Long-range interactions of hydrogen atoms in excited states. II. Hyperfine-resolved 2S-2S system

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by  U. D. Jentschura

Long-range interactions of hydrogen atoms in excited states. II.
Hyperfine-resolved 2S–2S system
U. D. Jentschura,1 V. Debierre,1 C. M. Adhikari,1 A. Matveev,2,3 and N. Kolachevsky2,3,4
1Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409-0640, USA
2P. N. Lebedev Physics Institute, Leninsky prosp. 53, Moscow, 119991 Russia
3Max–Planck–Institut fu¨r Quantenoptik, Hans–Kopfermann-Straße 1, 85748 Garching, Germany
4Russian Quantum Center, Business-center “Ural”, 100A Novaya street, Skolkovo, Moscow, 143025 Russia
7 The interaction of two excited hydrogen atoms in metastable states constitutes a theoretically
1 interesting problem because of the quasi-degenerate 2P1/2 levels which are removed from the 2S
0 states only by the Lamb shift. The total Hamiltonian of the system is composed of the van der
2 Waals Hamiltonian, the Lamb shift and the hyperfine effects. The van der Waals shift becomes
commensuratewiththe2S–2P fine-structuresplittingonlyforcloseapproach(R<100a ,where
n 3/2 0
a istheBohrradius)andonemaythusrestrictthediscussiontothelevelswithn=2andJ =1/2
a 0
J to good approximation. Because each S or P state splits into an F = 1 triplet and an F = 0
hyperfine singlet (eight states for each atom), the Hamiltonian matrix a priori is of dimension 64.
0
A careful analysis of symmetries the problem allows one to reduce the dimensionality of the most
3
involved irreducible submatrix to 12. We determine the Hamiltonian matrices and the leading-

order van der Waals shifts for states which are degenerate under the action of the unperturbed
]
h Hamiltonian (Lamb shift plus hyperfine structure). The leading first- and second-order van der
p Waals shifts lead to interaction energies proportional to 1/R3 and 1/R6 and are evaluated within
- thehyperfinemanifolds. Whenboth atoms aremetastable 2S states, we findan interaction energy
m
of order E χ(a /R)6, where E and are the Hartree and Lamb shift energies, respectively, and
h 0 h
o χ=Eh/ 6.22 106 is their ratio. L
L≈ ×
t
a
PACSnumbers: 31.30.jh,31.30.J-,31.30.jf
.
s
c
i
s I. INTRODUCTION without taking account of the fine and hyperfine struc-
y tures), with the intricacies of the hyperfine correctionto
h
thelong-rangeinteractionoftwoatoms,whichhavebeen
p Inspired by recent opticalmeasurements of the 2S hy-
studied in Refs. [6–9]. Indeed, it had been anticipated
[ perfinesplittingusinganatomicbeam[1],wehereaimto
in Ref. [3] that a more detailed study of the combined
carryoutananalysisofthehyperfine-resolved2S–2Ssys-
1 hyperfine and van der Waals effects will be required for
v temcomposedoftwohydrogenatoms. Thispaperfollows
the2S–2Ssystemwhenamoredetailedunderstandingis
3 a previous work of ours (Ref. [2]) in which we analyzed
sought. Themainlimitationofthemethodfollowedhere
1 the long-rangeinteraction between two hydrogen atoms,
isthatwewillonlyconsiderdipole-dipoletermsinthein-
8 one of which was in the 1S ground state, and the other
8 teratomic interaction, in contrast to Refs. [3, 4]. Hence,
one inthe metastable 2S state. Here we turnto the case
0 our analysis only yields reliable results for sufficiently
wherebothatomsareinanexcitedstate. Forthatweuse
. large interatomic separation. Inspection of the higher-
1 thesimplestcaseathand,namelythatwherebothatoms
0 are in the 2S state. The 2S–2S van der Waals interac- ordermultipoletermsobtainedinRefs.[3,4]clarifiesthat
7 the dipole-dipole approximation is already largely valid
tionhasbeenanalyzedbeforeinRefs.[3,4],butwithout
1 for interatomic separations of the order of R = 20a .
any reference to the resolution of the hyperfine splitting 0
: [This is true for the 2S–2S system, upon which we focus
v [5]. The entire problem needs to be treated using de-
Xi generateperturbationtheory,because the vanderWaals here. Judging from Fig. 2 in Ref. [4], for higher prin-
cipal quantum number (n = 4), the range of relevance
Hamiltonian couples the reference 2S state to neighbor-
r of higher-order multipole terms extends further out, but
a ing quasi-degenerate 2P states. The latter are displaced
these cases are beyond the scope of the current investi-
from the former only by the Lamb shift (in the case of
gation.]
