Long-range potentials and $(n-1)d+ns$ molecular resonances in an ultracold rydberg gas

0.37 MB·

by  J. Stanojevic

Long-range potentials and (n 1)d+ns molecular resonances in an ultracold Rydberg
−
gas
J. Stanojevic, R. Cˆot´e, D. Tong, E.E. Eyler, and P.L. Gould
Department of Physics, University of Connecticut, Storrs, CT 06269, USA
We have calculated long-range molecular potentials of the 0+, 0− and 1 symmetries between
g u u
highly-excited rubidium atoms. Strong np+np potentials characterized by these symmetries are
importantindescribinginteraction-inducedphenomenaintheexcitationspectraofhighnpRydberg
states. Long-range molecular resonances are such phenomena and they were first reported in S.M.
8 Farooqi et al., Phys. Rev. Lett. 91 183002. One class of these resonances occurs at energies
0 corresponding to excited atom pairs (n−1)d+ns. Such resonances are attributed to ℓ-mixing due
0
toRydberg-Rydberginteractionssothatotherwiseforbiddenmoleculartransitionsbecomeallowed.
2
Wecalculate molecular potentials in Hund’scase (c),usethem tofindtheresonance lineshape and

n compare to experimental results.
a
J PACSnumbers: 32.80.Rm,32.80.Pj,34.20.Cf

5
1
I. INTRODUCTION separations betwen the (n 1)d+ns resonances and the
−
] atomic np resonances are several times greater than the
h
correspondingenergyspacingsforthe(n 1)p+(n+1)pres-
p Rydbergatomshavelongbeenstudiedfortheirunique −
onances. As a result, much higher laser power is needed
- properties, such as long radiative lifetimes, large cross
m to excite the (n 1)d+ns resonances. Also, many more
sections and huge polarizabilities [1]. At high principal −
molecularstates areneeded for a detailedanalysisof the
o
quantum numbers, interaction forces between Rydberg
t interaction-induced ℓ-mixing, which is necessary for the
a atoms become extremely large. Manifestations of these
existence of the resonances.
. long-rangeinteractionscanbeseeninultracoldcollisions
s
c [2]ordensity-dependentlinebroadeningofresonancesin
si atomic beams[3]. InultracoldRydbergsystems thermal
y motion is greatly reduced so that the effects of strong II. THEORY
h interactions can be investigated and utilized in a more
p
controlable manner. The prospect of using ultracold
[ This paper consists mainly of two parts. In the first
Rydbergatomsinquantuminformationapplicationshas part we calculate long-range molecular potentials of the
1 stimulated a great deal of experimental and theoretical 0+, 0 and 1 symmetries since these symetries de-
v interest lately [4, 5]. It was proposed to use the effect of g −u u
scribe strong np + np potentials. In the second part
6
dipole blockade to realize scalable quantum gates [5]. In
8 we evaluate the excitation dynamics of these molecular
mesoscopic ensembles of atoms, one excited atom pre-
3 statesto investigatethe variouseffectsofinteractionsbe-
vents the Rydberg excitation of its neighbors because
2 tween the atoms. In this paper we primarily study the
. strong interactions shift many-atom excited states out (n 1)d+ns resonancesreported in [16]. They occur at
1
of resonance. Recently, large inhibition of Rydberg ex- −
0 the average energy of excited atom pairs (n 1)d and
citation due to van der Waals (vdW) interactions (for −
8 ns, and do not correspond to any single-atom transi-
n 70-80), have been observed using a pulse-amplified
0 ∼ tions. The contributions of diatomic potentials which
: single-mode laser [6] and cw excitation [7, 8]. Also, the coincide with the asymptotic (n 1)d + ns levels are
v
dipole-blockade of cw Rydberg excitation of Cs atoms −
i dominant, although, there are significant contributions
X has been achieved [9].
from (n 2)d+(n+1)s and (n 2)p +(n+2)p
1/2 1/2
− −
r There have been several proposals for weakly bound potentials, with approximately the same asymptotic en-
a
states involving Rydberg atoms but they have not yet ergies. In one-photon transitions from the 5s ground
been detected. The so called “trilobite” and “butterfly” state, dipole transitions to nd and ns states are not al-
states are molecular states formed by a pair of atoms, lowed. However, at high principal quantum numbers,
with one of the atoms in the ground state and the other long-rangeRydberg-Rydberginteractionscauseℓ-mixing
one in a Rydberg state [10, 11, 12, 13, 14]. Bound so that other molecular states, besides np+np, become
states of two Rydberg atoms have been also proposed accessible. Although the physics of this resonance and
[15]. The long-range molecular resonances are another the (n 1)p+(n+1)p one is verysimilar,the treatment
−
effect of Rydberg interactions. In the experiment [16], of (n 1)d + ns resonances is technically much more
−
(n 1)p + (n+1)p and (n 1)d + ns resonances have demanding. There are many more asymptotic states be-
− −
been observed in single photon UV excitation from the tween np+np and (n 1)d+ns asymptotes and they
−
5sgroundstatetohighnpRydbergstates(n=50 90). all have to be included in order to describe the ℓ-mixing
−
A detailed theoretical treatment of (n 1)p + (n+1)p correctly. To make the potentials accurate at short dis-
−
resonances was presented in [17]. However, the energy tances, many nearby asymptotes must also be included
2
in the asymptotic basis. In addition, the laser intensity asymptotic states are constructed as follows
used to excite them was almost two order of magnitude
greaterthanthat usedfor (n 1)p+(n+1)p resonances |nℓj,mj;n′ℓ′j′,Ω−mj;Ωg/ui
−
so there could be more power-dependent terms to con-
n,ℓ,j,m n,ℓ,j ,Ω m
j ′ ′ ′ j
sider, besides the two-photon Rabi frequency. We have ∼h| i| − i
used several different approaches to evaluate the results p( 1)(ℓ+ℓ′) n,ℓ,j ,Ω m n,ℓ,j,m . (2)
′ ′ ′ j j
oftheRydbergexcitationofatompairsandthey allgive − − | − i| ii
consistent results. No overlap of the charge distributions of different atoms
isassumedheresinceultracoldgasesareverydilute,even
considering these Rydberg atoms. All molecular states
A. Long-Range Rydberg-Rydberg potential curves are defined in the molecule-fixed reference frame. States
with Ω=0 have to be (anti)symmetric under the action
of the reflection operator σ . We have addopted the fol-
We calculate long-range potentials in Hund’s case (c) ν
lowing convention [18] for the action of σ
by diagonalization of an interaction matrix. There are ν
severalreasons why this Hund’s case is considered. Here σ Λ =( 1)Λ Λ , (3)
ν
we point out only the basic arguments since they are | i − |− i
explained in detail in [17]. We consider the states that σν S,MS =( 1)S−MS S, MS . (4)
| i − | − i
are directly or indirectly coupledto np+np asymptotes,
The first rule (3) is obviously not applicable if Λ = 0.
which can be directly excited from the 5s+5s ground
The correct result of σ on such states follows from its
ν
state. We can focus on strong potentials only because
action on atomic states σ ℓ,m =( 1)m ℓ, m .
