Table Of ContentMacrodimers: ultralong range Rydberg molecules
∗
Christophe Boisseau, Ionel Simbotin, and Robin Cˆot´e
Physics Department, University of Connecticut, 2152 Hillside Rd., Storrs, Connecticut 06269-3046
(Dated: February 2, 2008)
Westudylongrangeinteractionsbetween twoRydbergatomsandpredict theexistenceof ultra-
long range Rydberg dimers with equilibrium distances of many thousand Bohr radii. We calculate
2 thedispersion coefficients C5, C6 and C8 for two rubidiumatoms in thesame excited level np, and
0 find that they scale like n8, n11 and n15, respectively. We show that for certain molecular symme-
0 tries, these coefficients lead to long range potential wells that can support molecular bound levels.
2 Suchmacrodimerswouldbeverysensitivetotheirenvironment,andcouldprobeweakinteractions.
Wesuggest experimentsto detect these macrodimers.
n
a
J PACSnumbers: 34.20.Cf,32.80.Rm,32.80.Pj
0
1
Newtechniquesusedforcoolingandtrappingofatoms
] [1] and molecules [2], and which led to the realization of 0.00
m-ph aautpltoprmlaiceicodldBtooRseeyx–dpEbeiernrisgmteeainntotsmcowsnid[t5he]n.usalTttrhiaoecnoel[xd3a],gpglhaeasrvmaetaesadl[sp4or],obpaeenernd- −1) (cm)R −0.02 1Π −3Π3Σ−g−1Σ−u X 100
o ties of Rydberg atoms provide a fertile ground for new (V g u
−0.04
t physics. For example, transport properties of ultracold 1Σ+ −3Σ+ X 10−8
a g u
gases doped with ions were recently explored and ex-
s. tended to cold Rydberg samples [6], while entangled 102 103 104 105
c Distance R (a.u.)
i states relevant for quantum computing can also be pro-
s
duced with ultracold Rydberg atoms [7]. Finally, the
y FIG. 1: Comparison of the potential curves for the three
h creation of “trilobite” Rydberg molecules was proposed pairs of states that sustain a well for Rb(n=20). Inside the
p [8], where one atom of the dimer remains in its ground shaded area we haveR<RLR, and Eq. (1) is not adequate.
[ state while the second one is excited to a Rydberg state.
1 In this paper, we explore the interactions betweentwo
v Rydberg atoms. We show thatlong rangewells support- 3∆u,1Πg and3Πu,3Πg and1Πu,3Σ−g and1Σ−u,andtwo
2 ing several bound levels exist for certain molecular sym- pairs of 1Σ+ and 3Σ+ [12]. Note that the degeneracy of
g u
2 metries. We explore the sensitivity of these wells to the anypairofmolecularstatesisliftedbytheexponentially
0
particularasymptoticformofthepotentiallongrangeex- decaying exchange interactions, which can be neglected
1
0 pansionandshow thattheir existence is robust. We also at large enough R.
2 estimate the effect of retardation as well as the validity The numerical values of C , C and C [13] were cal-
5 6 8
0 of the Born–Oppenheimer approximation. We give nu- culated using the expressions of Marinescu [12]. The
/
s mericalexamplesfor the caseofrubidium (Rb), anddis- sums over the electronic states were evaluated directly,
c cuss experimental schemes to detect these macrodimers. and using the n-scalingof the dipole (∝n2), quadrupole
i
s Thesemoleculescouldleadtomeasurementsofveryweak (∝ n4), and octopole (∝ n6) matrix elements and the
y interactions, such as vacuum fluctuations, and provide a energydifferences (∝n−3)involved,we obtainedthe fol-
h
unique tool to study quenching in ultracold collisions. lowing scaling laws: C ∝ n8, C ∝ n11, and C ∝ n15.
p 5 6 8
: We consider two atoms each excited by one photon Themagnitudeandsignofthedispersioncoefficientsde-
v
from their ground state into the same Rydberg state np pend on the molecular symmetry considered, and it is
i
X [9], where n is the principal quantum number. For Rb, possible to obtain a long range potential well with an
r this correspondsto the 5s→np transition[10]. At large attractive long range R−5 contribution and a repulsive
a separation R, the potential energy between two atoms shorter range R−6 or R−8 contribution (see Fig. 1). For
canbeexpandedinpowersof1/R[11]. Fortwoidentical the np−np asymptote, we found that three pairs of de-
atoms in the same np state, it takes the form [12] generatemolecularstatesgivelongrangepotentialwells:
1Π -3Π , 3Σ−-1Σ−, and one of 1Σ+-3Σ+.
