Long-Range Atom-Wall Interactions and Mixing Terms: Metastable Hydrogen

0.3 MB·English

by  U. D. Jentschura

Long–Range Atom–Wall Interactions and Mixing Terms: Metastable Hydrogen
U. D. Jentschura
Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
Weinvestigatetheinteractionofmetastable2S hydrogenatomswithaperfectlyconductingwall,
including parity-breaking S–P mixing terms (with full account of retardation). The neighboring
2P1/2 and 2P3/2 levels are found to have a profound effect on the transition from the short-range,
nonrelativisticregime,totheretardedformoftheCasimir–Polderinteraction. ThecorrespondingP
state admixtures to the metastable 2S state are calculated. We find the long-range asymptotics of
theretardedCasimir–Polder potentialsandmixingamplitudes,forgeneralexcitedstates,including
afullyquantumelectrodynamictreatmentof thedipole-quadrupolemixingterm. Thedecay width
of the metastable 2S state is roughly doubled even at a comparatively large distance of 918atom
5
units(Bohrradii) from theperfect conductor. Themagnitudeofthecalculated effectsiscompared
1
tothe unexplained Sokolov effect.
0
2
PACSnumbers: 34.35.+a,31.30.jh,12.20.Ds,42.50.Ct
n
a
J
Introduction.—The investigationof atom-wallinterac- tum electrodynamic (QED) length-gauge interaction

1 tions for atoms in contactwith conducting materialshas
e
2 a long history. Starting from the works of Lennard– H = e~r E~ rirj∂Ei/∂rj +... , (1)
I − · − 2
Jones [1], Bardeen [2], Casimir and Polder [3], and Lif-
]
h shitz [4], research on related matters has found contin- follows naturally from the formalism of long-wavelength
p uously growing interest over the last decades [5–8]. In QEDinteractionHamiltonian[27,28](~rdenotestheelec-
- thenon-retardedregime(closerange),theinteractionen- troncoordinate). In contrastto the vectorpotential, the
t
n ergyscalesas1/ 3 withtheatom-walldistance ,while electric field strength (operator) is gauge-invariant (this
a Z Z
for atom-wall distances large in comparison to a typi- pointhasgivenrisetosomediscussion,seeRef.[29])and
u
cal atomic wavelength, the interaction energy scales as reads as [cf. Eq. (2.3) of Ref. [20]]
q
[ 1/ 4 (see Chap.8 ofRef. [9]). The leadingtermisgiven
1 byZvirtual dipole transitions, while multipole corrections E~(~r)= ∞dL d2kk√ω a (~k,L)(kˆ eˆ )sin(Lz)
v have recently been analyzed in Ref. [10]. The symmetry π 1 k× z
2 breaking induced by the wall leads to dipole-quadrupole Z0 Z2 n
R
mixing terms, which lead to admixtures to metastable
5
2 levels [11, 12]. While this effect has been analyzed in +a (~k,L) kˆ iLsin(Lz) eˆ kk cos(Lz)
5 the non-retarded van-der-Waals regime [11, 12], a fully 2 k ω − z ω
(cid:20) (cid:21)(cid:27)
0 quantum electrodynamic calculation of this effect would
1. be of obvious interest. ei~kk·~rk +h.c., (2)
0 This fact is emphasized by the curious observa- ×
5 tion of a long-range, and conceivably super-long-range where ~r = ~r +zeˆ with ~r = xeˆ +yeˆ , while ~k =
1 k z k x y k
v: (2mSicarotommetserw-sictahle)ainctoenrdauctcitoinngofsumrfeatcaesta(tbhlee hsyod-craolgleedn kxeˆx+kyeˆy, also ~k⊥ = kzeˆz, and L ≡ |~k⊥|. The com-
mutator relation is [a (~k ,L), a† (~k ,L)]=δ δ(2)(~k
Xi Stooksoulsopvecetffetchta,tsetheisReefffse.c[t13c–o1u6ld]).beItdiusentootafaqr-ufaetncthuemd ~k′)δ(L L′) for the asnnkihilatiosn′ ankd creatiosns′operatkor−s
r k −
a electrodynamicallyinducedtailofthe dipole-quadrupole as and a†s. In order to evaluate the interaction Hamilto-
mixing term in the atom-wall interaction. Namely, for nian (1), one shifts z +z where is the coordinate
the hydrogen2S atom, the neighboring 2P1/2 and 2P3/2 oftheatom’scenter(n→ucZleus). TheprZotonisat(0,0, ),
Z
levels are removed only by the Lamb shift and fine- while the atomic electron coordinates are (x,y, +z).