ν
they can significantly mix, at short internuclear separa- | i − | − i
After adding spin, there are many more different sym-
tions,withotherasymptoticstates. Suchnp+nppoten-
metries to consider. On the other hand, it makes the
tials are those coupled to nearby ns+(n+1)s states.
ℓ-mixing a smoother function of interaction energy, sim-
The asymptotic energy spacing between np + np and
ply because there are many more potential curves with
ns+(n+1)sstates is quite small, whichadditionallyin-
a given ℓ-component for a given range of energy. We
creasestheeffectoftheirstrongdipole-dipolecoupling. If
are primarily interested in how ℓ = 1 or p-character is
there were no atomic fine structure, the only candidates
distributed over different potential curves for each R. If
forstrongnp+nppotentialswouldbetheasymptotically
α(R;λ) is the fraction of a given asymptotic state φ in
degenerate 1Σ+ and 3Σ+ states [7]. After adding spin, | i
g u the molecular state ϕ , then the sum of α(R;λ) over
R;λ
thepossiblesymmetrieswithstrongpotentialsare0+,0 | i
g −u all molecular states has to be unity for each R. This
and1 . Finestructurecanbe,atleastinitially,neglected
u statement is just the normalization condition of φ in
only if its energy splitting is much less than the separa- | i
the eigenbasis of U(R). We use this α(R;λ) as a mea-
tion between adjacent nℓ+nℓ asymptotic levels, which
′ ′ sure of the character associated with φ . Therefore, for
is notthe casehere. Forinstance,the asymptotic energy | i
anynp+npasymptoticstate,the sumofthe p-character
spacingbetween 70p +70p and70p +70p is al-
1/2 1/2 3/2 3/2 of all molecular states has to be unity for any R.
most three times larger then the asymptotic separation
Wehaveassumedinthisanalysisthatthereisnoback-
between70s +71s and70p +70p . Thisobstacle
1/2 1/2 3/2 3/2 ground electric field. If a background electric field is in-
impedes a unique definition of the dispersion coefficient
cluded,thenthegroupsymmetryoftheelectronicHamil-
C . However,thesedispersioncoefficientscannotbeused
6 tonianis notD . Consequently,the quantumnumbers
h
becausetheinteractionenergyofpairsofRydbergatoms based on this s∞ymmetry are not good anymore and we
contributing to (n 1)d+ns resonances is about twenty
cannotreducetheinteractionmatrixinthebasisofstates
−
times greaterthanthe regionofenergyforwhichpertur- of a given symmetry. In general, one has to include all
bation theory is applicable. To obtain potential curves
possible basis states and use them to diagonalize the in-
one has to diagonalize simultaneously both interaction teraction matrix [19].
terms,thelong-rangeRydbergtermandtheatomicfine-
For non-overlapping charge distributions, the
structure one. Rydberg-Rydberg interaction may be expanded in
The interaction matrix includes long-range Rydberg- a series of inverse powers of nuclear separations R.
Rydberg interactions VRyd(R) and the atomic spin-orbit In this paper, we consider dipolar and quadrupolar
interaction Hfs interactions only. These interactions can be described
by the following form [20, 21]
U(R)=V (R)+H . (1)
Ryd fs
( 1)ℓ4πrLrL
V (R)= − 1 2
The eigenproblem of this interaction matrix is solved L − LˆR2L+1
for the 0+, 0 and 1 states only. The first step is (5)
to construgct a−usymptotiuc states of these symmetries and × B2Lℓ+mYLm(rˆ1)YL−m(rˆ2),
Xm
used them as a basis to represent U(R). The projection
Ω = m +m of the total angular momentum onto the where L=1 (L=2) for dipolar (quadrupolar) interac-
1 2
molecular axis is conserved and properly symmetrized tions, Bm m!/(n m)! is the binomial coefficient, ~r
n ≡ − i
3
2 where ℓˆ = 2ℓ +1, ˆj = 2j +1, and L is the radial
(a) 0 + part of tihe maitrix elemi ent ii rL j . Rij
1 g h | | i
The asymptotic basis used to represent U(R) consists
of all np+np, (n 1)d+ns and (n 2)d+(n+1)s asymp-
70s+71s
0 } totic states, as w−ell as all the st−ates in between with
70p+70p
a significant coupling to these asymptotes. These in-
Hz) -1 }69p+71p c(nlude2)tph+e(n(n+−2)1p)pa+sy(mnp+to1t)eps,. T(nhe−se1s)tsa+te(sna+re2o)usr apnrid-
G
69s+72s −
y (-2 maryinterest,buttodescribethemcorrectlyatshortin-
g
er ternucleardistancesoneshouldalsoincludeothernearby
En-3 statesstronglycoupledtothem. Includednearbyasymp-
totesarens+(n+1)s,(n 2)s+(n+3)s,(n 3)d+(n+2)s,
6688pp3/2++7722pp3/2 nd+(n 1)s,(n 3)f+npa−nd(n 2)f+(n 1−)p. While the
-4 68p3/2+72p1/2 − − − −
69d1+/720s 3/2 dipole-dipole interaction couples states belonging to dif-
68p +72p
1/2 1/2 ferent nℓ+nℓ asymptotes, the quadrupole interaction
68d+71s ′ ′
-5
is mainly relevant for states within the same nℓ+nℓ
′ ′
68s+73s asymptote. The only exceptions are the off-diagonal
-6 quadrupole matrix elements between (n 1)d+ns and
421/22ω/∆)>/(eff000...001482 (b) 666898pdd1++/772+017ss2p1/2 (t|tnnhh∗1ee−−ec2ffo)nuedc∗2p+t|l,iivn(wengh+pedrre1ienp)csetinhpadeasslyqmqounuapantn|tonutt1uemsm−.dnenAfu2esm|c,tbmoeδrerℓn−ttniosi∗ionibnaeecsdlufpbodrleeelfodcowirsieesn,:,
ω n = n δ . Since the difference in effective principal
< ∗i i − ℓ
quantum numbers of states (n 1)d and (n+1)s, as well
0
4 6 8 10 12 −
as (n 2)d and ns states, is only 0.22 for Rb (for high
Separation R (x 104 a.u.) −
Rydberg states δ 3.13 and δ 1.35 [22]), the
ℓ=0 ℓ=2
≈ ≈
off-diagonal quadrupole matrix element is several times
FIG. 1: (a) Potentials curves for the 0+ symmetry. We
g largerthanthediagonalones. Theseasymptotesarevery
present all the potential curvesbetween 70s+71s and 68s+
73s. Molecular states, besides those that coincide with the
asymptotic70p+70pstates,maybecomeaccessible duetoℓ-
2
mixinginducedbyinteractions. Theresonanceisdominantly
produced by the 69d+70s states, although significant con- (a) 0u-
tributions also come from the 68d+71s and 68p +72p 1
1/2 1/2
states. There are two 69d+70s and 68d+71s states of this
70s+71s
symmetry and only one from the68p1/2+72p1/2 asymptote. 0 }
The fine structure of 69d and 68d states is ignored. (b) Av- 70p+70p
erage radial dependencies of the two-photon Rabi frequency.