g u g u g u
C C C
5 6 8 For the system to be adequately described by Eq. (1),
V(R)=− − − , (1)
R5 R6 R8 the exchange energy must be negligible. To estimate
the region of validity of Eq. (1), we use the Le Roy
where the dispersion coefficients C , C , and C depend
5 6 8
radius R [14] as measure of the electron wavefunc-
on n. For the np − np asymptote of the homonuclear LR
tion overlap between the two atoms: it is given by
dimers,wehaveintotalsixpairsofdegeneratemolecular
states with identical coefficients for each pair: 1∆g and RLR = 2(cid:16)hr2i1A/2+hr2i1B/2(cid:17), where hr2i1A/,2B is the rms
2
−1pth D (cm)e 111000036 n=10 n=20 251530 401Σg+ −60 3Σu+ 95 (a) Coefficient (a.u.) 10111−0001012 − C8 /n15 C5 /n8 − C6 /n11 (a)
de n=10 20 1
Potential 1100−−63 3Σg− − 1Σ1u5− 20 3300 40 Πg6 0− 3Πu 95 −3nR (a.u.)e 000...223050 n−3Re n7De (b) 011...505 8−1cm)D (10e
95 95 7n
us (a.u.)RLR 104 n=20 1 Σ +g 2−5 3 Σ 3+u0 2040 30 4600 3060 31ΣΠg−g −−4 30 1ΠΣuu− F(bI)GR. 3e:(leSftcas2lc0inalge)ofan(Padri4)n0DcCipe5a,l( qCruig6ah,nt6tCu0ms8c nafoulemr)bt,ehare8 sn01aΠfgu-n3cΠtu1io0np0aoirf,na.nd
di 20
y ra 103 15 15
Ro n=10 (b) to zero, we find that the equilibrium distance scales as
Le n=10 Re ∼ 0.3 n3 a0, in good agreement with the numerical
102 102 103 104 105 values shown in Fig. 3(b). The scaling of De is 108n−7
cm−1, as expected (see Fig. 3(c)). Note that other pairs
Equilibrium distance R (a.u.)
e may have different R and D scaling, depending on the
e e
relativemagnitudeofthe dispersionterms. Thewellsfor
FIG. 2: In (a), comparison of D as a function of R for the
e e the1Π -3Π pair,althoughshallow,supportmanyvibra-
threepairssupportinglongrangewells, forvarious n. In(b), g u
Re iscompared toRLR for thesamestates. Theshadedarea tionalboundlevels. InTableI,welistthetwolowestand
is defined byRe <RLR. highestlevelsfoundforn=20,40and70[16],whichsup-
port 143, 125 and 107 bound levels, respectively. While
the wells for high n are much shallower than those with
position of the electron of atom A (B) [15]. If R<R , smallern(e.g.,seen=20and70inTableI),theirlarger
LR
exchangeandcharge-overlapinteractionsbecome impor- extension leads to denser energy levels, and hence they
tant and Eq. (1) is not adequate to describe the system. also support a large number of levels.