2P ) or by the fine structure (in the case of 2P ).
1/2 3/2
As was noted in Ref. [2], significant modifications of the Throughoutthisarticle,weworkinSImksAunitsand
long-range interactions between two atoms result from keep all factors of ~ and c in the formulas. In the choice
the presence of quasi-degenerate states, and the effects of the unit system for this paper, we attempt to opti-
lead to observable consequences. In a more general con- mize the accessibilityofthe presentationto twodifferent
text, one mayregardourinvestigationsasexample cases communities: the QED community in general uses the
foramoregeneralsetting,inwhichtwoexcitedatomsin- natural unit system with ~ = c = ǫ = 1, and the elec-
0
teract,while in metastablestates (with quasi-degenerate tron mass is denoted as m. The relation e2 = 4πα then
levels nearby). allows to identify the expansion in the number of quan-
The present work combines the challenges described tum electrodynamic corrections with powers of the fine-
in Ref. [3], where the 2S–2S interaction is studied (but structure constant α. This unit system is used, e.g., in
2
the investigationreportedin Ref. [10] on relativistic cor- Here, α is the fine-structure constant, m the electron
rectionstotheCasimir–Polderinteraction(withastrong mass, ~r , p~ and L~ are the position (relative to the re-
i i i
overlap with QED). In the atomic unit system, we have spective nucleus), linear momentum and orbital angular
|e|=~=m=1,and4πǫ0 =1. Thespeedoflight,inthe momentum operators for electron i; also, S~i is the spin
atomicunitsystem,isc=1/α 137.036. Thissystemof operator for electron i and I~ is the spin operator for
≈ i
unitsisespeciallyusefulfortheanalysisofpurelyatomic
proton i [both are dimensionless]. The electronic and
propertieswithoutradiativeeffects. Asthesubjectofthe
protonic g factors are g 2.002319and g 5.585695,
s p
currentstudy lies inbetween the twomentioned fields of ≃ ≃
whileµ 9.274010 10−24Am2 istheBohrmagneton
interest, we choose the SI mksA unit systemas the most and µ B ≃5.050784 ×10−27Am2 is the nuclear magne-
appropriatereferenceframeforourcalculations. Thefor- N ≃ ×
ton. The subscripts A and B refer to the relative coor-
mulas do not become unnecessarily complex, and can be
dinates within the two atoms, while R is the interatomic
evaluated with ease for any experimental application.
distance. The expression for H shifts S states rela-
We organizethis paperasfollows. The combinationof LS
tive to P states by the Lamb shift, which is given in the
the orbital and spin electron angular momenta, and the
Welton approximation [11], which is convenient within
nuclear spin, add up to give the total angular momen-
theformalismusedfortheevaluationofmatrixelements.
tum of the hydrogen atom; the conserved quantities are
(The importantpropertyof H is thatit shifts S states
discussedin Sec. II, together with the relevanttwo-atom LS
upward in relation to P states; the prefactor multiply-
productwavefunctions. In Sec.III, we proceedto inves-
ing the Dirac-δ can be adjusted to the observed Lamb
tigate the Hamiltonian matrices in the subspaces of the
shift splitting.) Indeed, for the final calculation of en-
spectrumofthetotalHamiltonianintowhichitnaturally
ergy shifts, we shall replace
decouples. Namely, the magnetic projection of the to-
tal angular momentum (summed over both atoms) com-
2S H 2S 2P H 2P
mutes with the total Hamiltonian, and this leads to ma- 1/2 LS 1/2 1/2 LS 1/2
h | | i−h | | i
trix subspaces with F = +2,1,0, 1, 2. For each one 4α α4
z − − = mc2 ln(α−2) , (3)
of these five hyperfine subspaces, we shall identify two 3π 8 →L
irreduciblesubspacesofequaldimensionality. Thisprop-
erty considerably simplifies the treatment of the prob- where = h1057.845(9)MHz is the “classic” 2S–2P1/2
L
lem. Finally, some relevant energy differences for the 2S Lamb shift [12]. The Hamiltonian H given in Eq. (1)
hyperfine splitting (with the spectator atom in specific defines the zero of the energy to be the hyperfine cen-
states, namely either 2S or 2P) are analyzed in Sec. IV. troid frequency of the 2P1/2 states. The result for HHFS
Conclusions are drawn in Sec. V. in the given form is taken from Ref. [13]. The Hamil-
tonians H and H are obtained from H by
HFS,A HFS,B HFS
specializing the coordinate ~r to be the relative coordi-
II. FORMALISM nate (electron-proton) in atoms A and B, respectively,
and correspondingly for H and H .