structure, respectively, while it is known that virtual The surface of the perfect conductor is in the xyZplane,
statesoflowerenergycaninducelong-rangetailsinatom- i.e., in the plane described by the points (x,y,0). The
wall interactions, as well as in the Lamb shift between unperturbed Hamiltonian H is the sum of the free ra-
0
plates (see Refs. [17–26]). The large admixtures typi- diation field and the unperturbed atom [see Eq. (2.1)
cally induced in atomic systems when a metastable level of Ref. [20] and Eq. (3.2) of Ref. [30]]. For a refer-
couplestonearlydegeneratestatesofoppositeparitysug- ence ground state n , second-order perturbation theory
gestthata closerinvestigationofthe hydrogensystemis leads to a known r|esuilt given in Eq. (8.41) of Ref. [9] or
warranted. Atomic units with~=4πǫ0 =1 andc=1/α Eq. (27) of Ref. [10], which involves the symmetric sum
are used throughout this Rapid Communication, where withimaginaryfrequencyintheargumentofthedynam-
α is the fine-structure constant. The electron charge is ics polarizability Π( iω). The Wick rotation of the vir-
explicitly denoted as e unless stated otherwise. tualphotonintegrati±oncontour,leadstothesymmetriza-
Retardation of the atom-wall interaction.—The quan- tion iω iω but cannot be done for excited reference
↔ −
2
states. We use second-orderperturbationtheoryto eval- obtain[cf.thediscussionfollowingEq.(2.12)ofRef.[20]],
uate ∆E = n( e~r E~)[1/(E H′)]( e~r E~)n and
h | − · n− 0 − · | i
. e2 ∞ ∞ L2 hn|~rk|qi 2+2 |hn|z|qi|2 +ω2 hn|~rk|qi 2−2 |hn|z|qi|2
∆E = (P.V.) dL dω cos(2L ) , (3)
2π Xq Z0 ZL Z (cid:16)(cid:12)(cid:12) (cid:12)(cid:12) Eq+(cid:17)ω−i(cid:16)δ(cid:12)(cid:12) (cid:12)(cid:12) (cid:17)
where the identity d3k =2 2πdϕ ∞dL ∞dωω , with ω = ~k2+k2, and L = k has been used in order to
3 0 0 0 k z | z|
R
transform the integration measure. The virtual states are denotedqas q , and their energy difference to the reference
R R R R | i
state is denoted as E E . In contrast to the velocity gauge [20], there is no seagull term to consider, and it
q q n
E ≡ −
is not necessary to add the electrostatic interaction with the mirror charges by hand [31]. It is an in principle well
known (see the remarks following Eq. (A.22) in Appendix A of Ref. [32]), but sometimes forgotten wisdom that the
Coulombinteractiondoesnotneedtobe quantizedinthevelocitygauge[31]. Theintegrationwithrespectto ω leads
to logarithmic terms [see the Appendix of Ref. [20]]. After the subtraction of -independent terms (the subtraction
. Z
is denoted by the = sign), one obtains
I (χ)=. ∞dL cos(2L ) ln( +L)= π[1−ε(Eq)] T(χ) πΘ( )2sin2(χ2) , (4a)
1 q q q
Z0 Z |E | E 2χ − χ − −E χ !
I (χ)=. ∂2I1 = π[ε(Eq)−1]+χ + 2−χ2 T(χ) 2 U(χ)+πΘ( ) ∂2 2sin2(χ2) , (4b)
2 − ∂χ2 Eq χ3 χ3 − χ2 −Eq ∂χ2 χ !