In Fig. 1, we illustrate the wells for the three pairs When using the expression for V(R), the Born–
of degenerate states 1Πg-3Πu, 3Σ−g-1Σ−u, and 1Σ+g-3Σ+u Oppenheimer approximation is assumed valid. To ver-
of Rb with n = 20. The Le Roy radius RLR is equal to ify that it is the case,we comparethe vibrationalperiod
1902a0,wherea0isaBohrradius. Thewellofthepairof τvib(v) of a given bound levelv with the typical time for
states1Σ+g-3Σ+u isverydeep(potentialdepthDe =4.5× the electron motion τe = h/|En| = (4π/αc)n2a0 where
106cm−1),butislocatedatamuchshorterdistancethan E istheelectronbindingenergy. Ifτ /τ ≪1,theRy-
n e vib
the Le Roy radius (equilibrium distance Re = 240 a0): dbergelectronsofbothatomseasilyfollowthe motionof
theinteractionsforthesestatesarenotwelldescribedby the ions, and the Born–Oppenheimer curve V(R) needs
Eq.(1),asopposedtothetwootherpairs. Thewellofthe no further diabatic corrections. Taking the levels in Ta-
pair1Πg-3Πu hasadepthDe =3.74×10−2cm−1 andan bleIasexamples,forn=20,wefindτe/τvib ∼1.5×10−6
equilibrium distance Re = 2509 a0. By comparison, the forthe lowestlevelv =0,and2.2×10−11 forthehighest
well of the pair 3Σ−-1Σ− is much shallower and farther level v =142: the electrons follow adiabatically the ions
g u
away, with De = 6.88×10−5 cm−1 and Re = 4956 a0. for all vibrational levels. For n = 70, the ratio becomes
In Fig. 2, we compare De, Re and RLR for the three 6.8×10−9 for v =0 and 4.5×10−14 for v =106. Natu-
pairs for various values of n, and find the same general rally, the Born–Oppenheimer approximation gets better
behavior. For the remainder of this letter, we focus our for the higher vibrational levels, since they are more ex-
attention on the deeper wells described by Eq. (1), i.e., tended and therefore take more time to make a full os-
the 1Πg-3Πu pair, since the much shallower wells of the cillation. However, the spontaneous decay of one of the
3Σ−-1Σ− pair may prove more difficult to detect. excitedRydbergatomswilllimitthelifetimeoftheselong
g u
In Fig. 3, we illustrate the scaling of the dispersion range molecules and prevent the existence of the upper
coefficients, equilibrium distance, and well depth of the lying vibrational bound levels. Although the lifetime of
1Π -3Π pair as a function of n. In atomic units, the the excited atoms is long, scaling as τ ∼ τ n3, where
g u at 0
coefficients are given approximately by C ∼ 3 n8, τ ∼ 1.4 ns [17], it is much shorter than the vibrational
5 0
C ∼ −0.7 n11, and C ∼ −50 n15 (see Fig. 3(a)). Ne- period for high v. E.g., for n = 20, we find τ ∼ 11.2
6 8 at
glectingC inEq.(1),andsettingthederivativeofV(R) µs and τ = 0.083 µs and 5.58 ms for v = 0 and 142,
8 vib
3
TABLE I: Sample of vibrational bound levels for the 1Πg- 0 0.5 Eq. (1)
3D2Π0e1u=4p1a3ai.r07.,4TD0he×e=1t0o4−p.02c6oc0rm×re−1s01p−,o4tnhdcesmmt−oi1dn,da=lned2tt0oh,newbi=toht4tRo0m,ew=tiot2hn50R=9e7a0=0,, −1)m −5 0 with C840p−40p 40p−40p
with Re =103380a0, De =1.042×10−5 cm−1. −30c −10 −0.5 Eq. (1)
1 with −1 C 36s−47p
v (cmE−b1) (cEm(−v)1) (Ra01) (aR02) τ(vsib) (() VR −15 −1104 2 8 105
0 −3.70[−2] 4.01[−4] 2454 2571 8.3[−8] −20 42s−36f
1 −3.62[−2] 1.19[−3] 2417 2623 8.4[−8]
. 103 104 105
.
. Distance R (a.u.)
141 −3.46[−8] 3.74[−2] 2142 49859 1.2[−3]
142 −3.89[−9] 3.74[−2] 2142 77220 5.6[−3] FIG. 4: Nearest asymptotes to 40p−40p for the 1Πg-3Πu
0 −4.01[−4] 5.04[−6] 19651 20701 6.6[−6] symmetries. Inset: convergence of Eq.(1) for n = 40. The
shaded area gives an estimate of thetruncation error.
1 −3.91[−4] 1.50[−5] 19331 21163 6.7[−6]
.
.
.
123 −3.27[−10] 4.06[−4] 17111 431262 0.1 regionoccur [19]. Fig.4 illustratesthe wellfor40p−40p
124 −1.91[−11] 4.06[−4] 17111 762169 0.8 and the two nearest curves correlated to 36s−47p and
42s − 36f, for the 1Π -3Π symmetries. The separa-
0 −1.03[−5] 1.52[−7] 100574 106629 2.2[−4] g u
tion between them is much larger than D and no cross-
1 −9.97[−6] 4.51[−7] 98762 109334 2.3[−4] e
ings occur. Although the various asymptotes become
.
.
. denser as n increases, the potentials become very shal-
105 −1.02[−11] 1.04[−5] 87306 2211658 3.1 low(De ∝n−7),andgenerallynocrossingisfound. Note
106 −3.57[−13] 1.04[−5] 87306 4337482 32.8 that avoided crossings from a lower asymptote repulsive
curve and a higher asymptote attractive curve (with the
same symmetry) could, in fact, produce deep wells.
