j,l)=V (r;l)+V (r;j,l). (11) nr
eff SO Hamiltonian, while the relativistic corrections are repre-
sented by the perturbation operators H and H . The
Certainly, within the first-order perturbation theory rs fs
perturbation term H contains all the relativistic per-
thereexistsnonoticeablediscrepancyinthespectrumof rs
turbations like mass correction, one- and two-body Dar-
the outermost electron on the fine-splitting scale, when
win terms, and further the spin-spin contact and orbit-
takingintoaccountthespin-orbitforceswithV (r;j,l)
SO orbit terms, which all commute with the total angular
instead of working with V(cid:101)SO(r;j,l). This is due to the momentum L and total spin S, thus effectuating only
differences being negligible for r > Zα2. However, since
smallshifts ofthespectrumofthenonrelativisticHamil-
VSO(r;j,l)eventuallydominateseventhecontributionof tonianH . TheperturbationoperatorH ontheother
nr fs
the centrifugal barrier term l(l+1) within the tiny region hand breaks the rotational symmetry. It consists of the
r2
0 < r (cid:46) α2Z, a subtle problem with a non-normalizable standardnuclear spin-orbit,thespin-other-orbit,andthe
radialwavefunctionU (r)emergeswhenattemptingto spin-spin dipole interaction terms, which all commute
n,j,l
solve the Schr¨odinger eigenvalue problem for any l > 0 withJ=L+S,butnotwithLorwithSseparately,thus
with the potential V (r;j,l). Such a problem is absent inducingthefinesplittingofthenonrelativisticspectrum.
SO
when one works with V(cid:101)SO(r;j,l) [4]. Although the proposed functional form of the poten-
A variational calculation with the potential (11) em- tial (11) is highly plausible on physical grounds out-
ploying N = 500 normalizable STOs as basis functions side the inner core region r > Z−31, prima facie it ap-
3
TABLE I. Fine splitting ∆En,l=1 for P states in [MHz].
State n,l=1 Exp. [11] Exp. [8] Theory [7] Theory (this work)
| (cid:105)
8P 565.1(4) 103 N/A 602.00 103 567.75 103
× × ×
10P 219.1(4) 103 N/A 231.87 103 218.77 103
× × ×
30P N/A 4246.30(5) 4500.50 4246.46
35P N/A 2566.41(32) 2717.41 2566.28
45P N/A 1144.09(13) 1217.24 1143.95
55P N/A 605.77(7) 644.81 605.68
60P N/A 460.76(5) 480.32 460.68
TABLE II. Fine splitting ∆En,l=2 for D states in [MHz].
State n,l=2 Exp. [11] Exp. [8] Exp. [10] Theory [7] Theory (this work)
| (cid:105)
8D 30.4(4) 103 N/A N/A 113.17 103 36.42 103
× × ×
10D 14.9(2) 103 N/A N/A 52.05 103 16.56 103
× × ×
30D N/A 452.42(18) 452.5(11) 1447.53 456.13
35D N/A 279.65(10) 280.4(11) 894.84 281.52
45D N/A 128.33(4) 127.8(11) 407.64 128.98
55D N/A 69.17(2) 69.4(11) 223.71 69.47
57D N/A 61.98(2) 62.2(11) 197.39 62.24
pearstobeinconsistenttolumptheaforementionedrela- fectivesingle-electronpotentialV (r;j,l)describingthe
eff
tivisticmany-bodyforcesintoaneffectivesingle-electron fine splitting of the spectrum of the outermost electron
potential of the functional form (11), so that it pro- in the alkali atoms, we introduce a cutoff at a distance
vides an accurate description also for small distances rso(l) with Z−1 < rso(l) < Z−13 so that the effective
Z−1 <r <Z−13. single-electron potential is now described by the follow-
Intheabsenceofabettermicroscopictheoryforanef- ing modified potential:

V (r;l) if 0≤r ≤r (l),
 eff so
V(cid:101)mod(r;j,l)= (14)
V (r;l)+V (r;j,l) if r >r (l).
eff SO so
The choice [2] of the atom. Because the higher the orbital angular mo-
mentum quantum number l, the lower the probability of
r (l=1)=0.029483×r (l=1)=0.0442825,
so c the electron being located near to the center, it is clear
(15)
that the quantum defect decreases rapidly with increas-
r (l=2)=0.051262×r (l=2)=0.2495720, ing orbital angular momentum l. Therefore, ∆ is only
so c j,l
notably different from zero for l=0,1,2.
gives a surprisingly accurate description of the fine split-
Writing∆ =δ +η withη (cid:28)δ ,thefinesplitting
ting in the spectroscopic data for all principal quantum j,l l j,l j,l l
to leading order in α2 is:
numbers n, see Tables I and II.