LS,A LS,B
A. Total Hamiltonian of the system We shall focus on the interatomic separation regime
where the van der Waals energy is commensurate with
thehyperfinesplittingandLambshiftenergies,butmuch
In orderto evaluate the 2S–2S long-rangeinteraction,
smallerthanthefinestructure(the2P –2P splitting
including hyperfine effects, one needs to diagonalize the 1/2 3/2
and likewise, the 2S–2P splitting). Hence,
Hamiltonian 3/2
E E E . (4)
H =H +H +H +H +H . (1) vdW HFS FS
LS,A LS,B HFS,A HFS,B vdW ∼ ∼L≪
Here, HLS is the Lamb shift Hamiltonian, while HHFS This is fulfilled for R > 100a0, as can be seen from
describes hyperfine effects; these Hamiltonians have to Eq.(2c)andwillbeconfirmedlater. Hence,weonlycon-
be added for atoms A and B. They are given as follows, sider 2S and 2P1/2 states. We shallneglectthe influence
of the 2P states, assuming that they are sufficiently
3/2
µ 8π displaced. Because the van der Waals interaction (2c)
HHFS = 0µBµNgsgp S~i I~iδ3(~ri) has nonvanishing diagonal elements between 2S and 2P
4π " 3 ·
i=A,B states, the interaction energy between the two 2S atoms
X
3 S~i·~ri I~i·~ri −S~i·I~i~ri2 L~i I~i canThbeezofcoormdperon1e/nRt3o.f the total angular momentum op-
+ + · (2a)
(cid:16) (cid:17)(cid:16) ~r 5(cid:17) ~r 3  erator of both atoms is
i i
| | | |
H =4α2mc2 ~ 3ln α−2 δ3(~r) , (2b) Fz =Fz,A+Fz,B =Jz,A+Jz,B +Iz,A+Iz,B
LS 3 mc i =L +L +S +S +I +I
(cid:18) (cid:19) i=A,B z,A z,B z,A z,B z,A z,B
H =α~cxAxB +yAyB(cid:0)−2z(cid:1)AXzB . (2c) =Lz,A+Lz,B+ 21σe,z,A+ 21σe,z,B
vdW R3 + 1σ + 1σ , (5)
2 p,z,A 2 p,z,B
3
where J~ = L~ + S~ is the total angular momen- states. At this stage, we remember that we discarded
tum of the electron. Let us investigate if F com- 2P states from our treatment because of their rel-
z 3/2
mutes with the total Hamiltonian H. In Eq. (5), atively large energy separation from 2S and 2P
1/2 1/2
the subscript e denotes the electron, while p de- states. Thus, we have a total of eight states per atom.
notes the proton. The following commutators vanish For the system of two atoms, we have 8 8=64 states.
×
separately, [S +S ,H ] = [S +S ,H ] = Due to the conservation of the total hyperfine quantum
z,a z,b LS z,a z,b vdW
[I +I ,H ]=[I +I ,H ]=0. Wethenturn number F = F + F , established above, the 64-
z,a z,b LS z,a z,b vdW z z,a z,b
tothenon-trivialcommutators. Forthat,itisveryuseful dimensional Hilbert space is decomposed into five sub-
to notice that the orbital angular momentum L~ of elec- spaces as
i
tronicommuteswithallsphericallysymmetricfunctions
Tofhtisheimramdeiadliaptoelsyitiyoineldopse[rLator+|~rLi| of,Hthe]sa=m0e.elWecetrcoann. Fz =Fz,a+Fz,b =±2 ⇒g =4, (8a)
z,a z,b LS F =F +F = 1 g =16, (8b)
also show that z z,a z,b ± ⇒
F =F +F =0 g =24. (8c)
z z,a z,b
⇒
[S +S ,H ]+[I +I ,H ] The most complicated case is the subspace for which
z,a z,b HFS z,a z,b HFS
F =0,inwhichcasetheHamiltonianmatrixis,apriori,
+[L +L ,H ]=0, z
z,a z,b HFS
24-dimensional. Thus, we have to generate the matrix,
i~
[L +L ,H ]=α~c [y x +x y diagonalize it and choose the eigenvalues which corre-
z,a z,b vdW R3 A B A B spondstotheunperturbed(withrespecttodipole-dipole
yAxB xAyB]=0. (6) interaction) states.