Theexcitationprobabilityofanatompairdependsonitssep- z) -1 }69p+71p
H
aration R since the fraction of p-character of any molecular G 69s+72s
stateisafunctionofR. Theradialdependenceisshownforall y (-2
g
states ofthethreeasymptotescontributingtotheresonance. er
n
E-3
68p +72p
is the position of the electron i around its center and -4 6688pp33//22++7722pp13//22
Lˆ=2L+1. 69d1+/720s 3/2
68p +72p
1/2 1/2
Defining a b n ,ℓ ,j ,m n ,ℓ ,j ,Ω m ,one 68d+71s
| i| i≡| a a a ai| b b b − ai -5
can show that the matrix elements of V (R) are
ℓ
68s+73s
-6
4
h1,2|VL|3,4i=(−1)L−1−Ω+iP=1jiqℓˆ1ℓˆ2ℓˆ3ℓˆ4ˆj1ˆj2ˆj3ˆj4 21/2∆)> 00..0068 (b) 6689dd++7710ss
× RR(1L32)LR+(21L4) (cid:18)ℓ01 L0 ℓ03(cid:19)(cid:18)ℓ02 L0 ℓ04(cid:19)(cid:26)ℓj13 L21 ℓj31(cid:27) 42ωω//(eff00..0024 68p1/2+72p1/2
j L j L j L j <
2 4 BL+m 1 3 0
×(cid:26)ℓ4 12 ℓ2(cid:27)mX=−L 2L (cid:18)−m1 m m3(cid:19) 4 Separ6ation R (x8 104 a.u.)10 12
j L j
2 4 , (6)
×(cid:18)−Ω+m1 −m Ω−m3(cid:19) FIG. 2: Same as Fig. 1 butfor the0−u symmetry
4
close in energy (separated by only 200 MHz for n=70) long-range interactions and a linearly polarized optical
so that at R 30000 a and n = 70, quadrupole off- field. For simplicity, we include only one molecular state
0
∼
diagonalcoupling is comparablewith the asymptotic en- ϕ and the corresponding molecular potential ǫ (R).
λ λ
| i
ergy spacing and these states become well mixed. This The Hamiltonian is (in the rotating frame and in the
off-diagonalquadrupolecouplingisrelevantfortheshape rotating-waveapproximation)
of (n 1)d+ns and (n 2)d+(n+1)sresonances.
− − 2 2 ω ω
H = ∆σeie+∆′σei′e′ + (cid:20)2σeig + 2′σei′g+h.c.(cid:21)
Xi=1(cid:2) (cid:3) Xi=1
B. Excitation of a pair of interacting Rydberg
+[∆ +ǫ (R)] ϕ ϕ , (7)
atoms λ λ λ λ
| ih |
where ω, ω , and ∆, ∆ are single-photon Rabi fre-
′ ′
Wetreatthe(n 1)d+nsmolecularresonancesastwo- quencies and detunings relative to the np and np
− 3/2 1/2
body phenomena, which is supported by the fact that fine-structure components, respectively. Here ∆ is the
λ
theirenergiescoincidewiththeaveragevalueofonlytwo two-photon detuning from the asymptotic ǫ (R )
λ
→ ∞
atomicenergies[16]. Theresonancesarefarred-detuned, molecular level. The operators σi and σi are de-
eg ee
sonormallythereshouldnotbemanyexcitedatomsand fined as follows: σi = e ,m g ,m and σi =
pnrisemsu.mTahbelygpeanierr-awlisperoebxlceimtattioonbiesstohleveddomisinthanetsmamecehaas- Ppermim|eein,mtailhveail,umes|.foTreogthcealocusPclaimltlae|tωoirasntirdhenωig′twh|ef3u/s2e[2th3]eeeaenxd-
in [17] for (n 1)p+(n+1)p resonances. However, the
− the ratio [24] ω/ω′ = f3/2/f1/2 2.3, which reflects
analysis of (n 1)d+ns resonances is technically much ≈
− the non-statistical chapracter of the oscillator strengths
more demanding and some approximations we have ap-
f and f .
3/2 1/2
plied before may not be satisfactory here. To check this,
The excitation process is essentially a three-level
we have used several different ways to evaluate the con-
scheme, although the number of states involved in the
tributions to the number of excited atoms from various
process is greater than three. We assume that a pair
molecularstates. We discuss them in the results section.
of ultracold Rb atoms is initially in one of the 5s+5s
We consider a two-body Hamiltonian that includes
ground states. There are four possible ground states
5s,m 5s,m ,correspondingtodifferentprojectionsof
| ji| ′ji
2 spin mj = 1/2. Ultimately, the total probability is av-
±
eraged over all possible initial states. In the first excita-
-
1
1 (a) u tionstep,oneoftheatomsisexcitedtoagivennpj state.