The calculation of the spectrum of the outermost ηl−1,l−ηl+1,l
∆E =2 2 2 (17)
bound electron is then reduced to solving the radial n,l (n−δ )3
l
Schr¨odinger equation (8) with the modified potential √
The quasiclassical momentum p≡ −Q of the bound
V(cid:101)mod(r;j,l). The resulting spectrum is actually hydro-
electron with orbital angular momentum l>0, total an-
gen like, that is,
gular momentum j =l± 1, and taking into account the
1 2
E =− , (16) Langershiftl(l+1)→(cid:0)l+ 1(cid:1)2 inthecentrifugalbarrier
n,j,l (n−∆ )2 2
j,l [15, 16], is then given by
where∆ denotesaquantumdefectcomprisingalsothe
j,l (cid:0)l+ 1(cid:1)2
fianreedsupclittitoinngo.fItnheacntuumalbfearctofthneodqeusanntuomftdheefercatddiaelscwraibvees- Q(r;j,l,E)= r22 +V(cid:101)mod(r;j,l)−E. (18)
r
function for l = 0,1,2 as a result of the short-range in- For l =0 the centrifugal barrier term and the spin-orbit
teraction of the outermost electron with the ionic core potential are absent.
4
100 20
80
15
)
E 60
Q
j,l, √−10
(
ν 40
5
20
0 0
0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 1.2
1/√ E r/r (l)
− c
FIG. 1. (Color online) The action integral ν(j,l,E) associ- FIG. 2. (Color online) The quasiclassical momentum
avtse.dscwailtehd ethneeregffyec√t−1ivEe fsoinrglle=-e0lec(tbrloune lpinotee)n,tlia=l 1V(cid:101)m(roedd(rl;inj,el)), (cid:112)bla−ckQ),(rl;=j,l1,E(g)revesn.),scla=led2 (driesdta),ncfoerrEcr(l=) fEorn,jl,l=co0rr(edspasohnedd-
l=2 (green line), all for j =l+ 1. The curves for j =l 1 ing to principal quantum number n = 57 and j = l + 1.
2 − 2 2
only differ by a tiny shift proportional to α2. The main contribution to the quantum defect values in (24)
originates from the inner core region r<rc(l).
Considering high excitation energies E < 0 of the where ν(j,l,E) denotes the action integral
boundoutermostelectron,i.e. aprincipalquantumnum-
ber n(cid:29)1, the respective positions of the turning points 1 (cid:90) r(+) (cid:112)
r(±) are given approximately by ν(j,l,E)= dr −Q(r;j,l,E)
π
r(−)
1 (cid:73)
= drp(r;j,l,E). (21)
(cid:0)l+ 1(cid:1)2 2π
r(−) = 2 if l≥3,
(cid:113)
1+ 1+(cid:0)l+ 1(cid:1)2E Plotting the function ν(j,l,E) versus √1 for l =
2 −E
0,1,2 clearly reveals a linear dependence of the form
(19)
ν(j,l,E)= √1 +c(j,l), see Fig. 1.
 (cid:115)  −E
1 (cid:18) 1(cid:19)2 According to [6], for A,B,C,D ∈R, with A>0, B >
r(+) (cid:39) −E 1+ 1+ l+ 2 E if l≥1, 0, C >0, and |D|(cid:28)C the following equality holds:
1 (cid:73) (cid:114) 2B C D B √ BD
dr −A+ − + = √ − C+ √
where 0 < l (cid:28) √1 . Of course for l = 0 only a single 2π r r2 r3 A 2C C
−E (22)
(large)turningpointr(+) = 2 existsduetotheabsence
−E For a pure Coulomb potential A ≡ −E, B ≡ 1, C ≡
of the centrifugal barrier. However, the lower turning (cid:0)l+ 1(cid:1)2 and D ≡ α2g(j,l). The corresponding action
points r(−) are strongly modified for l = 1,2 compared 2
integral then reads
to the pure Coulomb potential case taking into account
0th.0e2c×orercp(ol)larhiozladtsio;nt.haFtorisl,=r(1−,)2(lth=e1r)ela(cid:39)tio0n.0r3(4−7)2(la)n(cid:39)d √−1E if l=0,
r(−)(l=2)(cid:39)0.12827 [2]. Since the cutoff r (l) in (15) ν(C)(j,l,E)=
so
ipsosinutbssrta(−n)ti(al)ll,yaaqbuoavseicltahsossiceavlaclaulceusloaftiothneoflotwheerfintuer-snpinligt √−1E −(cid:0)l+ 12(cid:1)+ 2α(2lg+(j12,)l)3 if l>0.