− −
Letusaddangularmomentatoobtainthesingle-atom
ThecomponentF ofthetotalangularmomentumofthe
z states of definite hyperfine quantum number. First, we
two-atom system [see Eq. (5)] thus commutes with the
addtheelectronspinwithitsorbitalangularmomentum
total Hamiltonian H. We can classify states according
to obtain the J = 1/2 states within the n = 2 manifold
to the eigenvalues of the operator F =F +F .
z z,a z,b of hydrogen. These are given as follows,
B. Addition of Momenta and Total Hyperfine ℓ=0,Jz =±21 = |±ie |ℓ=0,m=0ie =|±ie |0,0ie ,
Quantum Number
(cid:12)(cid:12) (cid:11) (9a)
In order to calculate the matrix elements of the total 1
Hamiltonian (1), we first need to identify the relevant ℓ=1,Jz =±21 = ∓"√3 |±ie |1,0ie
states of the two atoms. For each atom, we easily iden-
(cid:12) (cid:11)
tify thefollowingquantumnumberswithinthe hyperfine (cid:12) 2
manifolds: −r3 |∓ie |1,±1ie# . (9b)
2S (F =0):ℓ=0, J = 1, F =0 g =1, (7a)
1/2 2 ⇒ F
2S1/2(F =1):ℓ=0, J = 12, F =1 ⇒ gF =3, (7b) tHheereS,c|h±r¨oiediinsgtehreeiegleecntsrtoantesp(winitshtoautet,sapnind).|ℓT,mheieprdienncoipteasl
2P1/2(F =0):ℓ=1, J = 21, F =0 ⇒ gF =1, (7c) quantum is n = 2 throughout. We also remember that
2P (F =1):ℓ=1, J = 1, F =1 g =3. (7d) the J = 3/2 states are displaced by the fine structure
1/2 2 ⇒ F shift and, therefore, far away in the energy landscape
Here ℓ, J, and F are the electronic orbital angular mo- given the scale of energies considered here. With the
mentum, the total (orbital+spin) electronic angular mo- help of Clebsch–Gordan coefficients, we add the nuclear
mentum and the total (electronic+protonic) atomic an- (proton)spin toobtaintheeightstatesinthesingle-
p
|±i
gular momentum, while g = 2F +1 is the number of atomhyperfinebasis. First,wehaveforthefourS states,
F
+ +
ℓ=0,F =0,F =0 = | ip |−ie−|−ip | ie 0,0 , (10a)
| z i − √2 | ie
+ + +
ℓ=0,F =1,F =0 = | ip |−ie |−ip | ie 0,0 , (10b)
| z i √2 | ie
ℓ=0,F =1,F = 1 = 0,0 . (10c)
| z ± i |±ip |±ie | ie
4
The P states are more complicated,
1 1 1 1
ℓ=1,F =0,F =0 = + + 1, 1 + 1,0 + 1,1 + 1,0
| z i √3| ip| ie| − ie− √6| ip|−ie| ie √3|−ip|−ie| ie− √6|−ip| ie| ie
(11a)
1 1 1 1
ℓ=1,F =1,F =0 = + + 1, 1 + + 1,0 + 1,1 + 1,0
| z i − √3| ip| ie| − ie √6| ip|−ie| ie √3|−ip|−ie| ie− √6|−ip| ie| ie
(11b)
1
ℓ=1,F =1,F = 1 = 1,0 √2 1, 1 . (11c)
| z ± i ∓ √3|±ip |±ie| ie− |∓ie | ± ie
h i
In the following, we shall use the notation ℓ,F,F for All the other matrix elements vanish. We define the pa-
z
| i
the eigenstates of the unperturbed Hamiltonian rameters
α4 m
H0 =HHFS,A+HHFS,B+HLS,A+HLS,B, (12) gN mc2 h59.1856114(22)MHz, (15a)
H≡ 18 m →
p
iwsitrhaitnhetrhient2uSit–i2vPe;1/t2hemfiarnsitfoelndt.ryTchlaerinfioetsatifiowne|ℓh,aFv,eFazni α5 ln(α−2)mc2 h1057.845(9)MHz, (15b)
L≡ 6π →
S (with ℓ = 0) or a P state (with ℓ = 1), the second
a2
entry specifies ifwe havea hyperfine triplet (F =1) ora 3α~c 0 , (15c)
V ≡ R3
hyperfine singlet (F =0) state, and the last entry is the
magnetic projection of the total angular momentum. wherethedatausedafterthereplacementsindicatesone-
third of the hyperfine splitting of the 2S state [1] and
the classic Lamb shift [12], respectively. These data are
C. Matrix Elements of the Total Hamiltonian usedinallfiguresforthe plotsofthedistance-dependent
energylevels. Notethat and obviouslyareconstants,
H L
whereas dependsontheinteratomicseparationR. The
We now turn to the computation of the matrix ele-
V
expectationvaluesofthehyperfineH andLambshift
ments of the total Hamiltonian (1) in the space spanned HFS
H Hamiltonians (for states of both atoms A and B)
by the two-atom states which are product states built LS
are given as follows
fromanytwostatesofthetypes(10)and(11). Wechoose
a basis in which the Lamb shift and hyperfine Hamilto- ℓ,F,M H ℓ,F,M = δ , (16a)
F LS F ℓ0
nians are diagonal, so that the only non-trivial task is h | | i L
3
to determine the matrix elements of the van der Waals 0,1,M H 0,1,M = , (16b)
F HFS F
h | | i 4H
interaction Hamiltonian.