There are two intermediate states if the ground-state
0 }70s+71s atomshavethe sameprojectionsmj,otherwisethereare
four of them. These intermediate states are further ex-
70p+70p
citedinthesecondstep. Thefinalstateinthisexcitation
Hz) -1 }69p+71p scheme can be a single molecular state |ϕλ(R)i or a su-
G perposition of states. We consider both cases but in the
69s+72s
y (-2 equations a single molecular state is assumed. Includ-
g
er ing more states in the equations is very straightforward
n
E-3 and there is no need to do it explicitly. Eventually, to
get the final excitation probability per atom for a given
68p 72p
68p3/272p3/2 optical frequency, the contributions from all molecular
-4 68p3/272p1/2
69d1+/720s3/2 potentials and all atom pairs that include a given atom
68p 72p
68d1+/721s1/2 are collected.
-5
To get pair excitation probabilities we solve the time-
68s+73s dependentSchr¨odingerequationforthe grounddiatomic
-6 state, all intermediate states and a given doubly-excited
1/2> 0.1 (b) mmoenletciuflatwrostmateo|ϕfλthi.eHdiearteowmeicpgrreoseunntdinstdateetaairlethdeiffterreeantt-
2∆) 69d+70s and give onlyjthe final result when they are the same
4/ 68d+71s
2ω/(eff0.05 68p1/272p1/2 (fathctistchaastesiysmcmonestirdicereadndinandteitsayiml imne[t1r7ic]).staUtetislizhianvgetihne-
ω
< dependent time evolutions, the first step is to construct
symmetric and antisymmetric combinations
0
4 6 8 10 12
1
4
Separation R (x 10 a.u.) ij = i,m j, m +q i, m j,m , (8)
| i √2{| i| − i | − i| i}
FIG.3: SameasFig. 1butforthe1u symmetry,exceptthat where q = 1( 1) for symmetric(antisymmetric) states
each nd+n′s asymptote has four states of thissymmetry. and m = 1/2−for linear laser polarization. In this way
5
the diatomic ground state gg , four intermediate states equivalent to the Bloch equations of a two-level system
| i
ge , eg , ge , eg ,andfourdoubly-excitedstates ee ,
′ ′
||ee′ii,||e′iei,||e′ei′i|areidefined. Hereeande′ referton|p3/i2 idc0 = ωg(t)c0 ωe∗ffc2, (16)
and np states, respectively. If there were no interac- dt − − 2
1/2
stitoantess,ddiiarteocmtlyicasctcaetsessib|eleeii,n|etew′io,-|peh′eoit,o|ne′eex′icwitaotuilodnb.eAtnhye iddct2 =[∆λ+ǫλ(R)−ωe(t)]c2− ω2effc0, (17)
molecular state ϕ is accessible if it has some compo-
λ
| i where the effective two-photon Rabi frequency is
nents of these diatomic states. Due to ℓ-mixing induced
by interactions, many ϕ (R) gaina significantnp frac-
tion at some finite inte|rnλucleair separation R. ωeff =ω2aee(λ)+ ωω′∆+∆′(aee′(λ)+ae′e(λ)),
∆ 2 ∆∆
The wave function is modeled as follows ′ (18)
ω2
′
+ ae′e′(λ)
|ψi=c0|ggi+c11|gei+c12|egi+c′11|ge′i (9) ∆′
+c′12|e′gi+c2|ϕλi. and the power-dependent terms ωg and ωe are
Solving the time-dependent Schr¨odinger equation ω 2 ω 2
′
i∂ψ/∂t = Hψ (~ = 1) leads to the coupled system for ωg = |2∆| + |2∆| , (19)
′
the excitation amplitudes c(t), ωaee+ω′ae′e 2 ωaee+ω′aee′ 2
ω = | | + | |
e
4∆ 4∆
iddct0=ω2∗(c11+c12)+ ω2′∗(c′11+c′12), (10) + |ω′ae′e′ +ωaee′|2 + |ω′ae′e′ +ωae′e|2. (20)
4∆ 4∆
′ ′
dc11 ω ω∗ ω′∗
i =∆c + c + eeϕ c + eeϕ c , (11)
dt 11 2 0 2 h | λi 2 2 h ′ | λi 2 Theseωg(t)andωe(t)canbeinterpretedaspowerdepen-
dent shifts of the diatomic ground and doubly-excited
dc ω ω ω
12 ∗ ′∗
i dt =∆c12+2c0+ 2 hee|ϕλic2+ 2 hee′|ϕλic2, (12) states, respectively. Also, ωg is much greater than ωe
because it does not depend on a , which measure the p-
dc ω ω ω ij
i ′11=∆c + ′c + ∗ ee ϕ c + ′∗ ee ϕ c , (13) character of the doubly-excite state. For the experimen-
dt ′11 2 0 2 h ′| λi 2 2 h ′ ′| λi 2
tal parameters, the peak value of ω (t) on a molecular
g
dc ω ω ω
i ′12=∆c + c + ∗ eeϕ c + ′∗ ee ϕ c , (14) resonance is about 280 MHz, which is comparable with
12 0 λ 2 ′ λ 2
dt 2 2 h | i 2 h | i the laser bandwidth.
dc2 ω ω′ In the vicinity of the molecular resonance, the second
i =(∆ +ǫ (R))c + ϕ ee + ϕ ee c
λ λ 2 λ λ ′ 11
dt (cid:16)2h | i 2 h | i(cid:17) term on the right-hand side of Eq. (16) is much smaller
ω ω ω than the first one, and thus can be ignored. After ne-
′
+(cid:16)2hϕλ|eei+ 2 hϕλ|ee′i(cid:17)c12+(cid:16)2hϕλ|ee′i (15) glecting that term, Eq. (16) can be solved
ω ω ω
′ ′
+ 2 hϕλ|e′e′i(cid:17)c′11+(cid:18)2hϕλ|eei+ 2 hϕλ|ee′i(cid:19)c′12. c0(t)=exp i tωg(t′)dt′ . (21)
(cid:20) Z (cid:21)
t0
The analogous system of equations for m = m = 1/2
j ′j This c can be used to find c after the phase trans-
wasconsideredin detail[17]in the analysisof(n 1)p+ 0 2
t
− formation c exp[ i(∆ +ǫ (R))t ω (t)dt]c¯ is
(n+1)p resonances. The projections onto the molecular 2 ≡ − λ λ − t0 e ′ ′ 2
statearedefinedas: aee(λ)= eeϕλ ,aee′(λ)= ee′ ϕλ , performed. The excitation amplitude c¯2Ris
h | i h | i
ae′e(λ) = he′e|ϕλi, and ae′e′(λ) = he′e′|ϕλi. We assume t ω (t )
that all of the aij coefficients are real. c¯2 =i dt1 eff 1
Onecanobtainatractablesystemaftertheelimination Zt0 2 (22)
of the excitation amplitudes of all intermediate states. ei(∆λ+ǫλ(R))t1+Rtt01(ωg(t2)−ωe(t2))dt2.