(23)
spectrum of the bound outermost electron is reliable.
Forachosenradialquantumnumbern ,theassociated It is thus found from WKB theory that the quan-
r
eigenvaluesE =E <0oftheoutermostelectronnow tum defect associated with the single-electron potential
n,j,l
follow from the WKB patching condition [17, 18]: V(cid:101)mod(r;j,l) is:
(cid:104) (cid:105)
 ∆j,l = lim ν(j,l,E)−ν(C)(j,l,E) (24)
nr+1 if l=0, E→0−
!
ν(j,l,E)= (20) Ignoring spin-orbit coupling, i.e. for α = 0 , one has
n + 1 if l>0, ∆ ≡ δ , the standard quantum defect. For l = 0 the
r 2 j,l l
5
TABLE III. The values of quantum defect ∆j,l associated with the Rydberg level n=57 for l=0,1,2.
Quantum defect ∆j,l Exp. [8] Exp. [10] Theory [7] Theory (this work)
∆ 3.1312419(10) 3.13125(2) 3.12791 3.13095
1/2,0
∆ 2.6549831(10) N/A 2.65795 2.65197
1/2,1
∆ 2.6417735(10) N/A 2.64399 2.63876
3/2,1
∆ ∆ 0.0132096(14) N/A 0.01396 0.01321
1/2,1− 3/2,1
∆ 1.3478971(4) 1.34789(2) 1.35145 1.34851
3/2,2
∆ 1.3462733(3) 1.34626(2) 1.34628 1.34688
5/2,2
∆ ∆ 0.0016238(5) 0.00163(3) 0.00517 0.00163
3/2,2− 5/2,2
centrifugal barrier and the spin-orbit coupling term (6) of the radial Schr¨odinger equation (8) [21].
are zero, so ∆ →∆ ≡δ .
j,l 1,0 0
2
The dependence of the quasiclassical momentum III. CONCLUSIONS
(cid:112)−Q(r;j,l,E) on the scaled distance r is shown for
rc(l)
l = 0,1,2 in Fig. 2. Clearly, it is the inner core re- In this work we reported a significant discrepancy be-
gion r(−)(l) < r < rc(l) that provides the main con- tween experiment [8, 10] and highly accurate variational
tribution to the quantum defect values. We find, for calculations[7]ofthespectrumofRydbergstatesof87Rb
l = 0,2, that changing the fitting parameter a3(l) in on the energy scale of the fine splitting. We discussed
(3) from its tabulated value in [5] according to the thattheusuala posteriori addingoftherelativisticspin-
scaling prescription a3(l=0) → 0.814×a3(l=0) and orbit potential to the effective single electron potential
a3(l=2) → 0.914×a3(l=2), leads to a slight down- governing the outermost electron of alkali atoms is in-
ward constant shift of the WKB-quantum defect. As a deed inconsistent inside the inner atomic core region.
result of this change, the calculated WKB-quantum de- In the absence of a full microscopic theory that lumps
fect ∆l±1,l then agrees well with the spectroscopic data, all many-body interactions together with the relativistic
2
see Table III. Such a change of a3(l) does not affect the corrections into an effective single-electron potential in
fine splitting values ∆En,l though. We also find that the a consistent manner, we suggested a modified effective
dependence of the fine splitting ∆En,l on the principal single-electron potential, see (14), that enables a cor-
quantumnumberniswelldescribedby(17)foralln≥8, rectdescriptionofthespectrumofRydbergstatesonthe
see Tables I and II. fine splitting scale in terms of a simple WKB-action in-
tegral for all principal quantum numbers n≥8. Modern
Inactualfact,forr(+) (cid:29)r(−),whichisacriterionthat
√ precision spectroscopy of highly excited Rydberg states
isalwaysmetforhighexcitationenergies −E (cid:39)0ofthe
thusenablestheprobingofthemulti-electroncorrelation
outermostelectron, theuniformLanger-WKBwavefunc-
problem of the ionic core of alkali atoms. This is cer-
tionU(WKB)(r)[19,20],withr(+) consideredastheonly
n,j,l tainly a fascinating perspective for further experiments
turning point, describes the numerical solution Un,j,l(r) and theoretical studies.
to the radial differential equation (8) under the influ-
ence of the effective modified single-electron potential
(14) rather accurately [21]. Only very near to the sec- ACKNOWLEDGMENTS
ondturningpointr(−),atadistancesmallerthanr (l),
so
the Langer-WKB wavefunction U(WKB)(r) ceases to be ThisworkwasfinanciallysupportedbytheFET-Open
n,j,l
agoodapproximationtothenumericalsolutionU (r) Xtrack Project HAIRS and the Carl Zeiss Stiftung.
n,j,l
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