9
With the definition of the spherical unit vectors [14], 0,0,0H 0,0,0 = , (16c)
HFS
h | | i − 4H
1 1
eˆ+ = − √2(eˆx+ieˆy), (13a) h1,1,MF|HHFS|1,1,MFi= 4H, (16d)
3
1
1,0,0H 1,0,0 = . (16e)
eˆ− = √2(eˆx−ieˆy), (13b) h | HFS| i − 4H
eˆ0 =eˆz, (13c) Thehyperfinesplittingenergybetween2P1/2(F =1)and
2P (F = 0) states thus amounts to , while the S-
1/2
H
and the states defined by (10) and (11), we obtain the state splitting is 3 . Additionally, the energies of the S
H
non-zeromatrix elementsofthe electronic positionoper- statesareliftedupwardby ,irrespectiveofthehyperfine
L
ator~r as follows: effects. FortheproductstateofatomsAandB,weshall
use the notation
0,0,0 ~r 1,1,0 =√3a eˆ , (14a)
0 z
h | | i (ℓ ,F ,F ) (ℓ ,F ,F ) , (17)
A A z,A A B B z,B B
0,0,0 ~r 1,1, 1 =√3a eˆ , (14b) | i
0 ±
h | | ± i which summarizes the quantum numbers of both atoms.
0,1,0 ~r 1,0,0 =√3a0eˆz, (14c) We anticipate that some of the eigenstates of the
h | | i
0,1, 1 ~r 1,0,0 =√3a (eˆ )∗ , (14d) combined and total Hamiltonian (Lamb shift plus hy-
0 ±
h ± | | i perfine effects plus van der Waals) do not decou-
0,1, 1 ~r 1,1, 1 = √3a0eˆz, (14e) ple into simple unperturbed eigenstates of the form
h ± | | ± i ±
h0,1,±1|~r|1,1,0i=±√3a0eˆ∓, (14f) o|(fℓAsu,pFeAr,pFozs,iAti)oAns(ℓoBf,FthBe,sFezs,tBa)tBesi,baustwmeayharedqualirreeatdhyeuexse-
0,1,0 ~r 1,1, 1 = √3a eˆ . (14g)
0 ± perienced for the (1S;2S) interaction in Ref. [2].
h | | ± i ∓
5
FIG. 1. (Color online.) Evolution of the energy levels within the F = +2 hyperfine manifold as a function of
z
interatomic separation. The eigenstates given in the legend are only asymptotic; for finite separation these states
mix. One has = 0.055949 according to Eq. (15). The unit of energy used for the ordinate axis is interaction
H L
energydividedbythePlanckconstanth(leftordinateaxis)andgiveninHertz(Hz). Ontherightordinateaxis,we
usetheLambshift asdefinedinEq.(3)asanalternativeunitoffrequency. TheBorn–Oppenheimerapproximation
L
is used in plotting theinteraction energy as a function of theinternuclear distance R.
III. HAMILTONIAN MATRICES IN THE where the Hamiltonian matrix reads
HYPERFINE SUBSPACES
H(II) = L+H −2V . (21)
A. Manifold Fz =+2 Fz=+2 (cid:18) −2V L+H (cid:19)
These subspaces are completely uncoupled. Namely, no
We have already pointed out that the n = 2, J =
state in subspace I is coupled to a state in subspace II.