For these extremely extended states, retardation ef-
respectively, and for n=70,τ ∼480µs and τ =220
at vib fects may become important. Again, to estimate the
µs and 32.8 s for v = 0 and 106, respectively. In ef-
importance of the photon time-of-flight, we compare the
fect, the upper vibrational levels can be considered as
electronictimeτ withthetimeittakesaphotontocover
e
quasi-continuum states since only a fraction of an entire
thedistancebetweenthetwocenters: τ ∼R/c. Forthe
ph
oscillation will take place before de-excitation.
highest levels, the relevant distance is the outer turning
Another effect limiting the existence of Rydberg long pointR ,butforthedeeperlevels,agoodestimateisob-
2
range molecules is auto-ionization. When two excited tained by using the equilibrium distance R . Using the
e
atoms interact, one atom can decay to a lower excited scaling R ∼ 0.3 n3, we have τ /τ ∼ (0.3α/4π)n ≪ 1
e ph e
state while the second atom is ionized, the free elec- for n < 100. For the highest vibrational levels v in Ta-
tron picking up most of the kinetic energy. However, ble I, we get τ /τ ∼ 0.06,0.07, and 0.11, for R of
ph e 2
if the separation between the Rydberg atoms is larger v = 140,141, and 142, respectively (for n = 20), and
than RLR, there is little overlapbetween their electronic τph/τe ∼0.19,0.26, and 0.51 for R2 of v =104,105,and
clouds, and one expects the auto-ionization probability 106, respectively (for n = 70). Although retardation ef-
to be small (the atoms interact via their electric dipoles fects are not important for the lower vibrational levels
at such distances). Note also that for alkali dimers in (with R slightly larger than R ), even for large n, one
2 e
general[18],andRbinparticular,avoidedcrossingswith needs to consider them for the higher vibrational levels.
the ionic curves (here correlated to Rb++Rb−) perturb Thesensitivityoftheinteractionpotentialtoretardation
the lowest asymptotes such as 5p−5p. However,for the effects couldactuallybe usedto detectthese effects with
case where the Le Roy radius condition is satisfied, i.e., high accuracy using high precision spectroscopy.
for n ≥ 20 (see Fig. 2(b)), the np−np asymptotes are Weincludedthreetermsintheasymptotic1/Rexpan-
much higher than the ionic curves, and no perturbation sion(1),butonehastoverifytheeffectofthetruncation
from avoided crossings will occur. on the well properties. The standard procedure to esti-
The density of electronic states in the Rydberg region matethe maximumerrordue tothe truncationofhigher
isverylarge,andavoidedcrossingsfrompotentialcurves termsistoincludehalfofthelastterm[20]: here,C . In
8
correlated to other asymptotes with the same symmetry Fig.4,weillustratetheeffectofsuchvariationforn=40.
could perturb the shallow wells. However, the closest The well becomes deeper, but is still located outside the
asymptotes are generally separated by several times the Le Roy radius, hence the existence of long range wells is
potential depth D , and no avoided crossing in the well robust to the truncation of the 1/R expansion.
e
4
To detect these Rydberg macrodimers,one could pho- [1] J. Weiner et al.,Rev.Mod. Phys. 71, 1 (1999).
toassociate them from the ground state (5s for Rb) to [2] A. Fioretti et al., Phys. Rev. Lett. 80, 4402 (1998);
the target molecular symmetry, using appropriate pho- T. Takekoshi, B. M. Patterson, and R. J. Knize, Phys.
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ton wavelength and polarization. After the photoassoci-
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J. Mol. Spectrosc.195, 236 (1999); R.Cˆot´eet al.,Phys.
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Rev. Lett.74, 3581 (1995).
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theatomicasymptote. Becausethespacingofthebound M. H. Anderson et al., Science, 269, 198 (1995);
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laser width smaller than the spacing ∆ is needed: for D. G. Fried et al.,Phys. Rev.Lett. 81, 3811 (1998).
n = 20, ∆ ∼ 10−3 cm−1 ∼ 300 MHz for the deepest [4] T. C. Killian et al.,Phys. Rev.Lett. 83, 4776 (1999).