π
T(χ)= sin(χ)Ci(χ) cos(χ)Si(χ)+ cos(χ), χ=2 , ε( )=Θ( ) Θ( ). (4c)
q q q q
− 2 |E |Z E E − −E
Here, Ci(χ)= ∞dtcos(t) and Si(χ)= χdtsin(t), and U(χ)= ∂ T(χ), while T(χ)=χ−1 ∂ U(χ). We confirm
− χ t 0 t ∂χ − ∂χ
the result given in Eq. (2.18) of Ref. [20] and represent the “distance-dependent Lamb shift” as
R R
∆E =. e2 3 n~r q 2 2 nz q 2 π[ε(Eq)−1] 1 + T(χ) +πΘ( )1−cos(χ) (5)
2π Eq h | k| i − |h | | i| 2χ − χ2 χ −Eq χ
Xq (cid:26)(cid:16)(cid:12) (cid:12) (cid:17) (cid:20) (cid:21)
(cid:12) (cid:12)
n~r q 2+2 nz q 2 π[ε(Eq)−1]+χ + 2−χ2 T(χ) 2 U(χ)+πΘ( ) ∂2 1−cos(χ) .
− h | k| i |h | | i| χ3 χ3 − χ2 −Eq ∂χ2 χ
(cid:16)(cid:12) (cid:12) (cid:17) (cid:20) (cid:21)(cid:27)
We should pe(cid:12)rhaps cl(cid:12)arify that the -independent contribution to the Lamb shift (the ordinary “free-space Lamb
Z .
shift”) is absorbed in the subtraction procedure denoted here by the “=” sign in Eqs. (4), (5), (7) and (8). The
-dependent position of the energy level is obtained after adding the “free-space Lamb shift” and “free-space fine
Z L
structure” given in Eq. (12) to the -dependent energy shifts given in Eqs. (5) and (8). In the nonretardation
F Z
limit, the -dependent results given in Eqs. (5) and (8) are replaced by the respective terms of the nonretarded
Z
potential (11). This (somewhat subtle) point is not fully discussed in previous works on the subject [17–21] and
therefore should be mentioned for absolute clarity.
The term χ−2 in the coefficient multiplying n~r q 2 2 n~z q 2 vanishes after summing over the entire
k
− h | | i − |h | | i|
spectrum of virtualstates; it is obtained naturally in the length gauge and otherwise cancels a term in the expansion
(cid:12) (cid:12)
oftheenergyshiftforlargeχ(evenbeforetheappli(cid:12)cationof(cid:12)thesumrule,whichiscrucialinvelocitygauge[20]). The
off-diagonal mixing term leads to to the matrix element ∆M = m( e~r E~)[1/(E H )′]( erirj(∂Ei/∂rj)n +
h | − · n− 0 −2 | i
mh.c.n ,
h | | i
∞ ∞
e2 L sin(2L )
∆M = (P.V.) dL dω Z L2 n m ω2 n m , (6a)
2 1
4π +ω iδ h |T | i− h |T | i
Xq Z0 ZL Eq − (cid:0) (cid:1)
m n = mz q q~r2 2z2 n + mh.c.n , (6b)
h |T1| i h | | ih | k − | i h | | i
m n = mz q q~r2 2z2 n 2 m~r q q~r z n + mh.c.n . (6c)
h |T2| i h | | ih | k − | i− h | k| i·h | k | i h | | i
3
∞
After the subtraction of -independent terms, the following two results for J (χ) = dLLsin(2L )ln( +L)
Z 1 0 Z |Eq |
and J (χ)= ∂2J (χ)/∂χ2 supplement the analytic integrals given in Eq. (4),
2 − 1 R
. π T(χ) U(χ) π 2sin2(χ) χsin(χ)
J (χ)= 2 ε( ) + +πΘ( ) 2 − , (7a)
1 Eq Eq (cid:18)2χ2 − χ2 χ (cid:19)− 2χ2 −Eq χ2 !
. 3π 4χ 3π 3(2 χ2) χ2 6 ∂2 2sin2(χ) χsin(χ)
J (χ)= 2 +ε( ) − + − T(χ)+ − U(χ) πΘ( ) 2 − . (7b)
2 Eq χ4 Eq (cid:20) χ4 χ4 χ3 (cid:21)− −Eq ∂χ2 χ2 !
We can finally give the complete result for the mixing term ∆M, with full account of retardation, as a sum over
virtual states q ,
| i
. e2 4+πχ T(χ) U(χ) π 2sin2(χ) χ sin(χ)
∆M = 4 m n ε( ) + +πΘ( ) 2 − (8)
4π q Eq (h |T1| i " Eq (cid:18) 2χ3 − χ2 χ (cid:19)− 2χ2 −Eq χ2 #
X
3π 4χ 3(χ2 2) 6 χ2 3π ∂2 2sin2(χ) χsin(χ)
+ m n ε( ) − + − T(χ)+ − U(χ) +πΘ( ) 2 − .