1/2Hilbertspacenaturallyseparatesintosubspaceswith
fixed total hyperfine quantum number F = F +F , The eigenvalues of H(I) are given by
z z,a z,b Fz=+2
according to Eq. (8). We can identify two irreducible
subspaces within the F = +2 manifold: the subspace I
z E(I) = + + 4 2+(1 + )2
is composed of the states + H L V 2H L
q 2
|φ(1I)i=|(0,1,1)A(0,1,1)Bi, (18a) = 32H+2L+4HV+2L +O(V4), (22a)
|φ(2I)i=|(1,1,1)A(1,1,1)Bi, (18b) E−(I) =H+L− 4V2+(12H+L)2
where the Hamiltonian matrix reads q 2
= 1 4 V + ( 4), (22b)
2H− +2 O V
H(I) = 2L+ 32H −2V . (19) H L
Fz=+2 2 1 with the corresponding eigenvectors
(cid:18) − V 2H (cid:19)
Subspace II is composed of the states 1
u(I) = a φ(I) +b φ(I) , (23a)
| + i √a2+b2 | 1 i | 2 i
||φφ(1(2IIII))ii==||((01,,11,,11))AA((10,,11,,11))BBii,, ((2200ba)) |u(−I)i= √a21+b2 (cid:16)b|φ(1I)i−a|φ(2I)i(cid:17) . (23b)
(cid:16) (cid:17)
6
Here the coefficients a and b are given by The eigenstates within the degenerate subspace II ex-
perience a shift of first order in the van der Waals in-
2 + + (2 + )2+(4 )2 teraction energy , because of the degeneracy of the di-
V
a= L H L H V , (24a) agonal entries + in Eq. (21); this pattern will be
− q 4 L H
observed for other subspaces in the following. In Fig. 1,
V
b=1. (24b) we plot the evolution of the eigenvalues (22) and (25)
with respect to interatomic separation. The two levels
(II)
The eigenenergies of H are given by within the subspace II noticeably experience a far larger
Fz=+2
interatomicinteractionshift from their asymptotic value
E(II) = 2 , (25) + , commensurate with the parametric estimate of
± H±L± V Lthe cHorresponding energy shifts.
with the corresponding eigenvectors,
u(II) = 1 (φ(II) φ(II) ). (26) B. Manifold Fz =+1
| ± i √2 | 1 i±| 2 i
We can identify two irreducible subspaces within the
For 0, which corresponds to the large separation F =+1manifold. SubspaceIiscomposedofthefollow-
V → z
limit R + , these eigenvalues tend toward the (de- ing states, with both atoms either being in S, or both in
→ ∞
generate) diagonal entries of the matrix H(II) . P states,
Fz=+2
ψ(I) = (0,0,0) (0,1,1) , ψ(I) = (0,1,0) (0,1,1) , ψ(I) = (0,1,1) (0,0,0) ,
| 1 i | A Bi | 2 i | A Bi | 3 i | A Bi
ψ(I) = (0,1,1) (0,1,0) , ψ(I) = (1,0,0) (1,1,1) , ψ(I) = (1,1,0) (1,1,1) , (27)
| 4 i | A Bi | 5 i | A Bi | 6 i | A Bi
ψ(I) = (1,1,1) (1,0,0) , ψ(I) = (1,1,1) (1,1,0) ,
| 7 i | A Bi | 8 i | A Bi
and the Hamiltonian matrix reads
2 3 0 0 0 0 2
L− 2H − V V −V
0 2 + 3 0 0 2 0
 L 2H − V −V V 
0 0 2 3 0 0 2
L− 2H V −V − V
H(I) =  0 0 0 2L+ 23H −V V −2V 0  . (28)
Fz=+1  0 2 1 0 0 0 
 − V V −V −2H 
 2 0 0 1 0 0 
 − V −V V 2H 
 0 2 0 0 1 0 
 V −V − V −2H 
 2 0 0 0 0 1 
 −V V − V 2H 
 
Subspace II is composed of the following states, where one atom is in a S, and the other, in a P state,
(II) (II) (II)
ψ = (0,0,0) (1,1,1) , ψ = (0,1,0) (1,1,1) , ψ = (0,1,1) (1,0,0) ,
| 1 i | A Bi | 2 i | A Bi | 3 i | A Bi
ψ(II) = (0,1,1) (1,1,0) , ψ(II) = (1,0,0) (0,1,1) , ψ(II) = (1,1,0) (0,1,1) ,
| 4 i | A Bi | 5 i | A Bi | 6 i | A Bi
ψ(II) = (1,1,1) (0,0,0) , ψ(II) = (1,1,1) (0,1,0) , (29)
| 7 i | A Bi | 8 i | A Bi
and the Hamiltonian matrix reads
2 0 0 0 0 2
L− H − V V −V
0 + 0 0 2 0
 0 L 0H 0 − V −0V V2 
L V −V − V
0 0 0 + 2 0
HF(IzI=)+1 =  0 2 L H −V V0 −0V 0  . (30)
 
 2 −0V V −V L0 + 0 0 
 − V −V V L H 
 0 2 0 0 2 0 
 V −V − V L− H 
 2 0 0 0 0 + 
 −V V − V L H 
 
These two submanifolds are, again, completely uncou- and P states. One observes that within the subspace I,
pled, as a consequence of the selection rules between S
7
FIG. 2. (Color online.) Evolution of the S–S and P–P energy levels of the submanifold I within the F = +1
z
hyperfinemanifold as a function of interatomic separation. The asymptotic eigenstates given in the legend mix for
finiteseparation. The labeling of theaxes is as in Fig. 1.