This is justified by the fact that the dominant frequen- ×
cies governing their time evolutions are ∆ and ∆′ which We assume that excitation laser pulses have a Gaus-
are much larger than the relevant Rabi frequencies. On sian time profile. It has been shown that the excess
amolecularresonance,∆and∆′ areabout2π 2.2GHz, bandwidth in this experimental setup is mainly due to
·
while the peak values of ω and ω′ are about 2.3 GHz a linear frequency chirp. Because of the constant ratio
and1 GHz, respectively(for the actualexperimentalpa- ω/ω , both ω and ω have the same time dependence
′ ′
rameters). Laserintensities usedinthisexperimentwere ω,ω exp( (1+iγ)t2/σ2, where σ measures the pulse
′
almosttwoorderofmagnitudehigherthantheonesused durat∼ion and−γ is a chirp parameter related to the laser
for the (n 1)p+(n+1)p resonances so that we have bandwith Γ and duration σ as follows
−
to check for the effects of higher laser fields. We adia-
batically eliminate c11, c12, c′11 and c′12, as in [17], but π2Γ2σ2 2ln2
γ = − . (23)
this time keeping power-dependent terms. The result is r 2ln2
6
Theprobabilitytoexcite ϕ (R) fromtheinitialstate where ρ is the sample density.
λ
5s,m 5s,m is P (λ)|= c¯ (it )2. According
1/2 1/2 1 2 Molecularpotentialscanbeveryclosetoeachotherfor
| i| i | → ∞ |
to Eqs. (19-20), ω and ω do not depend on the chirp.
g e some R, as shown in Figs. 1-3. For such R atom pairs
Ifω andω inEq. (22)canbeignored,asimpleformula
g e are excited into superpositions of molecular states. Al-
can be derived [17].
toughtheextensionofthesystem(10-15)toincludemore
molecular states ϕ is simple, the technical difficulties
λ
| i
|c¯2|2 = β2(2λ)πII22 πτln3Γ222−[∆λ+ǫλ(R)]2/2π2Γ2, (24)
sat
where Isat is the saturation intensity for ideal unchirped 1 (a)
light and isolated np atoms and τ is a Gaussianpulse
3/2
duration (FWHM). The approximation applied to get
this formula is equivalent to neglecting the ω -term in 0.8
e
Eq. (17) and both terms in the right-hand side of Eq.
2
(in1d6)ep(ein.ed.enutsipnagrtc0of=ωe1ff).deIfinneEdqv.ia(2ω4e)ff,(βt)(λ=)βis(λa)ωt2im(te).- ω)|eff0.6
If all a-coefficients are equal to zero, according to Eq. (
F
(18), the effective two-photon Rabi frequency is also | 0.4
zero. These coefficients measure different p-characters
in ϕ (R) . To evaluate them, we have to express the
λ
| i 0.2
doubly-excitedstates ee , ee ,and ee (definedinthe
′ ′ ′
| i | i | i
space-fixed frame) in the molecule-fixed reference frame,
whereall ϕ (R) arenaturallydefined[17]. Apparently,
λ 0
| i
all angular dependence related to different orientations -1 0 1
∆ /Γ
of the molecular axis is contained in these a-coefficients,
2ph mol
and the observable quantities have to be averaged over
all spatial orientations of the internuclear axis.
For the case m = m = 1/2, the ground and 0.08 (b)
j ′j ±
doubly-excitedstatesarerespectively gg = g,m g,m
| i | i| i n
and ee = e,m e,m . Intermediate states ge , ge
′ o
| i | i| i | i | i
a1nd di,omubljy,-mexc+itedj,mstatie,m|ee′.i Thhaveecotheffiecfioernmts a|ijiar=e acti0.06
√2{| i| i | i| i} ij r
defined as before. After the elimination of excitation f
n
amplitudes of all intermediate states, one gets again the o0.04
system (16-17) with different ω (t) and ω (t) ti
eff e a
t
ω2 ω2 ωω ∆+∆ ci
ωeff = aee(λ)+ ′ ae′e′(λ)+ ′ ′aee′(λ), (25) E0.02
∆ ∆ √2 ∆∆
′ ′
ω = |√2ωaee+ω′aee′|2 + |√2ω′ae′e′ +ωaee′|2. (26) 0
e 4∆ 4∆ -2.6 -2.4 -2.2 -2 -1.8
′
Detuning from 70p (GHz)
The solutions for these initial states are also given by 3/2
Eq. (22), or for low laser intensity by Eq. (24). These
excitation probabilities must also be averaged over all FIG. 4: (a) Fourier transforms of ω2ph(t) of a chirped Gaus-
sianpulse(dashedline)anditscounterpartsquarepulse(solid
spatial orientations of the internuclear axis. In (24) this
line). The pulseareas and Γ2ph (FWHM) of ω2ph(t)for both
is applied to β(λ) only.
pulses are matched to minimize the difference in the excita-
For an unpolarized sample of ultracold atoms, all ini- tionprobabilities. Thetailofthesquarepulsevanishesmuch
tialstatesareequallyprobable,sothatexcitationproba- slower so it is truncated at ∆2ph/Γ2ph = 1.6, where ∆2ph
bilitieshavetobeaveragedoverallinitialdiatomicstates is the two-photon detuning. The remaining small difference
andthenallcontributionstoagiven ϕ (R) aresummed betweenthepulsesisexpectedtodiminishfurtheraftersum-
λ
| i
up. This is done for many ϕ to include the possibility ming over all atom pairs in (35). (b) Comparison between
λ
of exciting different molecu|laripotentials. The result is calculatedmolecularsignals usingEq. (24)(dashedline)and
Eq. (30)(solidline). Thelatterisbasedontheexactsolution
the averaged excitation P (R) of an atom pair. The ex-
2
for a square pulse assuming that, in general, superpositions
citation probability per atom is the sum of all excitation
ofmolecularstatesareexcited. Thedifferencesaresomewhat
probabilities of atom pairs that include a given atom,
larger just above the molecular resonance, where there are
many very close molecular potentials, as shown in Figs. 1-3.
Pexc =4π ∞dRR2ρP2(R), (27) The assumed laser bandwidth in this plot is 200 MHz.