h |T2| i" Eq (cid:18) χ4 χ4 χ3 (cid:19)− χ4 −Eq ∂χ2 χ2 #)
The energy variable is defined with respect to the reference state; i.e., if one evaluates the m -state admixture to
q
E | i
the reference state n , then one sets =E E . For excited reference states, results for both ∆E givenin Eq.(5)
q q n
| i E −
and ∆M in Eq. (8) contain long-range retardation tails for excited reference states,
∆E =e2 Θ( ) n~r q 2 Eq2 cos(2EqZ) Eq sin(2EqZ) cos(2EqZ)
−Eq "|h | k| i| 2 − 4 2 − 8 3 !
q Z Z Z
X
sin(2 ) cos(2 ) 1 1
nz q 2 Eq EqZ + EqZ 2Π +Π , ,
−|h | | i| 2 4 3 − 8π 4 k ⊥ Z ≫
(cid:18) Z Z (cid:19)(cid:21) Z Eq
(cid:0) (cid:1)
1 2 2 e2 1
Π = n~r q q~r n , Π = nz q 2 , Π(ω)= n ri ri n , (9)
k 2 h | k| i·h | k| i ⊥ |h | | i| 3 ω
q,± Eq q,± Eq ± (cid:28) (cid:12) (cid:18)Eq± (cid:19) (cid:12) (cid:29)
X X X (cid:12) (cid:12)
(cid:12) (cid:12)
where Π and Π are the longitudinal and transverse static polarizabilities [for the grou(cid:12)nd state, Π =(cid:12)Π = Π(0)].
k ⊥ ⊥ k
The mixing term has the following long-range asymptotics,
∆M =e2 Θ( ) m~r q q~r z n Eq3sin(2EqZ) 3Eq2cos(2EqZ) + 3Eq sin(2EqZ) + 3Eq4 cos(2EqZ)
−Eq h | k| i·h | k | i − 4 − 8 2 8 3 16 4 !
q Z Z Z Z
X
+e2 Θ( ) mz q q~r2 2z2 n Eq2 cos(2EqZ) 3Eq sin(2EqZ) 3 cos(2EqZ) (10)
−Eq h | | ih | k − | i 8 2 − 16 3 − 32 4 !
q Z Z Z
X
e2 1 1 1 3 1
+ mz q q~r2 n + mz q q z2 n + m~r q q~r z n + mh.c.n , .
π 5 −8h | | ih | k| i 4h | | ih | | i 8h | k| i·h | k | i h | | i Z ≫
Z q Eq (cid:18) (cid:19) Eq
X
The results (5) and (8) will now be applied to metastable hydrogen.
Nonretarded admixtures to metastable hydrogen.—The tential(fromnowonwesettheelementarychargee=1),
results given in Eq. (5) and (8) have a rather involved
analyticstructure. Inthe short-rangelimit, these results 1 1 2 1
V = +
can be comparedto the static interaction of the electron
2 −2(z+ ) x2+y2+(z+2 )2 − 2 !
and proton [33, 34] with their respective mirror charges. Z Z Z
This interaction leads to the following nonretarded po- p
~r2+2z2 3z(~r2+2z2)
k k
= + + (11)
− 16 3 32 4 ···
Z Z
4
wherewe ignoreterms oforder1/ 5 andhigher[35,36].