notwodegeneratelevelsarecoupledtoeachother,result- Note that the designation of a degenerate subspace, for
ing in second-order van der Waals energy shifts. On the the F = +1 subspace, does not imply that there are
z
otherhand,thefollowingsubspaces,withinthesubspace no couplings to any other states within the manifold;
II, can be identified as being degenerate with respect to however,thecouplingsrelatingthedegeneratestateswill
theunperturbedHamiltonian,andhavingstatescoupled become dominant for close approach.
by nonvanishing off-diagonal elements. We first have a A second degenerate subspace is given as
subspace spanned by
ψ(B) = ψ(II) , ψ(B) = ψ(II) . (35)
ψ(A) = ψ(II) , ψ(A) = ψ(II) . (31) | 1 i | 3 i | 2 i | 5 i
| 1 i | 1 i | 2 i | 7 i
These states are composed of a triplet S and a singlet P
These states are composed of a singlet S and a triplet
state, and hence the diagonalentries in the Hamiltonian
P state, and hence the diagonal entries in the Hamilto- matrix are (3 + ) (3 ) = . The Hamiltonian
nian matrix are ( 9 + )+(1 ) = 2 + . The 4H L − 4H L
−4H L 4H − H L matrix is
Hamiltonian matrix is
H(A) = L−2H V . (32) HF(Bz=)+1 = L V . (36)
Fz=+1 2 (cid:18)V L (cid:19)
(cid:18) V L− H(cid:19)
The eigenvalues are
The eigenvalues are
E±(A) =L−2H±V, (33) E±(B) =L±V, (37)
with the corresponding eigenvectors, with the corresponding eigenvectors,
1 1
u(A) = ψ(A) ψ(A) . (34) u(B) = ψ(B) ψ(B) . (38)
| ± i √2 | 1 i±| 2 i | ± i √2 | 1 i±| 2
(cid:16) (cid:17) (cid:16) (cid:17)
8
FIG.3. (Color online.) Evolution oftheenergy levelsofthesubmanifold IIwithintheF =+1hyperfinemanifold
z
asafunction ofinteratomic separation. Theeigenstates giveninthelegend areonly asymptotic. Thecurveforthe
seventh state in the legend (counted from the top) has been slightly offset for better readability, in actuality it is
virtually indistinguishable from that for thesixth state in thelegend.
The most complicated degenerate subspace is given by In Figs. 2 and 3, we plot the evolutionof the eigenval-
the vectors ues of the matrices (28) and (30) with respect to inter-
atomic separation. The larger energy shifts within the
ψ(C) = ψ(II) , ψ(C) = ψ(II) , (39) subspace II are noticeable. A feature exhibited by the
| 1 i | 2 i | 2 i | 4 i
ψ(C) = ψ(II) , ψ(C) = ψ(II) . (40) Fz =+1manifold whichwasnotpresentinthe Fz =+2
| 3 i | 6 i | 4 i | 8 i manifold is that of level crossings: for sufficiently small
interatomic separation (R < 500a ), the eigenenergies
The Hamiltonian matrix is 0
of some of the states from the submanifolds I and II in
+ 0 0 fact cross (these crossings would be visible if one were
L H V
HF(Cz=)+1 = 00 L+H +V 00  , (41) tcorossusipnegrsimbeptowseeeFnigstsa.t2esabnedlo3n)g,inwghitloetthheerseamareesnuobmleavne-l
V L H
0 0 + ifold.
 V L H 
 
which again decouples into two 2 2 matrices, just like
×
we saw in the case of H . The eigenvalues are
Fz=+2
(C)
E = + , (42)
± H L±V
where the eigenvectors for u(C) (with i = 1,2 because
| ±,ii
of the degeneracy of the eigenvalues) are given by C. Manifold Fz =0
1
u(C) = ψ(C) ψ(C) , (43a)
| ±,1i √2 | 1 i±| 4 i
We can identify two irreducible subspaces within the
(cid:16) (cid:17)
1
u(C) = ψ(C) ψ(C) . (43b) Fz = 0 manifold: the subspace I is composed of states
| ±,2i √2 | 2 i±| 3 i with both atoms in S, or both atoms in P levels,
(cid:16) (cid:17)
9
FIG. 4. (Color online.) Evolution of the energy levels of the submanifold I within the F = 0 hyperfine manifold
z
as a function of interatomic separation. Energetically, the S–S states are above the P–P states. The eigenstates
given in the legend are only asymptotic; for finite separation these states mix. Some of the curves [namely, the
third (from the top), sixth and twelfth] have been slightly offset for better readability. Notice that, for sufficiently
close separation (R<1000a ),we witnesssome levelcrossings between levelswithin thesame submanifold I. The
0
coefficients α± and β± are determined by second-orderperturbation theory and given by Eq.(83).