[5] W.R.Anderson,J.R.Veale,andT.F.Gallagher,Phys.
levels), and the most probable signature would be an
Rev. Lett. 80, 249 (1998); I. Mourachko et al., ibid. 80,
overall red-detuned peak. The wells are shallow, and
253 (1998).
only those deep enough could be detected. To enhance [6] R. Cˆot´e and A. Dalgarno, Phys. Rev. A 62, 012709
the sensitivity of the detection scheme, one could also (2000); R.Cˆot´e, Phys. Rev.Lett.85, 5316 (2000).
use radio-frequency to excite a bound level to a contin- [7] D. Jaksch et al., Phys. Rev. Lett. 85, 2208 (2000);
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Phys. Rev.Lett. 85, 2458 (2000).
by ramp-field ionization) would be attributed to the ex-
[9] Other cases with np+n′p, ns+n′s, or ns+n′d could
istence of a long range bound molecular state, since the
beexplored.Forthedispersion coefficients, seeM.Mari-
RF field would not be resonant with free np atoms. nescu and A. Dalgarno, Phys. Rev. A 52, 311 (1995);
Wehavecomputedlongrangeinteractionsbetweentwo M. Marinescu and H. R. Sadeghpour, Phys. Rev. A 59,
Rb atoms in the same np state, and found that shallow 390 (1999).
long range wells supporting severalvibrationallevels ex- [10] A short intense laser pulse (e.g., 10MW/cm2 during 10
ist for certain molecular symmetries. Although specific ns) photoassociates all atom pairs withtheright separa-
tion. The number of macrodimers produced will depend
calculations were performed for Rb and np−np asymp-
only on thepairs’ spatial distribution.
totes, the existence of these macrodimers is general: one
[11] A. Dalgarno and W. D. Davison, Adv. Mol. Phys. 2, 1
can expect them for various asymptotes n ℓ −n ℓ and
1 1 2 2 (1966).
for all alkali atoms. Furthermore, avoided crossing be- [12] M. Marinescu, Phys.Rev.A 56, 4764 (1997).
tweencurvesofthesamesymmetrywithdifferentasymp- [13] The dipole, quadrupole, and octupole radial matrix ele-
totes couldalsoprovidedeeplong rangemolecularwells, mentswereobtainedusingthequantumdefectofp1/2 to
in a manner similar to the long range wells observed in evaluate electronic wavefunctions. See[17].
[14] R. J. Le Roy,Can. J. Phys. 52, 246 (1974).
manyalkalidimers[21]. The macrodimersareextremely
[15] Amodifieddefinitiontakingintoaccountthespatialori-
sensitivetotheirenvironment,andassuch,theycouldbe
entation of the atomic orbitals can be found in B. Ji,
used as probes for extremely weak interactions, e.g., to C.-C. Tsai, and W. C. Stwalley, Chem. Phys. Lett. 236,
measure retardation effects and vacuum fluctuations, or 242 (1995).
any weak electromagnetic interaction. Also, due to their [16] Special attention near the dissociation limit was neces-
very small rovibrational energy splittings, macrodimers sary to obtain the bound levels. See C. Boisseau et al.,
would provide a unique tool to study quenching in ul- Euro. Phys.Lett. 12, 199 (2000).
[17] T. F. Gallagher, Rydberg Atoms, Cambridge University
tracold collisions, since they would remain trapped (as
Press, Cambridge (1994).
opposed to usual dimers [22]). Finally, the detection of
[18] S. Magnier et al.,J. Phys.B 27, 1723 (1994).
such exotic molecules in itself would be a considerable
[19] Forrareaccidentalnear-degeneracieswhereotherasymp-
achievement. totesareclosecompared toD ,oneneedstodiagonalize
e
The authors thank A. Dalgarno, V. Kharchenko, P.L. the appropriate interaction subspace.
Gould, and E. Eyler for helpful discussions. This work [20] A. Dalgarno J. T. Lewis, Proc. Phys. Soc. A 69, 57
wassupportedbytheUniversityofConnecticutResearch (1956).
[21] W. C. Stwalley and Y. H. Uang, G. Pichler, Phys.
Foundation, the Research Corporation, and the Grant
Rev. Lett. 41, 1164 (1978); H. Wang, P. L. Gould,
ITR-0082913from the National Science Foundation.
W. C. Stwalley, Z. Phys.D 35, 317 (1996).
[22] R.C. Forrey et al., Phys.Rev.A 59, 2146 (1999).
∗
Electronic address: [email protected]