Z
0
7
ofsolving it arenotnegligible. Notonly do we havefour calculation we use a modification of Eq. (29) to account
parameters to vary (R, I, laser frequency and the ori- for the difference in excitation probabilities for large de-
entation of the molecular axis), but we also have many tunings. Excitation probabilities for a square pulse do
molecular potentials and several symmetry cases. Also, not vanish sufficiently fast. This happens because the
for a given optical frequency, different molecular states tails of the Fourier spectra of these two types of pulses
are excited at different distances R. This is especially are very different at large detunings. This is illustrated
trueatshortRforwhichthepotentialsvarysignificantly. in Fig. 4(a). We overcomethis by truncating the contri-
Instead of doing this, we choose a different approach to butions from large detunings
account for possible superpositions of molecular states.
This approach is in many ways as complete as the full ψe(t;R) = Θ(εi(R) 2∆ ηΓmol)
| i | − |−
numericalcalculationthatincludesthefullsetofdoubly- Xi (30)
excited states and at the same time is no more difficult φi(R)gg e−iεi(R)t φi(R) ,
×h | i | i
thanthecalculationofmolecularpotentialsitself. AsEq.
where Θ is the Heaviside function and η is the cut-off
(24) suggests, only a few parameters related to the exci-
parameter (our choice η = 1.6 is justified by Fig. 4(a)).
tation laser, such as the bandwidth and pulse duration,
We note that the radial dependence, shown in Fig. 5, is
are important. The details of the laser pulse cannot be
obtained using the last equation. One can find the same
of fundamental importance, so we can substitute for the
dependence using the simpler formula (22). Both ways
actualchirpedGaussianpulseasquarepulsewiththepa-
give essentially the same radial dependence. In the last
rameterschosentogiveprobabilitiesconsistentwith(24).
formulaallε (R)areexpressedwithrespecttotheenergy
Wehavetomatchthepulseareaandthewidth(actually i
of asymptotic np +np states.
FWHM) of the Fourier spectrum corresponding to the 3/2 3/2
two-photon Rabi frequency ω . Therefore, Γ and τ are
eff
not our direct concern, but rather Γ and T , which
2ph 2ph
C. n-scaling
characterize ω (t). For a Gaussian pulse Γ = √2Γ.
eff 2ph
This Γ would also be the FWHM of a square pulse if
2ph
Now we estimate the major contribution to the n-
its duration were chosen to be T = 2.783/πΓ . The
sq 2ph
scaling of the (n 1)d+ns resonance signal. For this
single-photon Rabi frequency ω of the square pulse is
sq −
purpose we use a rather simplified description of these
chosen to provide equal pulse areas of ω (t) of the ac-
eff
states and the resonances. First we ignore fine struc-
tual pulse and its substitute. A great adventage of this
ture. As Fig. 5 suggests, the major contribution to the
approach is that now we can easily write and calculate
molecularsignaldoesnotcomefromthe regionofstrong
the exact excitation probabilities of all molecular states.
ℓ-mixing,butratherfromthelong-rangeregionwithcon-
The totalHamiltonianH of the system consists ofthe
tot
siderably weaker mixing. In this estimate we completely
long-range interaction part U(R), given by Eq. (1) and
the optical field (~=1). neglectthe contributionof strongℓ-mixingat shortR to
the molecular signal. The wave function of (n 1)d+ns
−
2 states can be expanded as follows
Htot(R)=U(R)+ ∆σeie+∆′σei′e′ (n 1)dns;R =(n 1)dns(0) +α (R)npnp(0)
Xi=1(cid:2) (cid:3) | − i | − i pp | i (31)
(28)
+Xi=21(cid:20)ω2sqσeig + ω2s′qσei′g+h.c.(cid:21). awnhderend|nϕrse(s0;)R,i n+ipsn|pϕt(r0he)es;Rarriees,idausyempotfotitchestaetxepsan(snion
′
| i | i | −
TorepresentH weusethebasisofU(R)completedby 1)dns;R and npnp;R . In general, there are
the intermediatteotstates and the ground diatomic states. more tha→n o∞nei npn|p(0) stat→e∞inithe last expansion but
| i
The matrix elements of H , corresponding to these the scaling law, in this approximation, does not depend
tot
added basis states, are essentially given by Eqs. (10- of their number so keeping only one term is sufficient
15). The only modification is to replace ϕλ by a su- for our purpose. We want to find the function αpp(R)
perposition of different ϕ . The matrix o|f Hi is only because the two-photon Rabi frequency is directly pro-
λ tot
slightly bigger than the|maitrix of U(R), so solving the portionaltoit. TheHamiltonianisstillgivenbyEq. (1).
eigenproblemofHtot doesnotimposeadditionaldifficul- In the first approximation αpp(R) is
ties. If ε (R) are the eigenvalues related to the eigenvec-
i npnp(0) V (R)ndns(0)
tors |φi(R)i of Htot(R), then the solution of the time- αpp(R)= h E|dsRyd Ep|p ′ i, (32)
dependent Schr¨odinger equation i∂ ψ /∂t=H ψ is 0 0
tot −
| i | i
where Eds Epp is the asymptotic energy spacing of
0 0
ψ(t;R) = φi(R)gg e−iεi(R)t φi(R) . (29) diatomic(n −1)d+nsandnp+nplevels. Thelastformula
| i h | i | i −
Xi is valid in the region of weak ℓ-mixing for (n 1)d+ns
−
states. We conclude that
Unlike Eqs. (22,24), the last formula does not give un-
physicalprobabilitiesfor largelaserpower. Inthe actual α (R) n7/R3, (33)
pp
∼
8
findanestimateforit. Eventhoughthe(n 2)p+(n+2)p
−
Total states are very close to (n 1)d+ ns states, they are
0.3 −
very weakly coupled to them so they basically have no
5-110 Bohr) 0.02.52 0g+ i(wnnhfl−eur1ee)ndCc+e6noinsstsththaeitseCess,ct|oiUmedffias(tcRei.e0nI)tg|nsfhoororiutnlhgdetb(henee−fiq1nu)eadlst+torun|Ccst6us|t/raeRtoe06f.,
dR (x 0.15 STihneceesCti6matne11fo,rwtehfienloawllyerfilnimdit R0 is R0−3 ∼ pΓ/|C6|.
/ Pexc 0.1 1 ∼ P n8.5. (36)
d u exc ∼
0.05
0-
u
III. RESULTS AND DISCUSSION
0
2 4 6 8 10 12 14
Separation R (x 105 a. u.)