Ψ(I) = (0,0,0) (0,0,0) , Ψ(I) = (0,0,0) (0,1,0) , Ψ(I) = (0,1, 1) (0,1,1) ,
| 1 i | A Bi | 2 i | A Bi | 3 i | − A Bi
Ψ(I) = (0,1,0) (0,0,0) , Ψ(I) = (0,1,0) (0,1,0) , Ψ(I) = (0,1,1) (0,1, 1) ,
| 4 i | A Bi | 5 i | A Bi | 6 i | A − Bi
Ψ(I) = (1,0,0) (1,0,0) , Ψ(I) = (1,0,0) (1,1,0) , Ψ(I) = (1,1, 1) (1,1,1) ,
| 7 i | A Bi | 8 i | A Bi | 9 i | − A Bi
Ψ(I) = (1,1,0) (1,0,0) , Ψ(I) = (1,1,0) (1,1,0) , Ψ(I) = (1,1,1) (1,1, 1) (44)
| 10i | A Bi | 11i | A Bi | 12i | A − Bi
and the Hamiltonian matrix reads
2 9 0 0 0 0 0 0 0 0 2
L− 2H −V − V −V
0 2 3 0 0 0 0 0 0 2 0
 0 L−0 2H 2 + 3 0 0 0 2V − V −0V 
L 2H −V V V −V V
0 0 0 2 3 0 0 0 2 0 0
 0 0 0 L−0 2H 2 + 3 0 2 −0V −V 0 0 V 
HF(Iz)=0 =  00 00 0 00 L 022H 2L+ 32H −−3VV −0V V00 V0 V0 2V0V  .
 0 0 −V 2 −0V −V −02H 1 0 0 0 0 
 2V − V −0V 0 −02H 1 0 0 0 
 −0V V2 V −0V V0 0 0 20H 1 0 0 
 2 −0V −V 0 0 V 0 0 0 −02H 1 0 
 − V V0 2V 0 0 0 0 20H 1 
 −V −V V V V 2H 
 (45)
10
FIG. 5. (Color online.) Evolution of the energy levels of the 2S–2S states within the F = 0 hyperfine manifold
z
(subspace I) as a function of the interatomic separation (close-up of the “upper” levels in Fig. 4). The eigenstates
giveninthelegendareonlyasymptotic;forfiniteseparationthesestatesmix. Nooffsetsareusedhere. Noticethat
wewitness onelevelcrossing. Thecoefficients α± and β± are determinedbysecond-order perturbationtheory and
given by Eq.(83).
Subspace II is composed of the S–P and P–S combinations,
Ψ(II) =(0,0,0) (1,0,0) , Ψ(II) = (0,0,0) (1,1,0) , Ψ(II) = (0,1, 1) (1,1,1) ,
| 1 i | A Bi | 2 i | A Bi | 3 i | − A Bi
Ψ(II) =(0,1,0) (1,0,0) , Ψ(II) = (0,1,0) (1,1,0) , Ψ(II) = (0,1,1) (1,1, 1) ,
| 4 i | A Bi | 5 i | A Bi | 6 i | A − Bi
Ψ(II) = (1,0,0) (0,0,0) , Ψ(II) = (1,0,0) (0,1,0) , Ψ(II) = (1,1, 1) (0,1,1) ,
| 7 i | A Bi | 8 i | A Bi | 9 i | − A Bi
Ψ(II) = (1,1,0) (0,0,0) , Ψ(II) = (1,1,0) (0,1,0) , Ψ(II) = (1,1,1) (0,1, 1) , (46)
| 10 i | A Bi | 11 i | A Bi | 12 i | A − Bi
and the Hamiltonian matrix reads
3 0 0 0 0 0 0 0 0 2
L− H −V − V −V
0 2 0 0 0 0 0 0 2 0
 L− H V − V −V 
0 0 + 0 0 0 2 0
L H −V V V −V V
 0 0 0 0 0 0 2 0 0 
 0 0 0 L0 + 0 2 −0V −V 0 0 V 
 
HF(IzI=)0 =  00 00 0 00 L 02H L+H −−V3V −0V V00 V0 V0 2V0V  . (47)
 −V − V −V L− H 
 0 0 2 0 0 0 0 0 0 
 V − V −V L 
 2 0 0 0 + 0 0 0 
 −V V V −V V L H 
 0 2 0 0 0 0 0 2 0 0 
 − V −V V L− H 
 2 0 0 0 0 0 0 0 + 0 
 − V V V L H 
 0 2 0 0 0 0 0 + 
 −V −V V V V L H 
 
Again, we notice that within the subspace I, no two de- canbe identified as being degeneratewith respectto the
generate levels are coupled to each other. On the other unperturbed Hamiltonian, and having states coupled by
hand, the following subspaces, within the subspace II,