Z
After sometedious,butstraightforwardalgebra,onecan
convince oneself that the terms of order −3 and −4
Z Z
areinagreementwiththeshort-rangeasymptoticsofthe
results given in Eqs. (5) and (8), i.e., in the regime
Z ≪
1/ , which is equivalent to the limit χ 0.
q
E →
For close approach of the atom to the wall, the in-
teraction energy is well described by the static poten-
tial (11), which necessitates a diagonalization of the
Schr¨odinger Hamiltonian plus the nonretarded potential
V (both “diagonal” interaction and Lamb shift or fine
structure terms, as well as “mixing” terms) in the ba-
sisofthe 2S , 2P , and 2P Schr¨odinger–Pauli
1/2 1/2 3/2
| i | i | i
wave functions with magnetic projection µ = +1/2, to
FIG. 1. (Color.) The modulus-squared admixtures to the
form the manifestly coupled states , , and
|S1/2i |P1/2i coupled |S1/2i state are obtained from a diagonalization of
. Wedenotethe(free-space)fine-structureandthe
|P3/2i the potential (11) in the basis of |S1/2i, |P1/2i, and |P3/2i
Lamb shift interval as states, for close approach of the atom toward the wall. The
subscript j in Eq. (13) takes on the values j = S, as well as
=1.66 10−6a.u., =1.61 10−7a.u., (12) j = 1/2 and j = 3/2 and denotes the state responsible for
F × L × the admixture. As the |S1/2i state approaches the wall, the
respectively. Accordingtotheadiabatictheorem[37–39], initially dominant |S1/2i state contribution (solid curve, j =
S) gradually fades and the |P1/2i admixture (short-dashed
the state eigenvector has the form
1/2 curve, j = 1/2) increases, while a significant admixture of
|S i
the |P3/2i state (long-dashed curve, j = 3/2), is observed
1/2 aS 2S1/2 +a1 2P1/2 +a3 2P3/2 , (13a) only for close approach. The atom-wall interaction energy
|S i≈ | i 2 | i 2 | i becomescommensuratewiththeLambshiftandfinestructure
at Z ≈84 and at Z ≈184, respectively.
√3 15 3 15
aS =1, a21 = 2 4 , a32 = 2 4 , (13b)
LZ r FZ
reads as
1/ 1/ 1/4, 1/ 1/ 1/4, (13c)
L≫Z ≫ L Z ≫Z ≫ F b 2S +b 2P +b 2P , (16a)
1/2 S 1/2 1/2 1/2 3/2 3/2
|P i≈ | i | i | i
where we ignore higher-order terms in the expansion in
inverse powers of . The absolute square of the admix- 3 15 1 1
ture is given by Z bS = − 4 4 , b23 = 2√2 3 , (16b)
r LZ FZ
675 1 1 1 6.63 1015 and b1 =1. The 3/2 state reads as follows,
2 |P i
Ξ= + = × a.u.. (14)
2 2 2 2 8 8
(cid:18)F L (cid:19)Z Z 3/2 cS 2S1/2 +c1/2 2P1/2 +c3/2 2P3/2 , (17a)
|P i≈ | i | i | i
The one-photon decay width of the 2P state is given
as Γ = 6.27 108 rad = 1.51 10−8 a.u., whereas 3 15 1 1
the t2wPo-photon×decay wsidth of the×2S state reads Γ = cS = − 2 4, c12 = 2√2( + ) 3, (17b)
2S r FZ L F Z
8.229 rad = 1.99 10−16 a.u.. The effective decay rate
Γ at as distance× is and of course c3 = 1. For very close approach Z . 300,
eff 2
Z higher-order terms in the expansion of the potential
1.01 108 V [see Eq. (11)] gradually become important. [These
Γeff =Γ2S +Γ2PΞ= 1.99×10−16+ ×8 a.u.. are obtained by straightforwardexpansion of the poten-
(cid:18) Z (cid:19) tial(11).] Numericallydeterminedadmixturesofthecou-
(15)
pled are given in Fig. 1, they do not follow the
We have Γeff = 2Γ2S for Z0 = 918a.u.. The leading asymp|St1o/t2ici formulas for very close approach.