A. Numerical evaluation of excitation probabilities
FIG. 5: Pair excitation probability as a function of sepa-
ration R. We show contributions from all states associated Wehaveusedfourdifferentwaystoevaluateexcitation
with the three most relevant asymptotes. The total depen- probabilities in order to verify that the approximations
dence includes twice the contribution of 1u since this state are applicable for all conditions under which they are
is two-fold degenerate. According to Eq. (27), dPexc/dR = used. These conditions are different for different asymp-
4πρR2P2(R),where P2(R) is the excitation probability of an totes. One way is to use the method [17] (based on
atompair. NotethattheweightedfactorR2isincludedinthe
Eq. (24)) to get the lineshape of (n 1)d + ns reso-
shown dependence. This figure shows that pairs at shorter −
nances. However, it is assumed in this method that the
distances are difficult to excite, even though ℓ- mixing and
their hω2 i is much greater there (Figs. 1-3) because they effective two-photon Rabi frequency is sufficiently small
eff
are verydetunedfrom theresonance. Themain contribution and the molecular potentials are well separated. There-
comes from pairs at separations R≥40000 a0. fore, for any pair of atoms at a certain separation R,
only one doubly-excited molecular state is involved in
the excitation (not necessarily the same one for all R).
sinceonlythedipole-dipolepartofVRyd(R)couplesthose However,thetwo-photonRabifrequencyisalmostanor-
asymptotic states, and Eds0 Epp0 n−3. der of magnitude higher in this case so it is not obvious
− ∼
We proceed using the results of the previous sections, thatsomepower-dependenttermscanalwaysbeignored.
but ignoring many details which are not of great impor- Such terms are included in Eq. (22). We find that, for
tance for n-scaling. The n-scaling is well defined only if most potentials, the formulae (22)-(24) give overall very
thereisnosaturationofexcitationsothatthetwo-photon similar lineshapes, with the position of the resonance
absorption probability P2 per pair is slightly shifted, but the shape and amplitude are well
preserved. However, it turns out that the approximate
P ω 2 . (34)
2 eff Eqs. (24)and(22),fortheexperimentalparameters,can-
∼| |
not be used for all laser frequencies and all asymptotes.
We have ω ω2α (R)/∆, where ω is the single-
eff pp
∼ By varying the laser frequency, we actually vary the re-
atomRabifrequencydefinedintheprevioussectionsand
gion of internuclear separations R for which atom pairs
∆ (Eds Epp )/2isthedetuningfromtheatomicnp
0 0
≈ − areonresonance. The two-photonRabifrequencyω is
resonance. Note that ω2/∆ is in the first approximation eff
R-dependentand,iftheℓ-mixingistoolarge,theapprox-
n-independentbecause the singlephotonRabifrequency
imations may not be valid. To check if such parameters
ωscalesasthedipolematrixelementn 3/2and∆ n 3.
− −
∼ significantlyaffectthefinalprobabilities,wehavenumer-
To get the excitation probabilityP we use Eq. (34)
exc
ically solved the system (10)-(15).
P ω4 ∞dRR2 α (R)2 n14/R3. (35) Itissolvedformanydifferentinternuclearseparations,
exc ∼ ∆2 Z | pp | ∼ 0 spatial orientations of the molecular axis, laser intensi-
R0 tiesandlaserfrequencies. Thiscalculationisrepeatedfor
The lowerlimit R is set as follows. We assume that the thestates ϕ whichhavedominantcontributionstothe
0 λ
| i
laserfrequencycorrespondstothetwo-photonresonance, resonance lineshape. They coincide with the asymptotic
whose position coincides, to a very good approximation, 69d+70s,68d+71sand68p +72p states. Inthese
1/2 1/2
with the asymptotic energy of the (n 1)d+ns state. numerical calculations, only one molecular state ϕ is
λ
− | i
A pair of atoms will be out of two-photon resonance if consideredasafinaldoubly-excitedstateofanatompair.
its interaction energy U (R ) is greater than Γ Γ, As explained previously, at some R, one should consider
ds 0 2ph
∼
therefore U (R ) Γ. Because the laser bandwidth superpositions of molecular states ϕ . This is taken
ds 0 λ
| | ∼ | i
is considerably narrow (close to the Fourier transform into account in Eqs. (29) and (30). Even though we do
limit), U (R ) is not very large, and assuming weak ℓ- not consider the actual pulse shape in this case, the pa-
ds 0
mixing, we can use second order perturbation theory to rametersofthe substituted pulse arechosento minimize
9
any quantitative difference between the pulses. Since we 4
use exact solutions of (28), this approach should give a z) 3.5
rather fair description of the resonance phenomenon. GH
Itturnsoutthatallthesedifferentmethodsleadtothe 3 ( 3
same physical results, although, for some particular pa- *60)
rameters,theymaybesignificantlydifferent. Mostofthe *x (n/ 2.5
differencesvanishaftersummingoverallpotentialcurves 3/2 2
and all atom pairs. The remaining variations are mainly np
duetotheslightdifferenceintheresonancepositionsand m 1.5
o
lineshapes obtained using different methods. The real g fr
n 1
physical parameters, such as the linewidth, signal size, uni (a)
n-scaling or even the resonance position, are practically et 0.5
D
unchanged, as shown in Fig. 4. Here we present the re-
sult based on Eqs. (29)-(30), the most complete method 0
we use, and the results from the simplest method based
on Eq. (24). This gives an estimate of the variations u.)1.5
between these methods. Our calculation shows that the b.
ar
pofrombaagbniliittuiedseolfestshteh0en−u tahseymcopnttortibesutaiorenaobfotuhteoontheeorrtdweor *8.50) ( 1
symmetries. Interestingly, for (n 1)p+(n +1)p res- 6
onances, the contribution of the 0−+g symmetry was in- *x (n/ Weighted Average
significant [17]. e
z
si
al 0.5
n
g
Si (b)
B. Comparison with theoretical lineshapes
0
In Figs. 1(a)-3(a), we show three sets of molecular 60 70 80 90
n
potentials corresponding to the three symmetries con-
sidered. At short distances these potentials have very
FIG.6: (a)Scalingoftheposition ofmolecular(n−1)d+ns
complicated shapes due to multiple avoided crossings.
resonances with respect to the atomic np resonance. The
3/2
At these avoided crossings the ℓ-mixing is the strongest. position follows the characteristic n−3 scaling. (b) Scaling of
The np+np components of various nearby potentials are the molecular signal. Here we test the expected n8.5 scaling
the most important for the molecular resonance. The (36). This plot shows a reasonable agreement with the n8.5
relevant physical quantity which depends on the frac- scaling.
tion of np+np states is the two-photon Rabi frequency
ω , defined by Eq. (18). Asymptotes (n 1)d + ns,
eff
−
(n 2)d+(n+1)s and (n 2)p +(n+2)p give the overallvery similar lineshapes, and the difference is only
1/2 1/2
− −
essential contributions to the resonance. In parts (b) of in the details. The relative difference between them is
Figs. 1-3 we present the magnitude of the radial depen- somewhat larger just above the resonance, where there
dence ω for the three asymptotes and all their states. are many potentials very close to each other. For the
eff
For each of these states, the radial dependence is ob- positionof the molecularresonance,the method method
tained after averaging ω2 over different orientations of based on Eqs. (29-30) gives 2.18 GHz from the atomic
eff −
themolecularaxisandinitialstates. Eventhoughω for 70p resonance,which is in agreementwith the experi-
eff 3/2
atom pairs at short distances is larger due to stronger ℓ- mentalpositionof2.21(3)GHz. Itappearsthattheposi-
mixing,suchpairsaredifficulttoexcitebecausetheyare tionoftheresonanceobtainedusingthesimplestmethod
very detuned from the molecular resonance. The actual isclosertotheexperimentalvalue. However,thisislikely
pair excitation probability as a function of R, on exact just a coincidence because, in that calculation ∆ and ∆
′
molecular resonance, is shown in Fig. 5. The weighting were replaced by their averagevalue, so that fine details
factor R2 is included in the presented dependence. Sur- of the atomic level positions were not included.
prisingly,thereisasignificantcontributiontothe molec- We have tested the n-scaling of the molecular reso-
ular signal from the pairs at larger distances. nance, both numerically and experimentally. As men-
We illustrate different methods used to calculate the tioned, this scaling law makes sense only if, for a given
resonance lineshape in Fig. 4(b). We present the sim- laserintensity,thetwo-photontransitionisnotsaturated
plest method, given by Eq. (24), and the method based for all values of n for which the scaling law is used. The
onthe exactsolutionfor squarepulses (29-30). Only the calculated ratio (using Eq. (24)) of the signals for all
latter method allows superpositions of molecular states n = 70 and n = 60 is 3.915, while (70 /60 )8.5 = 3.920.
∗ ∗
to be excited. This possibility is relevant if potentials Also the same ratio for n = 90 and n = 70 is 9.274,
are very close to each other. These two methods give while (90 /70 )8.5 = 9.115. The experimental depen-
∗ ∗
10
dence for the n-scaling of the molecular signal is shown that molecular resonances at the average atomic nd and
in Fig. 6(b). The agreement is fairly good and the de- 45
viation could be explained by variations of experimen-
tal parameters between these three experimental scans. u.)40 210500 MMHHzz
The n-scaling of the resonance position is in an excel- b. 120 MHz
lent agreement with the expected scaling law, as shown ar35
in Fig. 6(a). al (
In Fig. 7(a) we present calculated lineshapes for sev- gn30
Si
elirnaelwliadstehrobfatnhdewriedstohnsa,nacseiisndsiigcnatifiedcanintlythlearggrearpthh.anTthhee cal 25
laser bandwidth and is primarily determined by the de- oreti
tails of the molecular potentials and ℓ-mixing. The con- e20
h
voluted lineshapes for a laser bandwidth of 200 MHz is T
shown in Fig. 7(b). We get the best agreement with the 15
experiment for this bandwidth. The actual laser band- 40
width was probably smaller than this one. For the ac-
tual experimentalconditions,this theory cannotbe used
35
to fit the portion of the red tail of the spectrum closer
to the atomic resonance because the excitation fractions
are much larger. The presence of excited atoms could b. u.)30
modify the pair excitation probability so that Eq. (27) ar
is not applicable. Also, the simple pair excitation model al (25
cannotbeusedtoexplaintheexcitationprocessatlarger n
g
excitationfractions. Atsuchfractions,ℓ-mixingcouldbe Si
20
more efficient because close molecular potentials could
additionally mix due to the interactions with nearby ex-
cited atoms. This could explain why the experimental 15
red tail, for the experimental conditions, is just a simple
-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
monotonic function even though there are many poten-
1/2 of Offset from 69d+70s resonance
tialcurvesandavoided-crossingsbetweenthenp+npand
(n 1)d+ns asymptotic levels.
FIG. 7: (a) Calculated molecular signal for different laser
−
We compare experimental and theoretical signal sizes bandwidths of the excitation laser, as indicated in the plot.
assuming a typical 5 1010 cm−3 density. The laser in- The linewidth of the resonance is dominantly determined by
×
tensity is typically in the range of 400-500 MW cm−2. the details of long-range interactions and it is significantly
Experimental signals are about 300 ions per shot. The larger than the laser bandwidth. (b) Comparison of the the-
details of the experimental setup are given in [16]. From oretical signals shownin Fig. 4(b)andtheexperimentalone.
the scaling factor introduced in order to compare the This is the best fit of the experimental data, although the
actualbandwidthwaslikelysmaller thantheoneweusedfor
theoreticalandexperimentallineshapes,wefindthatthe
thecomparison.
calculated number of atoms is about 6-7 times higher
than the experimental value. This is probably accept-
able considering the number of factors which influence
ns energies are expected to occur. The calculated prop-
this estimate. Besides uncertainties in the experimen-
erties of these resonances, such as position, linewidth,
tal parameters, this scaling factor may also suggest that
n-scaling and signal size are reasonably close to the ex-
there was some excitation blockade [6, 7].
perimentalobservation,butacompleteunderstandingof
the spectral features of Rydberg excitation requires im-
proved theoretical models.
IV. CONCLUSION
We have presented long-range doubly-excited molecu-
Acknowledgments
lar potentials of the 0+, 0 and 1 symmetries. These
g −u u
potentials are important in describing the effects of in-
This work was supported in part by the National Sci-
teractionsinsingle-photonexcitationtohighnpRydberg
ence Foundation.
states. We have illustrated the ℓ-mixing induced by in-
teractions over a broad range of internuclear distances.
Several methods to evaluate the spectrum of the excita-
tionofinteractingRydbergpairsofatomswerepresented References
and they all gave similar results. The analysis showed