(nonretarded) term in the atom-wall energy shift at this
Long-range tails.—The oscillatory repulsive-attractive
distance amounts to 7 −3/2 = 4.52 10−9 a.u. and
− Z0 − × dominantterminthelong-rangelimitoftheenergyshift,
approximates both the single-particle perturbative shift
for the 2S level, goes as [see Eqs. (5) and (9)],
given in Eq. (5) as well as the adiabatic energy of the
coupled|S1/2istateobtainedfromthediagonalizationof 9 2 cos(2 ) 1
the potential (11) to within 10%. The atom-wall energy ∆E L LZ , , (18)
2S
∼ 2 Z ≫
at 0 isequalto 29.7MHzandthusmuchsmallerthan Z L
Z −
the Lamb shift and fine structure. where we have isolated the leading term from Eq. (9),
The admixture formulas for the coupled state setting = and carrying the summation over the
1/2 q
|P i E −L
5
virtuallevels q = 2P withmagneticprojectionsµ= lengthtransition)thattheregioninwhichthe1/ terms
1/2
| i | i Z
1/2. Somewhat surprisingly, the oscillatory terms in dominateisrestrictedtoexcessivelylargeatom-wallsep-
±
Eq. (10) vanish for virtual 2P states, so that the arationswhere the single powerof inthe denominator
1/2
| i Z
long-range coupling to the lower-lying P state vanishes. is sufficient to make the interaction energy and admix-
The leading terms in the long-range asymptotics of the ture terms negligible. (We should add that the inclusion
admixture coefficients read as follows [see Eq. (13)], of additional mirror charges in a cavity as opposed to a
wall can be taken into account, in the short-range limit,
3√3 1 by summing the mirror charge interactions into a gener-
a , , (19a)
1/2 ∼ π 5 Z ≫ alized Riemann zeta function [40] and therefore cannot
LFZ L change the order-of-magnitude of the admixture terms.)
3 3 3 1 Ifthe observationsreportedinRefs.[13–16]hadfound
a L sin(2 ), . (19b)
3/2 ∼ − 2 LZ Z ≫ a natural explanation in terms of a QED effect, then
r FZ L this might have had significant implications for a typ-
The long-range asymptotic tail of the P -state ad- ical atomic beam apparatus [41] used in high-precision
3/2
mixture has an oscillatory (1/ )-form [see Eqs. (10) spectroscopyofatoms,potentially shifting the frequency
Z
and(19b)]. Ifthistailwerenotsuppressedbytheprefac- of transitions involving 2S atoms in a narrow tube. For
tor 3/ ,thenitcouldhaveeasilyprovidedatheoretical atom-wallseparationssmallerthan1000Bohrradii,sub-
L F
explanation for the Sokolov effect [13–16], because the stantial admixture terms are found, and the 1/ 8 scal-
Z
(1/ )-interactionhastherequiredfunctionalformtode- ing of the effective 2S decay rate predicted by Eq. (14)
Z
scribeasuper-long-rangeterm. Thetailiscreatedbyvir- could be tested against an experiment. The clarification
tual q = 2P states in Eq. (10), which are energeti- of the parity-breaking admixture terms also is impor-
1/2
| i | i
callylowerthanthereference 2S state. Theprefactorof tant for other precision measurements in atomic physics
| i
the super-long-rangetail of the admixture term depends whichinvolvemetastablestates,suchasEDMandweak-
ondetailsofthespectrumoftheatomicsystemandcould interaction experiments [42–47]. The fully retarded ex-
be larger for other atoms. For the P -state admixture pression for the mixing term, given in Eq. (10), formu-
1/2
(term a ), retardation changes the 1/ 4 asymptotics lateshigher-orderQEDcorrectionstoatom-wallinterac-
1/2
Z
for short range to a 1/ 5 asymptotics at long range. A tionsbeyonddipoleorder. Generalizationoftheformulas
full QED treatment ofZthe admixture terms is required to, e.g., the 23S1 mestable state of helium is straightfor-
for both results recorded in Eqs. (19a) and (19b). ward. One just sums the interactions over the electron
Conclusions.—Wecansafelyconcludethatthecurious coordinates.
observations reported in [13–16] regarding super-long-
range 2S–2P mixing terms near metal surfaces cannot
ACKNOWLEDGMENTS
find an explanation in terms of a long-range effect in-
volving quantum fluctuations. Both the energy shift (9)
aswellasthemixingterm(10)havelong-rangetailspro- SupportfromtheNationalScienceFoundation(grants
portional to 1/ , but the energy numerator for the 2S– PHY–1068547and PHY–1403973)and helpful conversa-
Z
2P transition is so small (Lamb shift, a 30cm wave- tions with M. M. Bush are gratefully acknowledged.
1/2
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